Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Hasan, Md Mehadi
Master of Applied Science - MASc
University of British Columbia
The reverberation-room method, which assumes a diffuse sound field, has long been used for various standardized room-acoustical determinations – e.g. of surface-absorption coefficients, power levels of sound sources, transmission losses of acoustical partitions, etc. In this regard, a number of standards have emerged to offer some help by outlining necessary reverberation-room design guidelines to achieve sufficient sound-field diffuseness. However, unsatisfactory opinions regarding the prediction accuracy of the method, especially at low frequencies, have been reported over the years. This might be due to deviations from the assumed diffuse-field concept, which is very challenging to implement from an application point of view; also there are no straight-forward ways to characterize the degree of sound-field diffuseness. To investigate the problem and propose solutions, diffuse-field theory and existing standards have been revisited. Using numerical, finite-element-based, modal prediction, their capacity/effectiveness to achieve a diffuse sound field is analyzed by means of a number of descriptors (room-acoustical parameters). Because of time limitations, the concept regarding the design of a reverberation-room structure – i.e. size, shape, etc. – is mainly explored, rather than the internal test-setup arrangements. The prediction accuracy of different room-acoustical parameters are also determined by the reverberation-room standard methods, with respect to both the Sabine and Eyring versions of the diffuse-field formulae, and both in octave and third-octave bands. The minimum approachable frequencies of predictions and the quality of sound-field diffuseness are discussed in terms of the prediction accuracy of different room-acoustical parameters. Considering three room volumes prescribed by standards, and four room shapes for each of the volumes, it has been found that the reverberation room of volume 150 m³, as prescribed by the ISO 354 standard, with the typical dimensional orientation (longest x-dimension/shortest vertical dimension) yields better field diffuseness than the other rooms of different volumes and shapes. To check the possibility of further improvement of the field diffuseness, a number of additional features are integrated into that reverberation room. It is found that the rooms with diffusers and absorbent corner treatments yield improved sound-field diffuseness, while the rooms with diffuse surface reflection yield poor field diffuseness due to the increased surface absorption.
Diffuse Sound Fields, Reverberation-Room Methods and the Effectiveness of Reverberation-Room Designs by Md Mehadi Hasan B.Sc., Bangladesh University of Engineering and Technology, Bangladesh, 2012 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate and Postdoctoral Studies (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2015 © Md Mehadi Hasan, 2015 ii Abstract The reverberation-room method, which assumes a diffuse sound field, has long been used for various standardized room-acoustical determinations – e.g. of surface-absorption coefficients, power levels of sound sources, transmission losses of acoustical partitions, etc. In this regard, a number of standards have emerged to offer some help by outlining necessary reverberation-room design guidelines to achieve sufficient sound-field diffuseness. However, unsatisfactory opinions regarding the prediction accuracy of the method, especially at low frequencies, have been reported over the years. This might be due to deviations from the assumed diffuse-field concept, which is very challenging to implement from an application point of view; also there are no straight-forward ways to characterize the degree of sound-field diffuseness. To investigate the problem and propose solutions, diffuse-field theory and existing standards have been revisited. Using numerical, finite-element-based, modal prediction, their capacity/effectiveness to achieve a diffuse sound field is analyzed by means of a number of descriptors (room-acoustical parameters). Because of time limitations, the concept regarding the design of a reverberation-room structure – i.e. size, shape, etc. – is mainly explored, rather than the internal test-setup arrangements. The prediction accuracy of different room-acoustical parameters are also determined by the reverberation-room standard methods, with respect to both the Sabine and Eyring versions of the diffuse-field formulae, and both in octave and third-octave bands. The minimum approachable frequencies of predictions and the quality of sound-field diffuseness are discussed in terms of the prediction accuracy of different room-acoustical parameters. Considering three room volumes prescribed by standards, and four room shapes for each of the volumes, it has been found that the reverberation room of volume 150 m3, as prescribed by the ISO 354 standard, with the typical dimensional orientation (longest x-dimension/shortest vertical dimension) yields better field diffuseness than the other rooms of different volumes and shapes. To check the possibility of further improvement of the field diffuseness, a number of additional features are integrated into that reverberation room. It is found that the rooms with diffusers and absorbent corner treatments yield improved sound-field diffuseness, while the rooms with diffuse surface reflection yield poor field diffuseness due to the increased surface absorption. iii Preface This thesis is an independent work of the author performed under the direct supervision of Professor Murray Hodgson, Director of the Acoustics and Noise Research Group at UBC. A part of Chapter 4 has been published in the Journal of the Canadian Acoustical Association [Md Mehadi Hasan and Murray Hodgson. Finite-element modeling of a reverberation room: effect of the room size and shape on measurement accuracy. J of the Canadian Acoustical Association, Vol. 43 No. 3 (2015)]. iv Table of Contents Abstract .................................................................................................................... ii Preface ..................................................................................................................... iii Table of Contents ................................................................................................... iv List of Tables .......................................................................................................... xi List of Figures ....................................................................................................... xiii Acknowledgements.............................................................................................. xxii Chapter 1 Introduction .............................................................................................................. 1 1.1 Research Objectives ...................................................................................... 3 1.2 Room-Acoustical Modeling .......................................................................... 4 1.3 Thesis Overview ............................................................................................ 6 Chapter 2 Diffuse-Field Concept and Room-Acoustical Modeling Techniques .................. 8 2.1 Diffuse-Field Concept and Reality ................................................................... 8 2.2 Geometric/Statistical Approaches: Diffuse-Field Theory ..............................10 2.2.1 Diffuse-Field Theories: Sabine and Eyring Formulae .............................11 2.2.1.1 Sabine’s Reverberation Equation .......................................................12 2.2.1.2 Eyring’s Reverberation Equation .......................................................13 2.2.1.3 Prediction Accuracy and Applicability of the Sabine and Eyring Formulae .........................................................................................................15 2.3 Theoretical Modal Approach: Analytical Analysis ........................................16 v 2.3.1 Wave Equation, Helmholtz Equation and Plane Waves ..........................17 2.3.2 Sound-Pressure Level ...............................................................................19 2.3.3 Impedance and Admittance: Locally-Reacting Surfaces .........................19 2.3.4 Eigenvalues and Normal Modes ...............................................................22 2.3.5 Mode Types ..............................................................................................27 2.3.6 Modal Sound Field ...................................................................................29 2.4 Numerical Modal Approach: Finite-Element (FE) Analysis ..........................34 2.4.1 FE Formulation of Sound Fields ..............................................................35 2.4.1.1 FEA in Time Domain .........................................................................36 2.4.1.2 FEA in Frequency Domain .................................................................36 2.4.2 FE Formulation of Room Eigenmodes .....................................................38 2.5 Definitions and Terminologies Regarding Modal Approaches ......................39 2.5.1 Modal Damping Constant and Reverberation Time .................................39 2.5.2 Modal-Decay Curves ................................................................................40 2.5.3 Energy-Decay Curves ...............................................................................41 2.5.4 Modal Width/Bandwidth ..........................................................................41 2.5.5 Modal Count .............................................................................................41 2.5.6 Modal Density ..........................................................................................43 2.5.7 Modal Spacing ..........................................................................................44 2.5.8 Modal Overlap and Modal-Overlap Factor ..............................................44 2.5.9 Mode-Propagation Angle .........................................................................46 2.5.10 Interference and the Waterhouse Effect .................................................47 vi Chapter 3 Characterization of Diffuse Sound Fields and Finite-Element Modeling ........48 3.1 Different Ways to Enhance Sound-Field Diffuseness ....................................48 3.1.1 Room Size .................................................................................................48 3.1.2 Room Shape ..............................................................................................49 3.1.3 Room Proportion ......................................................................................49 3.1.4 Hanging Diffusers/Volume Scatterers ......................................................50 3.1.5 Surface Diffusion ......................................................................................51 3.1.6 Low-Frequency Absorption ......................................................................53 3.2 Descriptors to Characterize Sound-Field Diffuseness ....................................53 3.2.1 Cut-off/Transition Frequency ...................................................................54 3.2.2 Directional Uniformity of Propagating Modes ........................................55 3.2.3 Reverberant-Field Uniformity ..................................................................55 3.2.4 Prediction of Sound-Pressure Level .........................................................56 3.2.5 Decay-Rate/ Reverberation-Time Uniformity ..........................................56 3.2.6 Prediction of Decay Rate/Reverberation Time.........................................57 3.2.7 Linearity of Temporal-Decay Curves .......................................................57 3.2.8 Prediction of Source-Power Level ............................................................58 3.2.9 Prediction of Absorption Coefficient .......................................................58 3.3 Methodology: Modeling and Characterizing a Diffuse Sound Field .............59 3.3.1 Finite-Element Modeling ..........................................................................59 3.3.2 Post-Processing .........................................................................................62 vii 3.3.2.1 Post-Processing with Eigenfrequencies .............................................63 3.3.2.2 Post-Processing with Frequency Responses (FR) ..............................66 3.4 Prediction Demonstration for the Given Reverberation Room ......................69 3.4.1 Third-Octave-Band Prediction without Test Absorber ............................69 3.4.2 Third-Octave-Band Prediction with Test Absorber .................................80 3.4.3 Octave-Band Prediction without Test Absorber ......................................89 3.4.4 Octave-Band Prediction with Test Absorber ............................................95 Chapter 4 Revisiting Existing Standards and Evaluating their Effectiveness .................100 4.1 Revisiting Standards: The Reverberation-Room Method ............................100 4.2 Numerical Simulation: The Reverberation Room ........................................103 4.2.1 ISO 354 Reverberation Room ................................................................103 4.2.1.1 Case #1: Regular Rectangular-Shaped with the Longest x-Dimension ......................................................................................................................103 4.2.1.2 Case #2: Regular Rectangular-Shaped with the Longest z-Dimension ......................................................................................................................107 4.2.1.3 Case #3: Irregular Oblique-Shaped with the Longest z-Dimension 111 4.2.1.4 Case #4: Irregular Oblique-Shaped with the Longest x-Dimension 114 4.2.1.5 Discussion .........................................................................................117 4.2.2 ASTM C423 Reverberation Room .........................................................131 4.2.2.1 Case #1: Regular Rectangular-Shaped with the Longest x-Dimension ......................................................................................................................131 viii 4.2.2.2 Case #2: Regular Rectangular-Shaped with the Longest z-Dimension ......................................................................................................................135 4.2.2.3 Case #3: Irregular Oblique-Shaped with the Longest z-Dimension 138 4.2.2.4 Case #4: Irregular Oblique-Shaped with the Longest x-Dimension 142 4.2.2.5 Discussion .........................................................................................145 4.2.3 ASTM E90 Reverberation Room ...........................................................159 4.2.3.1 Case #1: Regular Rectangular-Shaped with the Longest x-Dimension ......................................................................................................................159 4.2.3.2 Case #2: Regular Rectangular-Shaped with the Longest z-Dimension ......................................................................................................................163 4.2.3.3 Case #3: Irregular Oblique-Shaped with the Longest z-Dimension 166 4.2.3.4 Case #4: Irregular Oblique-Shaped with the Longest x-Dimension 170 4.2.3.5 Discussion .........................................................................................173 4.3 Summary .......................................................................................................188 Chapter 5 Alternative Reverberation-Room Configurations and their Effectiveness ....191 5.1 Oblique-Shaped, Empty Reverberation Room .............................................192 5.2 Oblique-Shaped, Empty Reverberation Room with Absorbent Corner Treatments ...........................................................................................................194 5.3 Oblique-Shaped Reverberation Room with Diffusers and Absorbent Corner Treatments ...........................................................................................................196 5.4 Oblique-Shaped, Empty Reverberation Room with Diffuse Surface Reflection ............................................................................................................198 ix 5.5 Oblique-Shaped Reverberation Room with Diffusers and Diffuse Surface Reflection ............................................................................................................200 5.6 Discussion .....................................................................................................202 Chapter 6 Conclusion .............................................................................................................207 6.1 Contributions .............................................................................................207 6.2 Summary of Results ..................................................................................208 6.3 Future Recommendations ..........................................................................211 References .............................................................................................................212 Appendix: A ..........................................................................................................217 Appendix: B ..........................................................................................................220 Appendix: C ..........................................................................................................223 Appendix: D ..........................................................................................................238 Appendix: E ..........................................................................................................247 Appendix: F ..........................................................................................................255 Appendix: G ..........................................................................................................269 Appendix: H ..........................................................................................................285 Appendix: I ...........................................................................................................299 Appendix: J ...........................................................................................................315 Appendix: K ..........................................................................................................329 Appendix: L ..........................................................................................................345 Appendix: M .........................................................................................................359 x Appendix: N ..........................................................................................................375 Appendix: O ..........................................................................................................389 Appendix: P ..........................................................................................................405 Appendix: Q ..........................................................................................................419 Appendix: R ..........................................................................................................435 Appendix: S ..........................................................................................................449 Appendix: T ..........................................................................................................465 Appendix: U ..........................................................................................................479 Appendix: V ..........................................................................................................495 Appendix: W .........................................................................................................509 Appendix: X ..........................................................................................................525 Appendix: Y ..........................................................................................................539 Appendix: Z ..........................................................................................................555 Appendix: AA .......................................................................................................569 xi List of Tables Table 3.1: Predicted modal frequencies and damping constants in the 100-Hz third-octave band........................................................................................................................................................ 64 Table 3.2: Predicted third-octave-band modal compositions without test absorber. .................... 70 Table 3.3: Predicted third-octave-band SPL statistics without test absorber. ............................... 70 Table 3.4: Predicted third-octave-band RT statistics without test absorber. ................................ 71 Table 3.5: Predicted third-octave-band decay-curve curvature statistics without test absorber. .. 71 Table 3.6: Predicted third-octave-band source-power level statistics without test absorber. ....... 72 Table 3.7: Predicted third-octave-band room absorption statistics without test absorber. ........... 72 Table 3.8: Predicted third-octave-band modal compositions with test absorber. ......................... 80 Table 3.9: Predicted third-octave-band SPL statistics with test absorber. .................................... 81 Table 3.10: Predicted third-octave-band RT statistics with test absorber. ................................... 81 Table 3.11: Predicted third-octave-band decay-curve curvature statistics with test absorber. ..... 82 Table 3.12: Predicted third-octave-band sample absorber absorption coefficient statistics of a test absorber. ........................................................................................................................................ 82 Table 3.13: Predicted octave-band modal compositions without test absorber. ........................... 89 Table 3.14: Predicted octave-band SPL statistics without test absorber. ..................................... 90 Table 3.15: Predicted octave-band RT without test absorber. ...................................................... 90 Table 3.16: Predicted octave-band decay-curve curvature statistics without test absorber. ......... 91 Table 3.17: Predicted octave-band source-power level statistics without test absorber. .............. 91 Table 3.18: Predicted octave-band room absorption statistics without test absorber. .................. 92 Table 3.19: Predicted octave-band modal compositions with test absorber. ................................ 95 Table 3.20: Predicted octave-band SPL statistics with test absorber. ........................................... 96 Table 3.21: Predicted octave-band RT statistics with test absorber. ............................................ 96 Table 3.22: Predicted octave-band decay-curve curvature statistics with test absorber. .............. 97 Table 3.23: Predicted octave-band sample absorber absorption coefficient statistics of a test absorber. ........................................................................................................................................ 97 Table 4.1: Comparison of ASTM E90, ASTM C423 and ISO 354 facility conditions. ............. 102 Table 4.2: Performance summary for the ISO 354 reverberation room with the longest x-dimension. ................................................................................................................................... 188 xii Table 4.3: Performance summary for the ASTM C423 reverberation room with the longest x-dimension. ................................................................................................................................... 188 Table 4.4: Performance summary for the ASTM E90 reverberation room with the longest z-dimension. ................................................................................................................................... 189 Table 5.1: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty), without test absorber. ....................................................................................... 193 Table 5.2: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty), with test absorber. ............................................................................................ 193 Table 5.3: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty), with absorbent corner treatments, without test absorber. ................................ 195 Table 5.4: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty), with absorbent corner treatments, with test absorber. ..................................... 195 Table 5.5: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 with diffusers and absorbent corner treatments, without test absorber. ......................... 197 Table 5.6: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 with diffusers and absorbent corner treatments, with test absorber. .............................. 197 Table 5.7: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty) with diffuse surface reflection, without test absorber. ...................................... 199 Table 5.8: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty) with diffuse surface reflection, with test absorber. ........................................... 199 Table 5.9: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 with diffusers and diffuse surface reflection, without test absorber. .............................. 201 Table 5.10: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 with diffusers and diffuse surface reflection, with test absorber. ................................... 201 xiii List of Figures Figure 2.1: Rectangular room geometry. ...................................................................................... 22 Figure 2.2: A typical axial modal field corresponding to eigenmode index of ‘300’. .................. 28 Figure 2.3: A typical tangential modal field corresponding to eigenmode index of ‘330’. .......... 28 Figure 2.4: A typical oblique modal field corresponding to eigenmode index of ‘333’. ............. 29 Figure 2.5: Modal overlapping for overlap factor, Mf =3. ............................................................ 45 Figure 2.6: Mode-propagation angle............................................................................................. 46 Figure 3.1: Lateral cross-section through a Schroeder quadratic residue diffuser with 𝑁 = 7. . 52 Figure 3.2: Lateral cross-section through a Schroeder quadratic residue diffuser with 𝑁 = 7. . 52 Figure 3.3: Domain discretization/meshing of the geometry. ....................................................... 60 Figure 3.4: Predicted mode propagation angles as function of frequency. ................................... 65 Figure 3.5: Theoretical modal-decay curves and the corresponding RTs, in third-octave bands and without test absorber. ............................................................................................................. 74 Figure 3.6: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #5. .................................................................................................................................... 75 Figure 3.7: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #4. .................................................................................................................................... 76 Figure 3.8: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #3. .................................................................................................................................... 77 Figure 3.9: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #2. .................................................................................................................................... 78 Figure 3.10: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #1. ................................................................................................................................ 79 Figure 3.11: Theoretical modal-decay curves and the corresponding RTs, in third-octave bands with test absorber. ......................................................................................................................... 83 xiv Figure 3.12: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #5. .................................................................................................................................... 84 Figure 3.13: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #4. .................................................................................................................................... 85 Figure 3.14: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #3. .................................................................................................................................... 86 Figure 3.15: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #2. .................................................................................................................................... 87 Figure 3.16: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #1. .................................................................................................................................... 88 Figure 3.17: Theoretical modal-decay curves and the corresponding RTs, in octave bands without test absorber. .................................................................................................................... 93 Figure 3.18: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #5. .................................................................................................................................... 93 Figure 3.19: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #4. .................................................................................................................................... 93 Figure 3.20: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #3. .................................................................................................................................... 94 Figure 3.21: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #2. .................................................................................................................................... 94 xv Figure 3.22: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #1. .................................................................................................................................... 94 Figure 3.23: Theoretical modal-decay curves and the corresponding RTs, in octave bands with test absorber. ................................................................................................................................. 98 Figure 3.24: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #5. .................................................................................................................................... 98 Figure 3.25: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #4. .................................................................................................................................... 98 Figure 3.26: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #3. .................................................................................................................................... 99 Figure 3.27: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #2. .................................................................................................................................... 99 Figure 3.28: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #1. .................................................................................................................................... 99 Figure 4.1: Rectangular ISO 354 reverberation-room setup with the longest x-dimension. ...... 104 Figure 4.2: Rectangular ISO 354 reverberation-room setup with the longest z-dimension........ 108 Figure 4.3: Oblique ISO 354 reverberation-room setup with the longest z-dimension. ............. 111 Figure 4.4: Oblique ISO 354 reverberation-room setup with the longest x-dimension. ............. 114 Figure 4.5: Distribution of number of modes in third-octave bands for ISO 354 reverberation rooms........................................................................................................................................... 118 Figure 4.6: Distribution of number of modes in octave bands for ISO 354 reverberation rooms...................................................................................................................................................... 118 Figure 4.7: SPL standard deviation in third-octave bands for ISO 354 reverberation rooms. ... 119 Figure 4.8: SPL standard deviation in octave bands for ISO 354 reverberation rooms. ............ 119 Figure 4.9: Predicted SPL in third-octave bands for ISO 354 reverberation rooms. .................. 119 xvi Figure 4.10: SPL-prediction error in third-octave bands for ISO 354 reverberation rooms. ...... 120 Figure 4.11: Predicted SPL in octave bands for ISO 354 reverberation rooms. ......................... 120 Figure 4.12: SPL-prediction error in different octave-bands for ISO 354 reverberation rooms. 120 Figure 4.13: RT standard deviation in third-octave bands for ISO 354 reverberation rooms. ... 121 Figure 4.14: RT standard deviation in octave bands for ISO 354 reverberation rooms. ............ 121 Figure 4.15: Predicted RT in third-octave-bands for ISO 354 reverberation rooms. ................. 122 Figure 4.16: RT-prediction error in third-octave bands for ISO 354 reverberation rooms. ....... 122 Figure 4.17: Predicted RT in octave bands for ISO 354 reverberation rooms. .......................... 122 Figure 4.18: RT-prediction error in octave bands for ISO 354 reverberation rooms. ................ 123 Figure 4.19: Curvature of decay-curve in third-octave bands for ISO 354 reverberation rooms...................................................................................................................................................... 123 Figure 4.20: Curvature of decay-curve in octave bands for ISO 354 reverberation rooms. ....... 124 Figure 4.21: Predicted source-power level in third-octaves for ISO 354 reverberation rooms. . 124 Figure 4.22: Power-level prediction error in third-octaves for ISO 354 reverberation rooms. .. 124 Figure 4.23: Predicted source-power level in octave-bands for ISO 354 reverberation rooms. . 125 Figure 4.24: Power-level prediction error in octave bands for ISO 354 reverberation rooms. .. 125 Figure 4.25: Sabine-predicted room-averaged absorption coefficients in third-octave bands for ISO 354 reverberation rooms. ..................................................................................................... 126 Figure 4.26: Sabine-prediction error for the room-averaged absorption coefficients in third-octave bands for ISO 354 reverberation rooms. ......................................................................... 126 Figure 4.27: Sabine-predicted room-averaged absorption coefficients in octave bands for ISO 354 reverberation rooms. ............................................................................................................ 126 Figure 4.28: Sabine-prediction error for room-averaged absorption coefficients in octave bands for ISO 354 reverberation rooms. ............................................................................................... 126 Figure 4.29: Sabine-predicted absorption coefficients of a sample absorber in third-octave bands for ISO 354 reverberation rooms. ............................................................................................... 127 Figure 4.30: Sabine-prediction error for the absorption coefficients of sample absorber in third-octave bands for ISO 354 reverberation rooms. ......................................................................... 127 Figure 4.31: Eyring-predicted absorption coefficients of a sample absorber in third-octave bands for ISO 354 reverberation rooms. ............................................................................................... 128 xvii Figure 4.32: Eyring-prediction error for the absorption coefficients of sample absorber in third-octave bands for ISO 354 reverberation rooms. ......................................................................... 128 Figure 4.33: Sabine-predicted absorption coefficients of a sample absorber in octave bands for ISO 354 reverberation rooms. ..................................................................................................... 128 Figure 4.34: Sabine-prediction error for the absorption coefficients of sample absorber in octave-bands for ISO 354 reverberation rooms. ..................................................................................... 129 Figure 4.35: Eyring-predicted absorption coefficients of a sample absorber in octave bands for ISO 354 reverberation rooms. ..................................................................................................... 129 Figure 4.36: Eyring-prediction error for the absorption coefficients of sample absorber in octave bands for ISO 354 reverberation rooms. ..................................................................................... 129 Figure 4.37: Rectangular ASTM C423 reverberation-room setup with the longest x-dimension...................................................................................................................................................... 131 Figure 4.38: Rectangular ASTM C423 reverberation-room setup with the longest z-dimension...................................................................................................................................................... 135 Figure 4.39: Oblique ASTM C423 reverberation-room setup with the longest z-dimension. .... 138 Figure 4.40: Oblique ASTM C423 reverberation-room setup with the longest x-dimension. ... 142 Figure 4.41: Distribution of mode number in third-octave bands for ASTM C423 reverberation rooms........................................................................................................................................... 145 Figure 4.42: Distribution of mode number in octave bands for ASTM C423 reverberation rooms...................................................................................................................................................... 146 Figure 4.43: SPL standard deviation in third-octave bands for ASTM C423 reverberation rooms...................................................................................................................................................... 146 Figure 4.44: SPL standard deviation in octave bands for ASTM C423 reverberation rooms. ... 147 Figure 4.45: Predicted SPL in third-octave bands for ASTM C423 reverberation rooms. ........ 147 Figure 4.46: SPL-prediction error in third-octave bands for ASTM C423 reverberation rooms...................................................................................................................................................... 147 Figure 4.47: Predicted SPL in octave bands for ASTM C423 reverberation rooms. ................. 148 Figure 4.48: SPL-prediction error in octave-bands for ASTM C423 reverberation rooms. ....... 148 Figure 4.49: RT standard deviation in third-octave bands for ASTM C423 reverberation rooms...................................................................................................................................................... 149 Figure 4.50: RT standard deviation in octave bands for ASTM C423 reverberation rooms. ..... 149 xviii Figure 4.51: Predicted RT in third-octave-bands for ASTM C423 reverberation rooms. .......... 149 Figure 4.52: RT-prediction error in third-octave bands for ASTM C423 reverberation rooms. 150 Figure 4.53: Predicted RT in octave bands for ASTM C423 reverberation rooms. ................... 150 Figure 4.54: RT-prediction error in octave bands for ASTM C423 reverberation rooms. ......... 150 Figure 4.55: Curvature of decay-curve in third-octave bands for ASTM C423 reverberation rooms........................................................................................................................................... 151 Figure 4.56: Curvature of decay-curve in octave bands for ASTM C423 reverberation rooms. 151 Figure 4.57: Predicted source-power level in third-octaves for ASTM C423 reverberation rooms...................................................................................................................................................... 152 Figure 4.58: Power-level prediction error in third-octaves for ASTM C423 reverberation rooms...................................................................................................................................................... 152 Figure 4.59: Predicted source-power level in octave-bands for ASTM C423 reverberation rooms...................................................................................................................................................... 152 Figure 4.60: Power-level prediction error in octave bands for ASTM C423 reverberation rooms...................................................................................................................................................... 153 Figure 4.61: Sabine-predicted room-averaged absorption coefficients in third-octave bands for ASTM C423 reverberation rooms. ............................................................................................. 153 Figure 4.62: Sabine-prediction error for the room-averaged absorption coefficients in third-octave bands for ASTM C423 reverberation rooms. .................................................................. 154 Figure 4.63: Sabine-predicted room-averaged absorption coefficients in octave bands for ASTM C423 reverberation rooms........................................................................................................... 154 Figure 4.64: Sabine-prediction error for room-averaged absorption coefficients in octave bands for ASTM C423 reverberation rooms. ........................................................................................ 154 Figure 4.65: Sabine-predicted absorption coefficients of a sample absorber in third-octave bands for ASTM C423 reverberation rooms. ........................................................................................ 155 Figure 4.66: Sabine-prediction error for the absorption coefficients of sample absorber in third-octave bands for ASTM C423 reverberation rooms. .................................................................. 155 Figure 4.67: Eyring-predicted absorption coefficients of a sample absorber in third-octave bands for ASTM C423 reverberation rooms. ........................................................................................ 156 Figure 4.68: Eyring-prediction error for the absorption coefficients of sample absorber in third-octave bands for ASTM C423 reverberation rooms. .................................................................. 156 xix Figure 4.69: Sabine-predicted absorption coefficients of a sample absorber in octave bands for ASTM C423 reverberation rooms. ............................................................................................. 156 Figure 4.70: Sabine-prediction error for the absorption coefficients of sample absorber in octave-bands for ASTM C423 reverberation rooms. ............................................................................. 157 Figure 4.71: Eyring-predicted absorption coefficients of a sample absorber in octave bands for ASTM C423 reverberation rooms. ............................................................................................. 157 Figure 4.72: Eyring-prediction error for the absorption coefficients of sample absorber in octave bands for ASTM C423 reverberation rooms. ............................................................................. 157 Figure 4.73: Rectangular ASTM E90 reverberation-room setup with the longest x-dimension. 159 Figure 4.74: Rectangular ASTM E90 reverberation-room setup with the longest z-dimension. 163 Figure 4.75: Oblique ASTM E90 reverberation-room setup with the longest z-dimension. ...... 166 Figure 4.76: Oblique ASTM E90 reverberation-room setup with the longest x-dimension. ...... 170 Figure 4.77: Distribution of mode number in third-octave bands for ASTM E90 reverberation rooms........................................................................................................................................... 174 Figure 4.78: Distribution of mode number in octave bands for ASTM E90 reverberation rooms...................................................................................................................................................... 174 Figure 4.79: SPL standard deviation in third-octave bands for ASTM E90 reverberation rooms...................................................................................................................................................... 175 Figure 4.80: SPL standard deviation in octave bands for ASTM E90 reverberation rooms. ..... 175 Figure 4.81: Predicted SPL in third-octave bands for ASTM E90 reverberation rooms. ........... 176 Figure 4.82: SPL-prediction error in third-octave bands for ASTM E90 reverberation rooms. 176 Figure 4.83: Predicted SPL in octave bands for ASTM E90 reverberation rooms..................... 176 Figure 4.84: SPL-prediction error in different octave-bands for ASTM E90 reverberation rooms...................................................................................................................................................... 177 Figure 4.85: RT standard deviation in third-octave bands for ASTM E90 reverberation rooms...................................................................................................................................................... 177 Figure 4.86: RT standard deviation in octave bands for ASTM E90 reverberation rooms. ....... 178 Figure 4.87: Predicted RT in third-octave-bands for ASTM E90 reverberation rooms. ............ 178 Figure 4.88: RT-prediction error in third-octave bands for ASTM E90 reverberation rooms. .. 178 Figure 4.89: Predicted RT in octave bands for ASTM E90 reverberation rooms. ..................... 179 Figure 4.90: RT-prediction error in octave bands for ASTM E90 reverberation rooms. ........... 179 xx Figure 4.91: Curvature of decay-curve in third-octave bands for ASTM E90 reverberation rooms...................................................................................................................................................... 180 Figure 4.92: Curvature of decay-curve in octave bands for ASTM E90 reverberation rooms. .. 180 Figure 4.93: Predicted source-power level in third-octaves for ASTM E90 reverberation rooms...................................................................................................................................................... 181 Figure 4.94: Power-level prediction error in third-octaves for ASTM E90 reverberation rooms...................................................................................................................................................... 181 Figure 4.95: Predicted source-power level in octave-bands for ASTM E90 reverberation rooms...................................................................................................................................................... 181 Figure 4.96: Power-level prediction error in octave bands for ASTM E90 reverberation rooms...................................................................................................................................................... 182 Figure 4.97: Sabine-predicted room-averaged absorption coefficients in third-octave bands for ASTM E90 reverberation rooms. ................................................................................................ 182 Figure 4.98: Sabine-prediction error for the room-averaged absorption coefficients in third-octave bands for ASTM E90 reverberation rooms. .................................................................... 183 Figure 4.99: Sabine-predicted room-averaged absorption coefficients in octave bands for ASTM E90 reverberation rooms. ............................................................................................................ 183 Figure 4.100: Sabine-prediction error for room-averaged absorption coefficients in octave bands for ASTM E90 reverberation rooms. .......................................................................................... 183 Figure 4.101: Sabine-predicted absorption coefficients of a sample absorber in third-octave bands for ASTM E90 reverberation rooms. ................................................................................ 184 Figure 4.102: Sabine-prediction error for the absorption coefficients of sample absorber in third-octave bands for ASTM E90 reverberation rooms. .................................................................... 184 Figure 4.103: Eyring-predicted absorption coefficients of a sample absorber in third-octave bands for ASTM E90 reverberation rooms. ................................................................................ 185 Figure 4.104: Eyring-prediction error for the absorption coefficients of sample absorber in third-octave bands for ASTM E90 reverberation rooms. .................................................................... 185 Figure 4.105: Sabine-predicted absorption coefficients of a sample absorber in octave bands for ASTM E90 reverberation rooms. ................................................................................................ 186 Figure 4.106: Sabine-prediction error for the absorption coefficients of sample absorber in octave-bands for ASTM E90 reverberation rooms. .................................................................... 186 xxi Figure 4.107: Eyring-predicted absorption coefficients of a sample absorber in octave bands for ASTM E90 reverberation rooms. ................................................................................................ 186 Figure 4.108: Eyring-prediction error for the absorption coefficients of sample absorber in octave bands for ASTM E90 reverberation rooms. ................................................................................ 187 Figure 5.1: Oblique-shaped, empty reverberation-room setup. .................................................. 192 Figure 5.2: Oblique-shaped, empty reverberation-room setup with absorbent corner treatments...................................................................................................................................................... 194 Figure 5.3: Oblique-shaped reverberation-room setup with diffusers and absorbent corner treatments. ................................................................................................................................... 196 Figure 5.4: Oblique-shaped, empty reverberation-room setup with diffuse surface reflection. . 198 Figure 5.5: Oblique-shaped reverberation-room setup with diffusers and diffuse surface reflection. .................................................................................................................................... 200 Figure 5.6: SPL deviations in third-octave bands for different reverberation-room configurations...................................................................................................................................................... 202 Figure 5.7: SPL-prediction errors in third-octave bands for different reverberation-room configurations. ............................................................................................................................ 202 Figure 5.8: RT deviations in third-octave bands for different reverberation-room configurations...................................................................................................................................................... 203 Figure 5.9: RT-prediction errors in third-octave bands for different reverberation-room configurations. ............................................................................................................................ 203 Figure 5.10: Curvatures of temporal-decay curves in third-octave bands for different reverberation-room configurations. ............................................................................................ 204 Figure 5.11: Source-power-level prediction errors in third-octave bands for different reverberation-room configurations. ............................................................................................ 204 Figure 5.12: Room-averaged absorption coefficients in third-octave bands for different reverberation-room configurations. ............................................................................................ 205 Figure 5.13: Test absorber absorption coefficients in third-octave bands for different reverberation-room configurations. ............................................................................................ 205 xxii Acknowledgements First and foremost, I would like to express my enduring gratitude to my supervisor, Dr. Murray Hodgson, Director of the Acoustics and Noise Research Group at UBC. I am very grateful to him for his time and kind effort that he committed over the last two years to complete this work, as well as his kind support during my hard times at UBC. Thank you very much, Murray! I would like to extend my gratitude to Dr. Ramani Ramakrishnan, Professor at Ryerson University, for his cordial collaborations regarding diffuse-surface-reflection modeling. I would like to thank Behrooz Yousefzadeh, a former Acoustic Lab member, for his suggestions regarding DIFT at an early stage of the work. I would also like to thank my Acoustic Lab colleague Vivek Shankar for his help regarding COMSOL modeling, and other colleagues for their inspiration and cordiality. A toast to all of my friends/well-wishers for their support and encouragement throughout my education at UBC. At the end, all praises go to the almighty God ALLAH (swt), who makes all these things possible, and for his mercy that he bestowed upon me and gave me strength to continue the work at times when I came up with some harsh reality of my life at UBC. 1 Chapter 1 Introduction In room-acoustical modeling, the accurate determination of different acoustical properties – i.e. absorption coefficients of surfaces, transmission losses of room surfaces and partitions, sound-power outputs of sources, etc. – is crucial to obtaining reliable predictions. For this reason, different methods, involving theory and measurements using test facilities, have evolved to determine different room-acoustical properties accurately. For instance, two techniques commonly used for determining surface-absorption coefficients are: 1) Reverberation-room method, and 2) Impedance-tube transfer-function method. Similarly, for source-power determination, commonly-used techniques are: 1) Reverberation-room method, 2) Anechoic-room method, 3) Direct intensity method, etc. Transmission suites, which consist of two reverberation rooms on either sides of a test partition, are used to determine transmission loss. Each of the methods has its own particular attributes that make it suitable for a particular application. In particular, the reverberation-room method involves a special test room in which a variety of standardised room-acoustical measurements are made. The experimental results are used in diffuse-field theory, to determine values of key parameters, among them sound-power levels of sources, transmission losses of building materials and absorption coefficients of noise-control products. The method has the advantages of simplicity of the measurement procedure and simplicity of the mathematics involved [1, 2]. In addition to this, a number of standards have emerged to ease the measurement procedure and improve the accuracy of the final results. For instance, the method for measuring the absorption coefficient of a specimen using the reverberation-room method is given in ASTM C423 [3] and ISO 354 [4], while ASTM E90 [5] and ISO 3741 [6] explain the methods for measuring the transmission loss of an acoustical partition and the sound-power output level of a source, respectively, using the same method. A central concept in this reverberation-room method, which makes the mathematics very easy and the final results relevant to reverberant rooms, is the ‘diffuse sound field’. Therefore, the prediction accuracy using the diffuse-field formulae is completely dependent on the degree of 2 sound-field diffuseness in the test facility (the reverberation room) [1, 2]. The concept of a special test facility like the reverberation room actually evolved to ensure that the applicability of the mathematical model – i.e. the diffuse-field formulae used in the reverberation-room method – is valid. In this respect, standards have emerged as tools to assist in ensuring that reverberation rooms create highly diffuse fields by outlining necessary design guidelines. The concept of a diffuse field, the most crucial assumption made in the reverberation-room method, has a long, controversial history. However, the well-defined characteristics of a diffuse field that have gained acceptance are: spatial uniformity of energy density, equal probability of energy flow in all directions, and directions of propagating waves having random phase relations and equal probability [7]. This provides an abstract concept of the diffuse field; however, it offers very little help from the operational point of view of designing a test facility to create one, as it is very challenging to ensure such a field over a wide frequency range from a practical point of view and, at the same time, there are no direct ways to characterize the degree of sound-field diffuseness in a room. In real-life applications, in order to achieve a diffuse sound field, the main design characteristics of a reverberation room that have been commonly used are: sufficient room volume, appropriate shape and/or diffusing elements, and insignificant absorption over the frequency range of interest [8]. Despite having a highly controlled environment in a reverberation room created and commissioned based on the standards’ prescriptions, significant deviations in measured quantities have been found in many round-robin tests [9, 10]. Therefore, the universal use of quantities measured in a specific reverberation room is questionable. Researchers have been investigating the problem over the years to find a solution, but the problem remains unresolved. However, researchers have unanimously agreed that the definition of the diffuse field in a reverberation room could be the main culprit behind this problem. The concept of ‘degree of field diffuseness’ has been widely studied; even though the concept of diffuse field is well understood, until now there is no straight-forward way to characterize the degree of field diffuseness in a reverberation room. Even though a number of definitions and descriptors of diffuseness have been used to characterize field diffuseness, many of them have been misused, if not misunderstood. For example, the knowledge of only the pressure-field uniformity does not 3 give sufficient information about the field diffuseness. In general, more than one descriptor must be used to correctly define the field diffuseness. All these phenomena regarding the reverberation-room method make the problem very interesting and important, and the focus of this thesis. 1.1 Research Objectives One point is very conspicuous from the above discussion – reverberation-room methods assuming a diffuse-field are crucial and, hence, require special attention. It is often forgotten that the diffuse-field concept is based on some strong assumptions, which are very challenging to implement and, eventually, may limit their applicability. Briefly, the diffuse-field approximation in a reverberation room may be misleading. Therefore, whenever we deal with a reverberation room, the following questions need to be answered:  What does the term ‘diffuse sound field’ actually mean?  How to characterize the degree of diffuseness of a sound field?  For a given reverberation room, does the room‘s sound field approximate a diffuse field?  If so, then what are the frequency limits and what is the quality of such a field?  What would be the optimal configuration of a reverberation room to create a diffuse sound field?  Since the room will be used for testing absorbent materials, what will be the effect of the test material on sound-field diffuseness? In this study, throughout our journey towards reverberation-room design, we will keep looking for answers to these questions. Concisely, how we can achieve the best approximation to a diffuse sound field in a reverberation room will be the ultimate objective of this study. Before diving into the detailed analysis, we will review existing standards and study their scopes and limitations (if any) to achieving a diffuse field. Then, we will try to characterize the concept of a diffuse field quantitatively. Next, we will try different room configurations to determine the optimal characteristics of a reverberation room that will approximate the diffuse field as closely as possible. 4 We can summarize the objectives of this study in the following points:  To make a comprehensive study of the theory of the diffuse sound field, and then to extend the study to formulate the mathematics – i.e. the Sabine and Eyring diffuse-field formulae – to understand the relevance/applicability of these theories in the context of reverberation-room methods.  To introduce some physical/mathematical descriptors to characterize the degree of sound-field diffuseness, quantitatively.  To review existing standards, and evaluate the effectiveness of their designs of reverberation rooms.  To perform comprehensive studies of reverberation-room modeling with different room configurations to determine the best configuration to approximate a diffuse sound field.  To calculate the uncertainty and level of accuracy associated with acoustical quantities – i.e. reverberation time, absorption coefficient, source power level, etc. – measured in a reverberation room.  Finally, to consider the possibility of writing a new standard outlining improved ways of designing reverberation rooms and conducting measurements using the reverberation-room method that will surpass current standards in terms of accuracy. 1.2 Room-Acoustical Modeling To achieve the objectives mentioned above, understanding the characteristics of the sound field in a reverberation room, to ensure whether it conforms to the assumption of a diffuse field, is of great importance. That is why computational modeling is fundamental to various room-acoustical applications like reverberation-room methods, since it helps in designing the rooms with the desired acoustical characteristics by predicting the performance before constructing the infrastructure. Sound propagation in a reverberation room with irregular geometry, complex surface impedance, wave interference and diffuse surface reflection are key features that need to be included in room modeling to simulate the best configuration of a reverberation room and approximate a diffuse sound field. For room-acoustical prediction, in general, two distinct approaches are dominant for studying sound fields in rooms: energy-based geometric theory and phase-based wave theory [11]. 5 The geometric theory is energy-based, that assumes sound propagates as rays and disregards phase effects – i.e. no interference, modes, diffractions, etc.; hence, it is accurate only at higher frequencies – i.e. above a certain cut-off frequency. This theory is based on sound-field decomposition and is computationally relatively efficient. Some notable geometric room-prediction approaches are: diffuse-field theory, image-source method, ray- and beam-tracing method, etc. Among them, diffuse-field theory is very popular and widely used due to its inherent simplicity [1, 2]. In diffuse-field theory, it is assumed that the field is diffuse, which means that the sound energy is distributed in such a way that the energy per unit volume in any region is nearly the same as in any other region; this theory at best applies to highly reverberant rooms – i.e. reverberation rooms, at frequencies at which the characteristic dimensions of which are large compared to the wavelength. Though geometric models are typically energy-based (without phase), models considering phase effects are also available. The ray-tracing model [12] and the beam-tracing model [13] developed by the UBC Acoustics and Noise Research Group are examples of this. The wave theory, on the other hand, is most reliable and appropriate from the physical point of view and, therefore, is essential for understanding the characteristics of reverberation-room sound fields. Wave-based methods solve the acoustical equations directly and, therefore, have the advantage of inherently modelling wave-related phenomena such as diffraction, interference, etc. Knudsen [14] showed experimentally that the reverberant sound field has the characteristic frequencies of the normal modes of vibration of the room, and that they are not necessarily the frequencies of the source which initiated the vibration. Wave-based methods are very helpful to resolve all the normal modes of any enclosure. Later, in a comprehensive review of room acoustics, Morse and Bolt [15] showed that the characteristics of the sound field in a room can be understood in terms of its normal modes and the associated decay constant of each of these modes. Wave-based techniques have the advantage of modelling acoustical spaces in great detail, which results in highly accurate simulations. However, this is achieved at the expense of the computational cost, which rises exponentially with increasing sampling frequency. It is, therefore, important to formulate these models as efficiently as possible. 6 This study mainly focuses on wave-based modeling, which is a good choice for the current reverberation-room modeling application for the following reasons. First, we are interested in the low-frequency range of sound fields in reverberation rooms, as the field is less diffuse in that range. At lower frequencies, the phase angles of plane waves are very important and, in this regard, wave-based models are inherently capable of capturing all wave-related phenomena that are significant in that frequency range. Second, wave-based techniques can be integrated with some other features of room-acoustical modeling, like diffuse-surface-reflection modeling, that could enhance field diffuseness, which is very important from our research point of view, as we are concerned with the low-frequency field diffuseness. As mentioned above, we preferred the wave-based approach and, therefore, selected both the theoretical modal approach and the numerical finite-element (FE) based modal approach as our study tools. Though both of them are wave-based, the numerical approach has many advantages over the theoretical modal approach, and will be discussed in subsequent chapters. The problem of developing accurate formulations of boundaries is an essential ingredient in creating realistic and accurate numerical simulations, especially given that realistic boundaries are generally frequency-dependent. Strictly speaking, complete physical models of boundaries should include the propagation of waves in the room surfaces. However, simulation results in previous studies [16, 17] have suggested that, in many practical cases, there is no significant difference if wave propagation in the surfaces is neglected. Therefore, in this study it is assumed that all room surfaces are locally reacting – i.e. the reflective properties at any point on the surfaces are completely characterised by the local surface impedance. 1.3 Thesis Overview The whole work can be divided into two major parts; first, background information, that can be found in the literature, along with theories and fundamentals necessary to conduct the reverberation-room modeling, are discussed in the first few chapters and, second, the modeling part, that includes the modeling of different room configurations and test setups, to determine the best configuration to obtain a diffuse sound field, are presented in the subsequent chapters. The chapter-wise anatomy is as follows: 7 In Chapter 2, the concept of the diffuse sound field is discussed along with theories that evolved based on this concept, in terms of their accuracy and applicability. At the same time, some fundamentals of modeling sound fields in reverberation rooms are briefly reviewed. A complete outline of the modal approach, both theoretical and numerical, along with different terminologies essential to characterize sound field diffuseness are presented. Room-acoustical fundamentals are reviewed, and the concept of a locally-reacting surface is discussed. Chapter 2 also provides an overview of the geometric/statistical modeling approach. In Chapter 3, some descriptors that have been used in this study to characterize the field diffuseness in a reverberation room – i.e. spatial uniformity of reverberation times and of sound-pressure levels (SPL), directional uniformity of propagating modes, degree of linearity of decay curves, etc. – are discussed and illustrated by a complete demonstration for a given reverberation room using the fundamentals discussed in Chapter 2. As our ultimate goal is to achieve the maximum possible field diffuseness, factors that play significant roles in controlling the degree of field diffuseness are also discussed. In Chapter 4, the existing standards used to create and commission reverberation rooms are reviewed. Then the concepts described in Chapters 2 and 3 are utilized to simulate room sound fields. Revisiting the standards and simulating room sound fields, as prescribed by standards, help us to understand and assess their effectiveness – i.e. the frequential extent of the field, degree of field diffuseness, etc. – to achieving a diffuse sound field and enable us to identify where we should focus our attention to achieve optimal field diffuseness. In Chapter 5, based on the conclusions drawn from the analyses done in Chapter 4, attempts to improve the degree of field diffuseness in terms of the accuracy of different quantities to be determined – i.e. the absorption coefficient of a specimen, sound-power level of a source, etc. – done using the reverberation-room method, by considering alternative room designs, are presented in details. Finally, in the subsequent chapters, an attempt is made to summarize the whole work by highlighting the success of the current work and its practical implications, as well as suggestions for future work. 8 Chapter 2 Diffuse-Field Concept and Room-Acoustical Modeling Techniques The aim of this chapter is to briefly review the basics of acoustics related to sound propagation in enclosed spaces. The main focus is on the prediction of the modal frequencies of the room and room sound fields, using both theoretical and numerical-modal approaches. All these analyses are based on either the Helmholtz equation or the wave equation, and on the boundary conditions and analytic formulae for the boundary impedance of locally-reacting surfaces. 2.1 Diffuse-Field Concept and Reality The most important and critical assumption made in the case of reverberation-room method is that the sound field is completely ‘diffuse’. Unfortunately, the term ‘diffuse sound field’ is not unambiguously defined in the literature. However, the following formulation appears to have gained acceptance: “In a diffuse sound field, there should be equal probability of energy flow in all possible directions, which means at a given position sound waves are approaching with random phases (incoherent), and the probabilities of approaching from all possible directions are equal [4]. It indicates that the time-averaged intensity is zero at all positions and, hence, the field is isotropic in nature. At the same time, there should be uniform energy density at all positions, ensuring homogenous characteristics of the field [4]. Once the field is homogenous and isotropic, another criterion of a diffuse sound field – constant irradiation from each volume element – will be fulfilled automatically [18].” Based on these criteria, in order to make the field homogenous and isotropic, we have to maintain at least three conditions: low absorption, complete diffuse reflection and total mixing of sound energy [18]. Under these circumstances, it can be assumed that the energy density is the same at all positions at any time, and only then are the diffuse-field formulae applicable. Unfortunately, achieving all conditions mentioned above is unrealistic. 9 This theoretical definition of ‘diffuse sound field’ is just a concept. Such a concept is valid for an unbounded space, which comprises an infinite number of plane progressive waves with random phase relations, coming from all equally-probable directions and generated by distant, uncorrelated sources – i.e. no interference – of random noise [7, 19]. The above-defined ‘diffuse sound field’ model is an idealistic construction, usually known as the Plane-Wave Model (PWM). It may be adequate from the conceptual standpoint, but hardly offers any help or criterion from an operational point of view, as we don’t have any practical, direct metric for this model [7]. All these points imply that a perfect (ideal) diffuse sound field can never exist in practice. For example, as we have to consider a confined space in practice, so there must be some energy dissipation in the space; hence there cannot be equal energy flow, but a net energy flow in the direction from the source towards the location where sound is absorbed – i.e. some part(s) of the confined space. Similarly, because of this confined nature of the space, Waterhouse (1955) stated that constructive and destructive interference patterns that happen at the room boundaries cause a non-uniform energy distribution throughout the room, and promote a non-diffuse nature of the field [20]. In addition, in the case of a diffuse-field formulation in a confined space, it is usual to refer to the surfaces as diffusely reflecting, where Lambert’s law of reflection is valid – i.e. random relation between angle of incidence and angle of reflection [21]. However, it is hard to ensure that all the surfaces reflect solely in a diffuse manner without any specular reflections. Despite having all these restrictions, the diffuse field is a very useful concept. It allows us to make many simplifying assumptions in the case of many room-acoustical predictions. For example, if somehow we can approximate the sound field in an enclosure as diffuse, then we can apply probabilistic/statistical geometric approaches like the diffuse-field model, which has been found to be more powerful than deterministic approaches like the modal approach; this is because of the generality of the theory and the simplicity of the mathematics involved – i.e. the Sabine and Eyring versions of diffuse-field formulae [1, 2]. There is a significant difference between the theoretical concept of a diffuse field and its real-life implementation; however, in a controlled environment – i.e. a reverberation room – we can create an approximation to a diffuse field in the central zone of the room. Here, it is also worth noting that the sound field in an enclosed space does not always show close resemblance to a 10 diffuse field over the entire acoustical frequency range. Specifically, in the low-frequency range, the sound field is hardly ever diffuse. This is primarily due to the fact that, at low frequency, axial and tangential modes of sound fields are dominant, which means the number of plane, progressive waves associated with them is small compared to the number associated with the oblique modes, which are dominant in the higher-frequency range. As a result, sound waves arriving at any given position cover only some particular directions. On the other hand, at mid and high frequencies, oblique modes are dominant, and so the number of plane waves associated with them is also high, which means waves approaching any position cover almost all possible directions. That’s why, at mid and high frequencies, it is reasonable to approximate the field as diffuse in the central part of the room. Based on all these observations, it is highly recommended to ensure all the following three criteria are met – irregular room shape without diffusing elements or regular room shape with diffusing elements; surfaces as diffusely reflecting; and uniform distribution of absorption over the surfaces – before assuming that there exists a diffuse sound field [21]. That means we need to be very careful before applying the diffuse-field concept to the real world. 2.2 Geometric/Statistical Approaches: Diffuse-Field Theory The geometric/statistical approach is a room-acoustical prediction tool, mostly used in mid- and high-frequency cases. It can account for irregular room geometry, furniture, and also for extended-reaction surface modeling with diffuse-reflection aspects. At the same time, it is very efficient at high frequencies, where the efficacy of the wave-based modal approaches is very poor. One of the limitations of statistical approaches is that it is not suitable at lower frequencies. This is because statistical approaches are typically energy-based (phase effect is neglected). In the low-frequency range, the sound field is hardly ever diffuse; therefore, applying this approach in the low-frequency range will yield inaccurate results. However, to overcome this low-frequency problem, wave-based statistical models (considering phase effect) have been developed in the recent past years, as mentioned in a previous section. Another limitation of statistical approaches is that, whether they are energy-based or wave-based, they are capable of predicting only room sound fields and reverberation times, but not the room eigen-modes. That’s 11 why, in order to define a low-frequency transition to a diffuse-field, modal approaches are the best candidates, rather than geometric/statistical models. A more realistic model of a diffuse sound field above the transition frequency can, however, be characterized very well with the help of geometric approaches. As discussed in the previous section, a practical diffuse sound field is composed of coherent plane waves, with random amplitudes and phases, generated by a single source in a bounded space [19]. Such a field takes into account the effect of interference. Due to interference, the field will no longer be homogenous and isotropic – i.e. time-averaged intensity assumes a finite value at all positions, and sound-pressure levels depend on the position, although the probability of the level being in a certain interval is the same at all positions. If the spatial second-order statistics of the statistical parameter – i.e. sound-pressure level, reverberation time, etc. – depend only on the separation vector between points, not on their positions, then the field can be said to be homogenous. If it depends on the magnitude of the separation vector, then the field can be said to be isotropic [19]. Among different geometric room-acoustical prediction theories, as mentioned in a previous section, diffuse-field theory will be used in this study – to perform predictions for absorption coefficient measurement of a sample absorber using the reverberation-room method – to justify the applicability of the theory, as well as the degree of diffuseness of the sound field, which is the ultimate objective of this study. 2.2.1 Diffuse-Field Theories: Sabine and Eyring Formulae Reverberation is heralded as having a “major role in every aspect of room acoustics” [22], and its attribute is always a distinct and necessary component when performing room-acoustical measurements. W. C. Sabine, in the early 20th century, is the pioneer of characterizing the reverberant quality of a room based on the average absorption quality of the room assuming a diffuse sound field; his contribution is still considered as the ground work of room-acoustical measurements and is known as the Sabine diffuse field formula [1]. Of the numerous studies done on Sabine’s work, the research done by Eyring is most notable. Eyring’s equation is based on Sabine’s, but it attempts to improve the accuracy of highly absorbent and differently-shaped 12 rooms [2]. Eyring’s equation is based on the averaging of a large number of rays’ mean free paths (MFP). Eyring’s new equation was successful in giving better predictions in high-absorption rooms, but he also stated that “…no one formula without modification is essentially all-inclusive” [2]. The Sabine and Eyring equations have been studied numerous times for prediction reliability [23-26], and although both equations are fairly accurate, there are cases that create theoretically invalid results. When using them in the reverberation-room method to determine, it is common for the Sabine and Eyring equations to produce values that exceed unity [25] because of edge-diffraction created by objects [18, 19], as well as the lack of sufficient field diffuseness. This has led to further attempts at deriving an all-inclusive reverberation-time equation. 2.2.1.1 Sabine’s Reverberation Equation As mentioned above, Sabine’s formula is based on the average-absorption criterion, according to which surfaces are exposed to a constant surface irradiation, and there exists only one energy value at any time instant over the whole domain. Under this assumption, the reverberation becomes dependent only on the dimensions and the amount of absorption in the room. The energy balance in the room as a function of time becomes: 𝐸𝑠𝑜𝑢𝑟𝑐𝑒 = 𝑉𝑑𝑤𝑑𝑡+𝑐𝐴4𝑤 (2.1) where 𝑉 and 𝐴 are the volume and ‘equivalent absorption area’ of the room, respectively, and c is the speed of sound in air (343 m/s). The differential equation becomes homogenous when the source is cut off, and the solution of Eq. (2.1) follows as a function of the damping constant, 𝛿: 𝑤(𝑡) = 𝑤0𝑒−2𝛿𝑡 =≫ 𝛿 =𝑐𝐴8𝑉 (2.2) which can be related to the reverberation time 𝑇60 by: 𝑇60 =6.91𝛿= 0.161𝑉𝐴= 0.161𝑉𝛼𝑆 (2.3) 13 This is the general form of Sabine’s reverberation-time equation. The simplicity of Sabine’s equation leaves some skepticism regarding its accuracy to predict reverberation times. This is because, maintaining the constant surface-irradiation condition is very difficult in practice, even when the surfaces are diffusely reflecting [18]. However, this equation is referenced in international standards ASTM C423 [3] and ISO 354 [4] as the reverberation-time equation to use in measuring absorption coefficients. 2.2.1.2 Eyring’s Reverberation Equation Sabine’s work was based on highly reverberant rooms (i.e. ‘live’ rooms) and, therefore, is not very helpful in smaller rooms where reverberation time needs to be predicted accurately. The basis of Eyring’s work was to introduce reverberation formula that will also be accurate in low-reverberation rooms (i.e. ‘dead’ rooms) [2]. In his formula, he used a constant mean-free-path definition for a representative sound particle, rather than considering constant energy as did Sabine [1]. In order to define mean free path, he considered the average number of reflections. Energy will be reduced gradually, since part of the energy will either be absorbed or transmitted, or both, at the surfaces during each reflection. So the decay of sound energy at time 𝑡, and after 𝑁𝑟𝑒𝑓 reflections, can be expressed as: 𝐸(𝑡) = 𝐸0 𝑒𝑁𝑟𝑒𝑓 ln(1−𝛼) (2.4) Now, we have an expression for energy decreasing as a function of time, and reflection 𝑁𝑟𝑒𝑓 will lose intensity by (1 – α) as time increases. When using Eyring’s expression for the average number of reflections: 𝑁𝑟𝑒𝑓 =𝑐𝑡𝑑𝑚𝑓𝑝 (2.5) where 𝑑𝑚𝑓𝑝 is the mean free path (MFP), determined by Sabine to be 𝑑𝑚𝑓𝑝 = 0.62 𝑉13⁄ . For a cubic-shaped room, 𝑑𝑚𝑓𝑝 = 3.7𝑉/𝑆, but generally the MFP is defined by the kinetic theory of 14 gases [2] as 𝑑𝑚𝑓𝑝 = 4𝑉/𝑆. So now Eq. (2.5) can be written to include the dimensions of the room and the MFP, and then the total average number of reflections becomes: 𝑁𝑟𝑒𝑓 =𝑐𝑆𝑡4𝑉 (2.6) This can be put back into Eq. (2.4) and the total energy becomes as follows as a function of the damping constant, 𝛿: 𝐸(𝑡) = 𝐸0𝑒−2𝛿𝑡 =≫ 𝛿 =𝑐𝑆 ln(1 − 𝛼)8𝑉 (2.7) which can be related to the reverberation time 𝑇60 by: 𝑇60 =6.91𝛿= 0.161𝑉−𝑆 ln(1 − 𝛼) (2.8) This is the general form of Eyring’s reverberation-time equation. The Eyring equation incorporates Sabine’s equation for live-room absorption; when 𝑙𝑛(1 − 𝛼) is expanded out, then 𝑙𝑛(1 − 𝛼) ≈ −𝛼 which, when used in the denominator of Eq. (2.8), becomes Sabine’s Eq. (2.3). When these two situations are evaluated in Eq. (2.8):  for high absorption, 𝛼 ≥ 0.3 (Eyring version): 𝐸(𝑡) = 𝐸0 𝑒(𝑆 ln(1−𝛼)4𝑉 𝑐𝑡) (2.9)  for low absorption, 𝛼 ≤ 0.3 (Sabine version): 𝐸(𝑡) = 𝐸0 𝑒(−𝑆𝛼4𝑉 𝑐𝑡) (2.10) 15 Therefore, it is obvious that Sabine’s equation is a special case of Eyring, and that both ‘dead’ and ‘live’ rooms can be represented by Eyring’s formula. This leads to the conclusion that Eyring’s equation offers a more complete representation of reverberation. This approximation was proven by subsequent experiments undertaken by Eyring [2]. Another noticeable fact in the case of measuring absorption coefficients using Sabine’s formula, there is the following benefit – even though the absorption coefficient approaches unity, the reverberation time will always be non-zero. This is useful in practical applications, since reverberation cannot be totally removed. However, the same is not true when using the Eyring formula, though it is more accurate than the Sabine formula. 2.2.1.3 Prediction Accuracy and Applicability of the Sabine and Eyring Formulae Though the mathematical formulation of both the Sabine and Eyring formulae is straight-forward and based on the assumption of a diffuse sound field, still, however, an unresolved question remains about their prediction accuracy. Researchers have done many experiments to come to a conclusion about the prediction accuracy of these formulae. Hodgson [24] conducted experiments with surfaces of varying absorption with an aim to predict how accurate the Sabine and Eyring predictions are in the case of non-low surface absorption. He found that Eyring prediction of reverberation times and steady-state sound-pressure levels – is much closer to the experimental results, while Sabine predictions become significantly too high as the frequency increases. This study concludes that Eyring predictions are better than Sabine’s, though they are based on the same diffuse-field assumption. Sum [27] also focused on the Sabine reverberation formula, and concluded that its over-prediction of absorption coefficients is misleading, and that it is better to consider this absorption coefficient as the Sabine-absorption factor, rather than the Sabine absorption coefficient, which will vary from room to room. Mange [28] also tried to solve this absorption-coefficient problem by introducing shorter mean free path values; for this he used some hanging panels as diffusers to ensure diffuseness. Unfortunately, it didn’t work, and he concluded that Sabine’s over-prediction is not due to the longer mean free path. Another review of the factors that affect the repeatability of absorption measurements was 16 conducted by Cops [29]. He focused on how to achieve sufficient diffuseness, and tried placing low-frequency absorbers in the corners of the room, instead of hanging diffusers. It was concluded that this approach helps to increase field diffuseness by removing the axial and tangential modes that are associated with low frequencies. It is apparent from different studies, as discussed, that there remains a lingering problem of prediction consistency with the Sabine reverberation formula. Much of this research gives new ideas about how to create sufficient diffuseness, as well as how to deal with high absorption coefficients. The most practical implication of these results is the confirmation that prediction by Eyring’s equation is the most accurate in practical situations [30]. To investigate the applicability of the diffuse-field theories – i.e. the Sabine and Eyring versions – to predict the decay rate/reverberation times and steady-state sound-pressure levels (SPL), Hodgson [23] considered four room-acoustical parameters – room shape, fitting density of volume scatterers, surface absorption and surface reflection. Finally, he concluded that Eyring decay-rate/reverberation-time predictions are highly accurate for a quasi-cubic room containing an optimum fitting density, diffusely reflecting surfaces, and for any magnitude of the absorption coefficient, distributed uniformly; steady-state sound-pressure-level predictions are accurate for an empty, quasi-cubic room with specularly-reflecting surfaces, and for any magnitude of absorption coefficient, distributed either uniformly or non-uniformly. It can be implied from this finding that the Eyring formula can be used to predict both reverberation times and the SPLs accurately only in the case of an empty room with quasi-cubic dimensions, specularly-reflecting surfaces and uniformly-distributed surface absorption. 2.3 Theoretical Modal Approach: Analytical Analysis Having looked at the concept of a diffuse sound field, we will now take a look into the modal sound field in a room, which is also an idealized model; this will be the starting point in our journey towards the modeling of diffuse fields in reverberation rooms. The modal approach provides a useful tool to characterize the sound-pressure field at any frequencies, as well as the eigenfrequencies, in reverberation rooms of any dimensions, especially at lower frequencies and 17 with smaller dimensions [31, 32]. The main advantage of this approach is the ability to implement it analytically by means of a two-index Green’s function [33]. These descriptors are also reliable, and easy to compute in the framework of the plane-wave model (PWM) [34]. Thus, it may be possible to compare the modal-approach solution to that of the ‘perfect’ diffuse-field model, the so-called PWM. Despite having various good aspects, this theoretical modal approach also suffers from a number of severe constraints. Most importantly, it is a pure-tone excitation model, which means the accuracy of the prediction is driving-frequency sensitive. In addition to this, it is capable of handling only empty, rectangular rooms with locally-reacting impedance boundary conditions. At the same time, it cannot account for diffuse surface reflections. However, this modal approach is a very helpful tool to start with and it helps gain some preliminary notions about the complexity of sound fields that are encountered in practice. In the next chapters, we will make a detailed study of the different aspects of this modal theory. 2.3.1 Wave Equation, Helmholtz Equation and Plane Waves To gain an insight into sound fields in rooms, it is often assumed that the sound field is comprised of plane waves. This is due to the fact that both the sound pressure and particle velocity are always in phase in any plane perpendicular to the direction of propagation [21, 35]. These planes are referred to as wavefronts. In a Cartesian coordinate system, the wave equation that governs the propagation of sound in a three-dimensional space is: 𝜕2𝑝𝜕𝑥2+𝜕2𝑝𝜕𝑦2+𝜕2𝑝𝜕𝑧2=1𝑐2𝜕2𝑝𝜕𝑡2 (2.11) where 𝑝 is the acoustic pressure, which is the difference between the instantaneous and static pressures in the fluid. 18 Since the wave equation (Eq. 2.11) is a second-order partial-differential equation (PDE) and, therefore, considering the plane-wave condition, the solution of Eq. (2.11) can be represented by two arbitrary functions [35]. The classical d’Alembert solution of the wave equation is: 𝑝(𝑥, 𝑦, 𝑧, 𝑡) = 𝐹 (𝑡 −𝒔𝑐) + 𝐺 (𝑡 +𝒔𝑐) (2.12) where 𝐹 and 𝐺 are two arbitrary functions representing plane waves traveling in the positive and negative s directions (𝒔 ≡ 𝑠(𝑥, 𝑦, 𝑧)) with a velocity c, respectively. In room acoustics, it is common practice to characterize a sound wave in terms of its frequency and pressure amplitude. These waves are known as harmonic waves [35]. From a mathematical point of view, it is very convenient to express the propagation of harmonic waves in terms of complex functions. Under this condition, the above plane-wave solution of Eq. (2.12) becomes: 𝑝(𝑥, 𝑦, 𝑧, 𝑡) = ?̂? 𝑒−𝑖𝒌𝒔𝑒𝑖𝜔𝑡 (2.13) where ?̂? is an arbitrary constant for the peak value, and 𝒌 is a constant relating to the wavenumber with which the wave propagates in the s direction. As we are considering harmonic time-dependence, 𝑒𝑖𝜔𝑡, the wave equation (Eq. 2.11) can now be written in terms of the wavenumber vector, 𝒌, which results in the classical Helmholtz equation: 𝜕2𝑝𝜕𝑥2+𝜕2𝑝𝜕𝑦2+𝜕2𝑝𝜕𝑧2+ 𝑘2𝑝 = 0 (2.14) The Helmholtz equation (Eq. 2.14) governs the sound propagation in a three-dimensional space. However, it does not describe the sound field in an enclosed space like a room, because it does not take account of the boundary-condition effect – i.e. waves impinging on the room surfaces. Without considering the effect of boundary conditions, the plane-wave solution is known as PWM, as mentioned earlier. 19 2.3.2 Sound-Pressure Level As has been mentioned, sound/acoustic pressure is the difference between the pressure caused by the passing wave and the ambient pressure at a particular point in space. The effective sound pressure is the root mean square (RMS) of such a pressure difference measured over a period of time at the point in space, according to: 𝑝𝑟𝑚𝑠 = √𝑝12 + 𝑝22 +⋯+ 𝑝𝑛2𝑛 (2.15) where 𝑝1, 𝑝2, … 𝑝𝑛 is the sound/acoustic pressure in Pascals measured over 𝑛 samples. Due to the nature of the human hearing system, the sound pressure is often expressed on a logarithmic scale relative to a reference pressure value 𝑝0 by: 𝐿 = 20 log |𝑝𝑟𝑚𝑠𝑝0| (2.16) where 𝐿 denotes the level difference between two sound pressures, measured in dB. If the reference pressure is taken as 𝑝0 = 2 × 10−5 Pa, which is the threshold of hearing at a frequency of 1 kHz, the resulting value 𝐿 is then referred to as the ‘sound pressure level (SPL)’. 2.3.3 Impedance and Admittance: Locally-Reacting Surfaces As we are dealing with the sound field in an enclosure, we must implement boundary conditions. The impedance boundary condition corresponding to the wave equation (Eq. 2.11) is usually expressed as: 𝜕𝑝𝜕𝒏= −𝜌𝑍𝑛,𝑗𝜕𝑝𝜕𝑡 (2.17) 20 where 𝑍𝑛,𝑗 is the normal acoustic impedance of the 𝑗𝑡ℎ surface. So, we need to know the acoustic surface impedance as seen by the impinging sound waves. The normal acoustic impedance, 𝑍𝑛, of a surface is defined as the ratio of the complex sound pressure, 𝑝, at a point on a surface to the component of the complex particle velocity, 𝑢𝑛, normal to the surface at that point: 𝑍𝑛 =𝑝𝑢𝑛 (2.18) The specific impedance, 𝜁, of a surface is defined as the ratio of the normal impedance, 𝑍𝑛, to the characteristic impedance, 𝜌𝑐, of the propagating medium – i.e. air: 𝜁 =𝑍𝑛𝜌𝑐 (2.19) Sometimes, it is convenient to use the specific admittance, 𝜂, of a surface rather than the specific impedance, 𝜁, where: 𝜂 =1𝜁 (2.20) For the case of harmonic time dependence, the impedance boundary condition corresponding to the Helmholtz equation (Eq. 2.14) is expressed as: 𝜕𝑝𝜕𝒏= −𝑖𝜌𝑐𝑍𝑛,𝑗𝑘𝑝 = −𝑖𝜂𝑗𝑘𝑝 (2.21) where 𝜂𝑗, is the specific acoustic admittance of the 𝑗𝑡ℎ surface. Surface impedance, as well as admittance, are generally functions of both driving frequency and angle of incidence, and have complex values. Consequently, to characterize a surface with uniform properties, one needs to know the (complex) value of the impedance for every angle of incidence, and at every frequency of interest. However, making simplifying assumptions may 21 lead to a significant reduction in the amount of data required for the analysis of the problem. For instance, if someone assumes that surfaces are locally-reacting, then impedance is no longer a function of angle of incidence, but only of frequency [32]. This is a huge gain over extended-reaction surfaces, where impedances vary with both angle of incidence and frequency [32]. There is a very special case for heavy, rigidly-constructed surfaces – i.e. concrete walls, as well as for absorbent surfaces like porous materials, where impedances are independent of angle of incidence [13]. Considering surface impedances – both for rigid and porous surfaces – frequency invariance is a valid assumption in the low-frequency range, where surface absorption is hardly governed by frequency. In the case of a locally-reacting surface, particle velocity normal to the surface is affected by the pressure only at that point, and unaffected by the pressures at adjacent points on the surface. However, same is not true in the extended-reaction case, where particle velocity normal to the surface is affected by the pressures at adjacent points on the surface [21, 22]. This means that, in the case of an extended-reaction surface, there will be reacting waves propagating in directions parallel to the surface like vibration or bending waves in a panel. This phenomenon also reveals another aspect – that if any surface is sufficiently rigid, which results in an insignificant bending motion in the surface, then considering that surface as locally-reacting is a reasonable assumption. In many room-acoustical calculations, it is often assumed that surfaces are completely rigid – i.e. the particle velocity normal to surface is zero – indicating infinite impedance or zero admittance and, hence, no damping. Though this assumption makes the mathematics very simple, it is not a realistic assumption, as there will always be some damping, even though the surfaces are very hard. In this study, as we are dealing with sound fields in reverberation rooms, where surfaces are made very rigid, then based on the above discussion, we will consider the surfaces as locally-reacting. At the same time, instead of considering surfaces with infinite impedance, we will consider large, finite, frequency-invariant values for the impedances to take account of the realistic effects. 22 2.3.4 Eigenvalues and Normal Modes The modal sound field in an enclosure is nothing but the superposition of all the normal modes of the room. Therefore, calculation of all the eigen-modes of the room is the prerequisite to resolve for the modal sound field. Every room has an infinite number of resonant (natural) frequencies, and the sound field in the room at any resonant frequency is called a mode or normal mode. The spatial distribution of the sound pressure resulting from the standing wave corresponding to the modal (normal) frequency is known as the mode shape. The governing equation, along with the boundary conditions imposed by the walls, form a non-self-adjoint Sturm-Liouville boundary-value problem, which gives rise to a characteristic eigenvalue problem. The mathematical solution of this equation leads to the eigenvalues and the eigenfunctions representing the natural frequencies of the room and their corresponding normal modes, respectively. In room acoustics, it is common practice to represent different room-acoustical parameters as functions of frequency. That’s why it would be reasonable to work with the Helmholtz equation, which evolves from the wave equation under a time-harmonic assumption as explained in a previous section. Normal modes of a rectangular room are obtained by solving the homogenous form of the Helmholtz equation (Eq. 2.14). Figure 2.1: Rectangular room geometry. 23 We now follow the trajectory of a sound wavefront that travels across a rectangular room (Fig. 2.1). At some point in its journey, it will hit one of the surfaces and reflect back, before continuing on its way to be reflected from other surfaces. In this way, it is possible to trace the propagation paths of many other wavefronts, travelling in other directions. Out of all these wavefronts, some of them will end up with the same starting points with the same directions as when they first left, while the rest of them will have a different end positions from the starting points. That means that, for wavefronts having the same starting and ending points, the lengths of their journies in terms of phase will correspond to an integer multiple of 2𝜋, such that there will be continuity of phase, indicating that these wavefronts will present a closed line, which we call phase-closure [21]. Wavefronts demonstrating this phase closure are termed eigen-modes of the room. This phase-closure property is the spatial extension of the ‘principle of phase matching’, which is a useful principle to characterize the room eigen-modes. Each and every wavefront must have to fulfill this condition to be an eigen-mode [36]. To calculate the eigen-modes of this rectangular room, it is necessary to calculate the wavenumbers, 𝑘. To calculate the wavenumbers, we need to satisfy both the Helmholtz equation (Eq. 2.14) and the finite-impedance boundary conditions of Eq. (2.22). The domain is Ω = (0, 𝐿𝑥) × (0, 𝐿𝑦) × (0, 𝐿𝑧), and the uniform wall impedances on each of its surfaces, Γ𝑗, are: 𝜕𝑝𝜕𝑥= 𝑖𝜂𝑥1𝑘𝑝 𝑎𝑡 𝑥 = 0𝜕𝑝𝜕𝑥= −𝑖𝜂𝑥2𝑘𝑝 𝑎𝑡 𝑥 = 𝐿𝑥 𝜕𝑝𝜕𝑦= 𝑖𝜂𝑦1𝑘𝑝 𝑎𝑡 𝑦 = 0𝜕𝑝𝜕𝑦= −𝑖𝜂𝑦2𝑘𝑝 𝑎𝑡 𝑦 = 𝐿𝑦 𝜕𝑝𝜕𝑧= 𝑖𝜂𝑧1𝑘𝑝 𝑎𝑡 𝑧 = 0𝜕𝑝𝜕𝑧= −𝑖𝜂𝑧2𝑘𝑝 𝑎𝑡 𝑧 = 𝐿𝑧 (2.22a) (2.22b) (2.22c) Using the separation-of-variables technique, the solution to Eq. (2.14) can be factored – that is, written as a product of a complex exponential and a function of 𝑥, a function of 𝑦 and a function of 𝑧: 24 𝑝(𝑥, 𝑦, 𝑧, 𝑡) = 𝑝𝑥(𝑥)𝑝𝑦(𝑦)𝑝𝑧(𝑧)𝑒𝑖𝜔𝑡 (2.23) Inserting Eq. (2.23) into Eq. (2.14), and dividing by 𝑝 yields: 1𝑝𝑥(𝑥)𝑑2𝑝𝑥(𝑥)𝑑𝑥2+1𝑝𝑦(𝑦)𝑑2𝑝𝑦(𝑦)𝑑𝑦2+1𝑝𝑧(𝑧)𝑑2𝑝𝑧(𝑧)𝑑𝑧2+ 𝑘𝑥2 + 𝑘𝑦2 + 𝑘𝑧2 = 0 (2.24) Equating the first three terms of Eq. (2.24) with the separation constants 𝑘𝑥2, 𝑘𝑦2 and 𝑘𝑧2 yields: 𝑑2𝑝𝑥(𝑥)𝑑𝑥2+ 𝑘𝑥2 𝑝𝑥(𝑥) = 0 𝑑2𝑝𝑦(𝑦)𝑑𝑦2+ 𝑘𝑦2 𝑝𝑦(𝑦) = 0 𝑑2𝑝𝑧(𝑧)𝑑𝑧2+ 𝑘𝑧2 𝑝𝑧(𝑧) = 0 (2.25a) (2.25b) (2.25c) Here, separation constants are subjected to the wavenumber vector, 𝒌, (eigenvalue) by: 𝒌2 = 𝑘𝑥2 + 𝑘𝑦2 + 𝑘𝑧2 (2.26) The solutions to Eq. (2.25) can be written in the following forms: 𝑝𝑥(𝑥) = 𝐴𝑒𝑖𝑘𝑥𝑥 + 𝐵𝑒−𝑖𝑘𝑥𝑥 𝑝𝑦(𝑦) = 𝐶𝑒𝑖𝑘𝑦𝑦 + 𝐷𝑒−𝑖𝑘𝑦𝑦 𝑝𝑧(𝑧) = 𝐸𝑒𝑖𝑘𝑧𝑧 + 𝐹𝑒−𝑖𝑘𝑧𝑧) (2.27a) (2.27b) (2.27c) Consequently, the final solution to Eq. (2.14) can be written as: 𝑝(𝑥, 𝑦, 𝑧, 𝑡) = (𝐴𝑒𝑖𝑘𝑥𝑥 + 𝐵𝑒−𝑖𝑘𝑥𝑥)(𝐶𝑒𝑖𝑘𝑦𝑦 + 𝐷𝑒−𝑖𝑘𝑦𝑦)(𝐸𝑒𝑖𝑘𝑧𝑧 + 𝐹𝑒−𝑖𝑘𝑧𝑧)𝑒𝑖𝜔𝑡 (2.28) 25 Eq. (2.28) must satisfy the boundary conditions of Eq. (2.22). Satisfying Eq. (2.27a) with the first condition in Eq. (2.22a) yields the 𝑙𝑡ℎ eigenfunctions in the 𝑥 direction; similarly, satisfying Eq. (2.27b) with the first condition of Eq. (2.22b), and Eq. (2.27c) with the first condition of Eq. (2.22c), yields the 𝑚𝑡ℎ and 𝑛𝑡ℎ eigenfunctions in the 𝑦 and 𝑧 directions, respectively, as follows: 𝑝𝑥𝑙(𝑥) = 𝑃[(𝑘𝑥𝑙 + 𝜂𝑥1𝑘)𝑒𝑖𝑘𝑥𝑙𝑥 + (𝑘𝑥𝑙 − 𝜂𝑥1𝑘)𝑒−𝑖𝑘𝑥𝑙𝑥] 𝑝𝑦𝑚(𝑦) = 𝑄[(𝑘𝑦𝑚 + 𝜂𝑦1𝑘)𝑒𝑖𝑘𝑦𝑚𝑦 + (𝑘𝑦𝑚 − 𝜂𝑦1𝑘)𝑒−𝑖𝑘𝑦𝑚𝑦] 𝑝𝑧𝑛(𝑧) = 𝑅[(𝑘𝑧𝑛 + 𝜂𝑧1𝑘)𝑒𝑖𝑘𝑧𝑛𝑧 + (𝑘𝑧𝑛 − 𝜂𝑧1𝑘)𝑒−𝑖𝑘𝑧𝑛𝑧] (2.29a) (2.29b) (2.29c) Now, again, satisfying Eq. (2.27a) with the second condition in Eq. (2.22a) yields the characteristic eigenvalue equation for the 𝑥 direction, which will result in the 𝑙𝑡ℎ eigenvalue (𝑘𝑥𝑙) for which Eq. (2.29a) is valid; similarly, satisfying Eq. (2.27b) with the second condition of Eq. (2.22b), and Eq. (2.27c) with the second condition of Eq. (2.22c), yields the eigenvalue equations, which will result in the 𝑚𝑡ℎ and 𝑛𝑡ℎ eigenvalues (𝑘𝑦𝑚 and 𝑘𝑧𝑛) for the 𝑦 and 𝑧 directions, respectively, as follows: 𝑒𝑖2𝑘𝑥𝑙𝐿𝑥 =(𝑘𝑥𝑙 − 𝜂𝑥1𝑘)(𝑘𝑥𝑙 − 𝜂𝑥2𝑘)(𝑘𝑥𝑙 + 𝜂𝑥1𝑘)(𝑘𝑥𝑙 + 𝜂𝑥2𝑘) 𝑒𝑖2𝑘𝑦𝑚𝐿𝑦 =(𝑘𝑦𝑚 − 𝜂𝑦1𝑘)(𝑘𝑦𝑚 − 𝜂𝑦2𝑘)(𝑘𝑦𝑚 + 𝜂𝑦1𝑘)(𝑘𝑦𝑚 + 𝜂𝑦2𝑘) 𝑒𝑖2𝑘𝑧𝑛𝐿𝑧 =(𝑘𝑧𝑛 − 𝜂𝑧1𝑘)(𝑘𝑧𝑛 − 𝜂𝑧2𝑘)(𝑘𝑧𝑛 + 𝜂𝑧1𝑘)(𝑘𝑧𝑛 + 𝜂𝑧2𝑘) (2.30a) (2.30b) (2.30c) The characteristic eigenvalue equations of Eq. (2.30) are of nonlinear type, having exponentials. Because of the transcendental nature of these equations, their analytical solution is not feasible. This is why the modal sound field in an enclosure with finite-impedance boundary conditions is hardly available in the literature. Most literature covered only the rigid-wall (infinite-impedance) 26 condition; some of them, though, considered the finite-impedance condition, but then applied simplifying assumptions to make life easier [22, 32, 36]. So, a general solution of the modal field is hardly found and, in this study, a general solution has been developed that can handle uniform surface impedances, where any surface can have any value of impedance. Moreover, this has been done numerically, using a method called Interval-Newton/Generalized-Bisection (IN/GB), which can yield all the eigenvalues in a specified interval [37]. Since we are using finite-surface-impedance (real or complex) boundary conditions in our study, there must be damping in the room to a certain extent. The eigenvalues resulting from Eq. (2.30), which are typically complex numbers, give a very clear notion about the damping, along with the eigenfrequencies (normal modes) of the room. The real part of the eigenvalues quantifies the normal frequencies in terms of the wave numbers, and the imaginary part quantifies the damping phenomenon [15, 22, 32]. Mathematically, any eigenvalues resulting from Eq. (2.30) can be expressed as: 𝑘𝑥𝑙 𝑜𝑟 𝑦𝑚 𝑜𝑟 𝑧𝑛 = {𝜔𝑐}𝑚𝑜𝑑𝑒 𝑤𝑎𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟+ 𝑖 {𝛿𝑐}𝑑𝑎𝑚𝑝𝑖𝑛𝑔 (2.31) where the variables 𝑙, 𝑚 and 𝑛 can take zero or positive integer values, 𝜔 is the angular eigenfrequency of a mode, and 𝛿 is the damping constant of that mode. Eigenvalues can be real for a very special case: for rigidly constructed room surfaces with infinite-valued impedance boundary conditions – i.e. no surface absorption, hence no damping, due to infinite rigidity of the surfaces [36]. Each combination of 𝑙, 𝑚 and 𝑛 describes a room mode for which the eigenvalue or the wavenumber, 𝑘𝑙𝑚𝑛, can be obtained from Eq. (2.26): 𝑘𝑙𝑚𝑛 = √𝑘𝑥𝑙2 + 𝑘𝑦𝑚2 + 𝑘𝑧𝑛2 (2.32) Therefore, the modal frequency/normal frequency/eigenfrequency, 𝑓𝑙𝑚𝑛: 27 𝑓𝑙𝑚𝑛 =𝑐2𝜋√𝑘𝑥𝑙2 + 𝑘𝑦𝑚2 + 𝑘𝑧𝑛2 (2.33) can be obtained from the real part of Eq. (2.33), and also the corresponding imaginary part will represent the damping associated with that mode. Normal modes are actually ‘local’ to the space that is confined by its boundaries; hence they are also called as ‘local modes’. They also have other names, such as natural modes or pure standing waves. They are functions or properties of the space only, rather than combined functions of both space and excitation frequency. Basically, local or normal modes, and resonant frequency, are from the same concept, but when modes are connected to the excitation frequency, they are referred to as resonance [21]. 2.3.5 Mode Types As discussed in a previous section, the eigenfrequencies are completely determined by the nature of the room. The modes are labeled by the triple number set (𝑙, 𝑚, 𝑛) corresponding to the three coordinate directions – 𝑥, 𝑦 and 𝑧, respectively. Depending on different possible combinations of these numbers, modes can be separated into three groups and seven classes [32]. The three groups of modes are axial, tangential and oblique:  Axial mode: If two of the numbers are zero – i.e. (𝑙, 0, 0) or (0,𝑚, 0) or (0, 0, 𝑛) – then the propagation vectors of the standing waves are parallel to one of the surface axes – i.e. parallel to one of the room edges – and then we are speaking about ’axial modes’ (Figure 2.2). They are composed of two planar waves and categorized into three classes [36]. 28 Figure 2.2: A typical axial modal field corresponding to eigenmode index of ‘300’.  Tangential mode: If one of the three numbers equals zero – i.e. (𝑙, 𝑚, 0) or (𝑙, 0, 𝑛) or (0,𝑚, 𝑛) – then the direction of propagation angle with the axis corresponding to the zero-valued number is 900, which indicates that the propagation vectors of the standing waves are perpendicular to the respective axis – i.e. parallel to one pair of the room surfaces which are perpendicular to that axis – and the corresponding standing waves are termed ‘tangential modes’ (Figure 2.3). They are composed of four planar waves and also categorized into three classes [22, 36]. Figure 2.3: A typical tangential modal field corresponding to eigenmode index of ‘330’.  Oblique mode: If all of the three numbers are non-zero, then the oscillation patterns are termed ‘oblique modes’ (Figure 2.4) and, in this case, wave propagation occurs in a direction that is oblique to all surfaces [21]. 29  Figure 2.4: A typical oblique modal field corresponding to eigenmode index of ‘333’. Oblique modes are composed of eight planar waves and all of them are in one class [36] and, mathematically, they can be represented by decomposing Eq. (2.27) as follows: 𝑝(𝑥, 𝑦, 𝑧, 𝑡) = 𝑎1𝑒−𝑖𝑘𝑥𝑥𝑒−𝑖𝑘𝑦𝑦𝑒−𝑖𝑘𝑧𝑧 + 𝑎2𝑒−𝑖𝑘𝑥𝑥𝑒𝑖𝑘𝑦𝑦𝑒𝑖𝑘𝑧𝑧+ 𝑎3𝑒𝑖𝑘𝑥𝑥𝑒−𝑖𝑘𝑦𝑦𝑒𝑖𝑘𝑧𝑧 + 𝑎4𝑒𝑖𝑘𝑥𝑥𝑒𝑖𝑘𝑦𝑦𝑒−𝑖𝑘𝑧𝑧+ 𝑎5𝑒−𝑖𝑘𝑥𝑥𝑒−𝑖𝑘𝑦𝑦𝑒𝑖𝑘𝑧𝑧 + 𝑎6𝑒𝑖𝑘𝑥𝑥𝑒−𝑖𝑘𝑦𝑦𝑒−𝑖𝑘𝑧𝑧+ 𝑎7𝑒−𝑖𝑘𝑥𝑥𝑒𝑖𝑘𝑦𝑦𝑒−𝑖𝑘𝑧𝑧 + 𝑎8𝑒𝑖𝑘𝑥𝑥𝑒𝑖𝑘𝑦𝑦𝑒𝑖𝑘𝑧𝑧 (2.34) where 𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, 𝑎6, 𝑎7 and 𝑎8 are arbitrary constants. From Eq. (2.34), it is obvious that an oblique mode can be decomposed into eight interfering plane waves that propagate in the directions (−𝑘𝑥, −𝑘𝑦, −𝑘𝑧 ), (−𝑘𝑥, 𝑘𝑦, 𝑘𝑧 (𝑘𝑥, −𝑘𝑦, 𝑘𝑧 ), (𝑘𝑥, 𝑘𝑦, −𝑘𝑧 ), (−𝑘𝑥, −𝑘𝑦, 𝑘𝑧 ), (𝑘𝑥, −𝑘𝑦, −𝑘𝑧 ), (−𝑘𝑥, 𝑘𝑦,−𝑘𝑧 ) and (𝑘𝑥, 𝑘𝑦, 𝑘𝑧 ). These plane waves may be interpreted as one obliquely incident plane wave, and its reflections from the walls of the room. The eight waves interfere in such a manner that the boundary conditions (Eq. 2.2) are satisfied. 2.3.6 Modal Sound Field The modal sound-pressure field for the domain shown in Fig. 2.1 can be obtained by solving the inhomogeneous Helmholtz equation, given by: 𝜕2𝑝𝜕𝑥2+𝜕2𝑝𝜕𝑦2+𝜕2𝑝𝜕𝑧2+ 𝑘2𝑝 = −Δ(?̅? − 𝑟?̅?) (2.35) 30 where Δ is the Kronecker delta function, 𝑟?̅? is the source position and ?̅? is the receiver position. Once all the eigenvalues in the 𝑥, 𝑦 and 𝑧 directions are known from Eq. (2.30), the modal solution of Eq. (2.35) can be expressed by means of Green’s function, 𝐺(𝑥, 𝑦, 𝑧, 𝑡), using series summation, as follows: 𝐺(𝑥, 𝑦, 𝑧, 𝑡) =∑∑∑?̃?𝑙𝑚𝑛∞𝑛=1∞𝑚=1∞𝑙=1 𝑝𝑥𝑙(𝑥) 𝑝𝑦𝑚(𝑦) 𝑝𝑧𝑛(𝑧)𝑒𝑖𝜔𝑡 (2.36) It is not difficult to show that eigenfunctions are orthogonal to each other, which means, for the 𝑥 direction: ∫ 𝑝𝑥𝑙 𝑝𝑥𝑙 𝑑𝑥𝐿𝑥0= 0 (2.37) unless 𝑙 = 𝑙. Here, 𝑙 and 𝑙 are two different eigenvalues in the 𝑥 direction. It is customary to normalize the eigenfunctions so that: ∫ 𝑝𝑥𝑙 𝑝𝑥𝑙 𝑑𝑥𝐿𝑥0= Λ𝑥𝑙Δ𝑙𝑙 (2.38) where Δ𝑙𝑙 is known as the Kronecker delta function, which can have only two values: Δ𝑙𝑙 = 0, when 𝑙 ≠ 𝑙 and Δ𝑙𝑙 = 1, when 𝑙 = 𝑙. Using the latter condition, Eq. (2.38) yields: Λ𝑥𝑙 = ∫ 𝑝𝑥𝑙2 𝑑𝑥𝐿𝑥0 (2.39) Introducing Eq. (2.29a) into Eq. (2.39) yields: 31 Λ𝑥𝑙 = 2𝐿𝑥[𝑘𝑥𝑙2 − (𝜂𝑥1𝑘)2]−𝑖2𝑘𝑥𝑙[(𝑘𝑥𝑙 + 𝜂𝑥1𝑘)2(𝑒𝑖2𝑘𝑥𝑙𝐿𝑥 − 1)− (𝑘𝑥𝑙 − 𝜂𝑥1𝑘)2(𝑒−𝑖2𝑘𝑥𝑙𝐿𝑥 − 1)] (2.40a) Here, Λ𝑥𝑙 is the normalization constant for the eigen-function in the 𝑥 direction. It is also known as the Robin factor related to the Robin (impedance) boundary condition. Similarly, normalization constants for eigenfunctions in the 𝑦 and 𝑧 directions can be formulated as follows: Λ𝑦𝑚 = 2𝐿𝑦[𝑘𝑦𝑚2 − (𝜂𝑦1𝑘)2]−𝑖2𝑘𝑦𝑚[(𝑘𝑦𝑚 + 𝜂𝑦1𝑘)2(𝑒𝑖2𝑘𝑦𝑚𝐿𝑦 − 1)− (𝑘𝑦𝑚 − 𝜂𝑦1𝑘)2(𝑒−𝑖2𝑘𝑦𝑚𝐿𝑦 − 1)] (2.40b) Λ𝑧𝑛 = 2𝐿𝑧[𝑘𝑧𝑛2 − (𝜂𝑧1𝑘)2]−𝑖2𝑘𝑧𝑛[(𝑘𝑧𝑛 + 𝜂𝑧1𝑘)2(𝑒𝑖2𝑘𝑧𝑛𝐿𝑧 − 1)− (𝑘𝑧𝑛 − 𝜂𝑧1𝑘)2(𝑒−𝑖2𝑘𝑧𝑛𝐿𝑧 − 1)] (2.40c) Using these normalization constants of Eq. (2.40), we have the normalized, orthogonal eigenfunctions in the 𝑥, 𝑦 and 𝑧 directions, respectively, as: 𝜓𝑥𝑙(𝑥) =𝑝𝑥𝑙(𝑥)√Λ𝑥𝑙 𝜓𝑦𝑚(𝑦) =𝑝𝑦𝑚(𝑦)√Λ𝑦𝑚 𝜓𝑧𝑛(𝑧) =𝑝𝑧𝑛(𝑧)√Λ𝑧𝑛 (2.41a) (2.41b) (2.41c) 32 The modal solution then can be represented in terms of these normalized, orthonormal eigenfunctions as: 𝐺(𝑥, 𝑦, 𝑧) =∑∑∑𝐴𝑙𝑚𝑛∞𝑛=1∞𝑚=1∞𝑙=1𝜓𝑥𝑙(𝑥)𝜓𝑦𝑚(𝑦)𝜓𝑧𝑛(𝑧) (2.42) Here, 𝐴𝑙𝑚𝑛 are obtained by substituting series expansion of Eq. 2.42 into the inhomogeneous Helmholtz equation (Eq. 2.35); making use of the orthonormal property of the eigenfunctions finally yields: 𝐴𝑙𝑚𝑛 =∑∑∑1(𝑘𝑙𝑚𝑛2 − 𝑘2)∞𝑛=1∞𝑚=1∞𝑙=1 𝜓𝑥𝑙(𝑥𝑠)𝜓𝑦𝑚(𝑦𝑠)𝜓𝑧𝑛(𝑧𝑠) (2.43) Substituting Eq. 2.43 into Eq. 2.42 yields the modal solution in terms of normalized, orthonormal eigenfunctions as follows: 𝐺(𝑥, 𝑦, 𝑧) =∑∑∑𝜓𝑥𝑙(𝑥)𝜓𝑦𝑚(𝑦)𝜓𝑧𝑛(𝑧)𝜓𝑥𝑙(𝑥𝑠)𝜓𝑦𝑚(𝑦𝑠)𝜓𝑧𝑛(𝑧𝑠)(𝑘𝑙𝑚𝑛2 − 𝑘2)∞𝑛=1∞𝑚=1∞𝑙=1 (2.44) The modal solution can also be presented in terms of the eigenfunctions and their corresponding normalization constants, as follows: 𝐺(𝑥, 𝑦, 𝑧) =∑∑∑ 𝑝𝑥𝑙(𝑥) 𝑝𝑦𝑚(𝑦) 𝑝𝑧𝑛(𝑧) 𝑝𝑥𝑙(𝑥𝑠) 𝑝𝑦𝑚(𝑦𝑠) 𝑝𝑧𝑛(𝑧𝑠)(𝑘𝑙𝑚𝑛2 − 𝑘2)(Λ𝑥𝑙Λ𝑦𝑚Λ𝑧𝑛)∞𝑛=1∞𝑚=1∞𝑙=1 (2.45) The modal solution is expressed by the infinite series given in Eq. (2.45). However, in practice, this series is truncated at a finite number of normal modes. Actually, as the number of mode increases, the contribution to the pressure field becomes insignificant. That’s why, after a certain number of modes, a truncation criterion is set up. The truncation criterion ℛ is introduced such that all normal modes satisfying: 33 𝑟𝑙𝑚𝑛 ≡ √|(𝑘𝑙𝑚𝑛2 − 𝑘2)| ≤ ℛ (2.46) are included in the summation. With this criterion, the truncated modal solution is expressed as: 𝐺ℛ(𝑥, 𝑦, 𝑧) = ∑ 𝑝𝑥𝑙(𝑥) 𝑝𝑦𝑚(𝑦) 𝑝𝑧𝑛(𝑧) 𝑝𝑥𝑙(𝑥𝑠) 𝑝𝑦𝑚(𝑦𝑠) 𝑝𝑧𝑛(𝑧𝑠)(𝑘𝑙𝑚𝑛2 − 𝑘2)(Λ𝑥𝑙Λ𝑦𝑚Λ𝑧𝑛)𝑙,𝑚,𝑛𝑟𝑙𝑚𝑛≤ℛ (2.47) The modal solution expressed by Green’s function (Eq. 2.47) represents the free modal field in the room consisting of modes. These are standing waves that exist only at certain frequencies, and represent only the shape of the modal field, but do not carry any information about the size/magnitude of the field. That’s why we need to introduce a sound source as a forcing function, to excite or force the sound waves by the radiation from the source. Considering a harmonic, point sound source, 𝑄0(𝑥𝑠, 𝑦𝑠 , 𝑧𝑠)𝑒𝑖𝜔𝑡, the inhomogeneous Helmholtz equation can be expressed as: 𝜕2𝑝𝜕𝑥2+𝜕2𝑝𝜕𝑦2+𝜕2𝑝𝜕𝑧2+ 𝑘2𝑝 = −iωρ 𝑄0(𝑥𝑠, 𝑦𝑠, 𝑧𝑠) (2.48) The solution of Eq. (2.48) will represent the actual sound field in the room generated by a point source of volume velocity 𝑄0 and located at a position (𝑥𝑠, 𝑦𝑠, 𝑧𝑠), and can be expressed by means of above derived Green’s solution as: 𝑝(𝑥, 𝑦, 𝑧) = −iωρ 𝑄0 𝐺ℛ(𝑥, 𝑦, 𝑧)= −iωρ 𝑄0 ∑ 𝑝𝑥𝑙(𝑥) 𝑝𝑦𝑚(𝑦) 𝑝𝑧𝑛(𝑧) 𝑝𝑥𝑙(𝑥𝑠) 𝑝𝑦𝑚(𝑦𝑠) 𝑝𝑧𝑛(𝑧𝑠)(𝑘𝑙𝑚𝑛2 − 𝑘2)(Λ𝑥𝑙Λ𝑦𝑚Λ𝑧𝑛)𝑙,𝑚,𝑛𝑟𝑙𝑚𝑛≤ℛ (2.49) Instead of considering the volume flow rate, 𝑄0, of the source, we consider the power radiation, 𝑃0 , which is related to the volume flow rate, 𝑄0, by Eq. (2.50) [35] as: 𝑃0 =12 𝑅𝑒(𝑍𝑟𝑎𝑑)𝑄02 (2.50) 34 where 𝑍𝑟𝑎𝑑 is known as the radiation impedance, and can be calculated using Eq. (2.51) [35] as: 𝑍𝑟𝑎𝑑 = −iωρ ∑𝑝𝑥𝑙2 (𝑥𝑠) 𝑝𝑦𝑚2 (𝑦𝑠) 𝑝𝑧𝑛2 (𝑧𝑠)(𝑘𝑙𝑚𝑛2 − 𝑘2)(Λ𝑥𝑙Λ𝑦𝑚Λ𝑧𝑛)𝑙,𝑚,𝑛𝑟𝑙𝑚𝑛≤ℛ (2.51) Finally, the pressure field due to this frequency-invariant, constant-power-radiation point source can be expressed as: 𝑝(𝑥, 𝑦, 𝑧) = −iωρ√2𝑃0𝑅𝑒(𝑍𝑟𝑎𝑑)∑𝑝𝑥𝑙2 (𝑥𝑠) 𝑝𝑦𝑚2 (𝑦𝑠) 𝑝𝑧𝑛2 (𝑧𝑠)(𝑘𝑙𝑚𝑛2 − 𝑘2)(Λ𝑥𝑙Λ𝑦𝑚Λ𝑧𝑛)𝑙,𝑚,𝑛𝑟𝑙𝑚𝑛≤ℛ (2.52) 2.4 Numerical Modal Approach: Finite-Element (FE) Analysis Like the theoretical modal approach described in a previous section, the numerical modal approach is another very important tool from the room-acoustical modeling point of view. A number of numerical methods are available that can be used for room-acoustics modeling purposes: i.e. finite-difference, finite-element and boundary-element analyses. Among them, finite-element (FE) and boundary-element analyses are very popular in the field of acoustics. Boundary-element analysis is used mainly in the case of exterior acoustical modeling, while finite-element analysis (FEA) is very popular for interior or room-acoustical modeling [38]. They are both applied to integral forms of the wave equation and represent a continuous system (infinite degrees of freedom) by a discrete system (finite degrees of freedom). In our study, we will use COMSOL finite-element analysis (FEA) as reverberation-room modeling tool to predict sound fields, as well as different modal compositions, in rooms. Unlike the theoretical modal approach, numerical modal approach is equipped with some special features and can be used as an efficient room-acoustical modeling tool. For example, the theoretical modal approach is a pure-tone-excitation-based model, valid only for an empty, rectangular room with locally-reacting surfaces, and can account only for the specular-reflection 35 case, as discussed before; the FE-based numerical modal approach can be used as an broadband-noise-based excitation model, and is suitable for irregular room geometry with extended-reaction surfaces; it can account for the case of diffuse surface reflection as well. That’s why the FE-based numerical modal approach is of primary interest here and was chosen for our current reverberation-room modeling purposes. Despite all those special features, numerical modeling using FEA also has some limitations. As mentioned before, FEA represents a discrete solution (finite degrees of freedom) of a system, rather than a continuous one; hence care should be taken regarding the accuracy of the solution, which is directly linked to the degrees of freedom of the system being solved. At the same time, as the frequency of interest increases, the more degrees of freedom are required to achieve an acceptable level of accuracy. Consequently, numerical methods like FEA become inefficient at higher frequencies [13]. In particular, for rooms with large volumes, this problem is severe. As the finite-element model must discretize space into a number of elements per wavelength, large volumes involve a large number of elements, and quickly become computationally expensive at higher frequencies [39]. 2.4.1 FE Formulation of Sound Fields As outlined in previous sections, the mathematical bases for both the theoretical modal approach and the numerical modal approach are the same – i.e. the Helmholtz equation or wave equation, the only difference is that the FE-based numerical modal approach can be extended to some special features, as outlined before, while the analytical-based theoretical modal approach is very restrictive in terms of its applicability. Like other numerical methods, FEA is also based on the discretization of the computational domain and, numerically, it is very intensive; therefore, it can adapt itself easily to complex geometries. FEA can be used to predict both the frequency response/transfer function and the impulse response of a room. When FEA is performed under steady-state conditions – i.e. in a frequency-domain study using the Helmholtz equation – it will result in the transfer function of the room; on the other hand, transient analysis – i.e. a time-domain study using the wave equation – will result in the impulse response of the room. 36 2.4.1.1 FEA in Time Domain Assuming that energy is lost only by overcoming damping at boundaries, a FE formulation of the wave equation (Eq. 2.11) and its corresponding impedance boundary condition (Eq. 2.17) can be shown to yield [40]: [M]{?̈?} + [D]{?̇?} + [K]{p} = {f} (2.53) Solving this equation will result in the impulse response of the room; i.e. sound pressure as a function of time. Here, in Eq. (2.53), [M], [D] and [K] are symmetric matrices, [M] being found from the potential energy and known as the mass matrix, [D] found from the energy dissipated at the boundaries and known as the damping or dissipation matrix, and [K] from the kinetic energy and known as the stiffness matrix; {f} is the forcing function, typically known as the source term, and {p} is the acoustic-pressure response vector. 2.4.1.2 FEA in Frequency Domain Like the FEA time-domain analysis, with the same assumption that energy is dissipated only at the boundaries, a FE formulation of the Helmholtz equation (Eq. 2.14) and its corresponding impedance boundary condition (Eq. 2.21) can be shown to yield [40]: −𝜔2[M]{p} + iω[D]{p} + [K]{p} = iωρ{f} (2.54) Solving Eq. (2.54) will yield the frequency response or the transfer function of the room; i.e. sound pressure as a function of frequency. Here, in Eq. (2.54), i, ω and ρ represent the imaginary unit, angular frequency and air density, respectively. Using FEA, sound pressure in the steady state (frequency domain) can usually be found by two different techniques: direct frequency-response analysis and modal frequency-response analysis [41]. Generally, if absorption is small, the direct frequency-response analysis has mathematical uncertainty at points close to the eigenfrequencies, but it requires less computational efforts. On 37 the other hand, the modal frequency-response analysis has no such uncertainty, but is computationally very expensive. 2.4.1.2.1 Direct Frequency-Response Analysis In this case, a system of linear equations is solved directly, and the linear system of equations can be obtained by rewriting Eq. (2.54) as: [A]{p} = {f} (2.55) The coefficient matrix, [A], becomes a complex, sparse, symmetric and banded matrix – i.e. a non-Hermitian matrix. The {p} can be obtained directly by applying a direct solver or an iterative solver to Eq. (2.55). 2.4.1.2.2 Modal Frequency-Response Analysis This FEA technique is very similar to the modal-superposition technique used while developing the sound field using the theoretical modal approach, as outlined in a previous section. That means we have to compute the eigen-modes of the room first, then we construct the sound field by means of the superposition of these modes. After obtaining the eigenvalues 𝜔𝑙𝑚𝑛 (discussed in the next section) and, therefore, the corresponding eigenvectors, {𝜑𝑙𝑚𝑛}, the steady-state pressure, {𝑝(𝜔)}, can be obtained by the modal-superposition technique as [41]: {𝑝(𝜔)} = ∑{𝜑𝑙𝑚𝑛}𝑇 {f} {𝜑𝑙𝑚𝑛}(𝜔𝑙𝑚𝑛2 −𝜔2) {𝜑𝑙𝑚𝑛}𝑇 [M] {𝜑𝑙𝑚𝑛}𝑙,𝑚,𝑛 (2.56) where the damping or dissipation constant, 𝛿𝑙𝑚𝑛, associated with the eigen-modes of the room, are represented by the imaginary parts of the eigenvalues, 𝜔𝑙𝑚𝑛, and can be expressed as [42, 43]: 𝛿𝑙𝑚𝑛 =12{𝜑𝑙𝑚𝑛}𝑇 [D] {𝜑𝑙𝑚𝑛}{𝜑𝑙𝑚𝑛}𝑇 [M] {𝜑𝑙𝑚𝑛} (2.57) 38 2.4.2 FE Formulation of Room Eigenmodes In previous sections, we outlined the modal frequency-response analysis using FEA. There, it was mentioned that eigenvalue calculation is the prerequisite for that analysis. Now, we will discuss the calculation of room eigenvalues using FEA. To calculate the room eigenvalues, we consider the discretized, but homogenous (no forcing vector, {f}), form of the Helmholtz equation (Eq. 2.54). After a bit of rearrangement, Eq. 2.54 can be expressed in terms of the eigenvectors, {φ}, corresponding to the eigenvalues, ω , as: ([K] − 𝜔2[M] ){φ} = iω[D]{φ} (2.58) Like the traditional eigenvalue problem, Eq. (2.58) can be rewritten in terms of 𝜆, representing the eigenvalues of the system, which are equal to the squares of the natural or resonant frequencies, 𝜔, of the system as: ([K] − 𝜆[M] ){φ} = √−𝜆 [D]{φ} (2.59) It is easy to conclude that Eq. (2.59) is a non-linear eigenvalue problem, because it includes a non-zero right-hand-side expression coming from the damping associated with the room boundaries. The linear version of Eq. (2.59) can be deduced by making the right hand side value equal to zero as: ([K] − 𝜆[M] ){φ} = 0 (2.60) In order to have a unique solution of the linear eigenvalue system of Eq. (2.60), the determinant of the coefficient matrix must be zero, i.e.: |[K] − 𝜆[M] | = 0 (2.61) Then, we can resolve all the eigenvalues of the system from Eq. (2.61). However, for a non-linear type problem, as in Eq. (2.59), the procedure is not that straight-forward. In order to solve it, we have to assume an initial eigenvalue of the system and then iterate the procedure until it 39 converges to the actual eigenvalues [40]. This is why the computation of eigenvalues for a non-linear system is computationally very expensive. In this study, we will use the COMSOL non-linear eigenvalue solver to resolve the room eigenvalues. 2.5 Definitions and Terminologies Regarding Modal Approaches Regardless of whether it is the theoretical modal approach or the numerical modal approach, the following concepts are related to both approaches. They are very helpful to understand the characteristics of sound fields in rooms and, therefore, can be used to characterize a diffuse sound field in a reverberation room. 2.5.1 Modal Damping Constant and Reverberation Time Room eigenvalues can be used to predict sound decays in rooms. Each normal mode (standing wave) has a natural frequency given by Eq. (2.33) (real part), and it decays according to the damping constant given by the same equation (imaginary part). The damping constants, 𝛿, of normal modes are very important from the reverberation point of view, because they determine the decay rate, which is directly related to the reverberation time of the room. In room acoustics, it is more convenient to talk in terms of the reverberation times of different modes, rather than their damping constants. The reverberation time of each individual mode can be expressed in terms of its time constant, 𝜏, as follows: 𝑇60 = 6 ln(10) 𝜏 (2.62) Here, time constant 𝜏 is related to the damping constant, 𝛿, by the following relation: 𝛿𝜏 =12 (2.63) Then, the reverberation time of a mode can be expressed in terms of its damping constant, 𝛿, as follows: 40 𝑇60 =3 ln(10)𝛿 (2.64) 2.5.2 Modal-Decay Curves It was discussed in a previous section that modes are of different types and classes, so modal reverberation times also vary for different modal groups and classes; i.e. oblique modes have the highest decay rates, hence the lowest reverberation times, and axial modes have the lowest decay rates, hence the highest reverberation times. In room acoustics, it is customary to express the reverberation time in frequency bands, rather than for individual modes. In each frequency band, there are certain numbers of normal modes that determine the decay process. The decay in each frequency band of width, ∆𝑓, is characterized by the collective modal-decay curve [44, 45], expressed in terms of modal damping constant as: Modal decay (∆𝑓, t) = 10 log [∑1𝛿𝑐 𝑒−2𝛿𝑡∆𝑓∑1𝛿𝑐∆𝑓] dB (2.65) The same expression in terms of modal reverberation time can be expressed as: Modal decay (∆𝑓, t) = 10 log [∑ 𝑇 10−6𝑡 𝑇⁄∆𝑓∑ 𝑇∆𝑓] dB (2.66) where 𝑡 is time, and the summation is over all eigenvalues whose eigenfrequencies are within the band, ∆𝑓. The driving wave number, 𝑘, was set such that it corresponds to the center frequency of the band ∆𝑓. The band modal reverberation time, 𝑇60 – the time corresponding to a decay of 60 dB – was estimated from the collective-modal decay expression (Eq. 2.65). It was observed that, at low frequencies, decay curves show a monotonic curvature [44, 45]. As an explanation, it was mentioned that, at lower frequencies, the number of modal classes is high within a given frequency band, ∆𝑓; since the modal reverberation times vary for different modal classes, the 41 decay curves show monotonic curvatures. However, as frequency increases, a particular modal class – especially oblique modes – dominates in a given frequency band, ∆𝑓; hence the decay curves become linear. This implies that, as the field goes towards diffuse, the decay becomes linear. 2.5.3 Energy-Decay Curves Unlike the modal-decay curves, energy-decay curves are based on the acoustic pressure, 𝑝, which is derived from the impulse response of a room. To do that, the Schroeder backward-integral formula expressed in Eq. (2.67) is used: Energy decay (∆𝑓, t) = 10 log10∫ 𝑝2(𝑡) 𝑑𝑡∞𝑡∫ 𝑝2(𝑡) 𝑑𝑡∞0 dB (2.67) 2.5.4 Modal Width/Bandwidth Modal width is an important term related to the modal-overlap concept (discussed later). The width, 𝛿𝑤, of a resonance peak corresponding to the half-power bandwidth (3 dB point) can be estimated from the modal-damping constant, 𝛿, as follows: 𝛿𝑤 =𝛿𝜋 (2.68) In terms of modal reverberation time, 𝑇60, Eq. (2.68) can be written as: 𝛿𝑤 =3 ln(10)𝜋𝑇60 (2.69) 2.5.5 Modal Count Modal count in a given frequency band, especially at low frequencies, has many aspects that are important from a diffuse-field point of view. This is because, at mid and high frequencies, room 42 sound fields are almost diffuse, but the scenario is different at low frequencies. So we need to define a transition point before applying the concept of diffuse field. In this regard, the number of modes concept is important, because many transition-point definitions are based on this band modal-count [7] concept. The approximate number of modes, 𝑁(𝑓), with eigenfrequencies lying below a certain frequency, 𝑓, can be calculated theoretically using the formula: 𝑁(𝑓) ≈43𝜋𝑉𝑓3𝑐3+14𝜋𝑆𝑓2𝑐2+18𝐿𝑓𝑐 (2.70) Eq. (2.70) is valid only for a rectangular room and provides overestimation of the number of modes at low frequencies due to the consideration that it is a continuous function of frequency instead of discrete one [46]. For rooms with irregular shape, the modal-count formula can be rewritten using statistical concepts [32] as: 𝑁(𝑓) ≈43𝜋 𝑉 𝑓3𝑐3+64𝜋 𝑉23 𝑓2𝑐2+128 𝑉13 𝑓𝑐 (2.71) where it is assumed that 𝑆 ≈ 6 𝑉23 and 𝐿 ≈ 12 𝑉13 [19, 31]. Then the number of modes in a given band, ∆𝑓 = 𝑓𝑚𝑎𝑥 − 𝑓𝑚𝑖𝑛, can be calculated as: ∆𝑁 = 𝑁(𝑓𝑚𝑎𝑥) − 𝑁(𝑓𝑚𝑖𝑛) (2.72) So, modal count can be done either using Eq. (2.72), and will give the theoretical values, or by using the eigenfrequency Eq. (2.33) to calculate individual modal frequencies, and then by counting their numbers within the given band to obtain the actual values. 43 2.5.6 Modal Density Like the modal count, the modal-density concept is also very useful to gain an insight into the mode distribution among different frequency bands, especially at low frequencies. Therefore, modal density, 𝑛(𝑓), is also an important parameter to characterize sound fields, whether they are diffuse or not. It is defined as the number of normal modes per unit frequency bandwidth, and it can be obtained by differentiating Eq. 2.70 with respect to frequency, 𝑓, as: 𝑛(𝑓) =𝑑𝑁(𝑓)𝑑𝑓=4𝜋𝑉𝑐3𝑓2 +𝜋𝑆2𝑐2𝑓 +𝐿8𝑐 (2.73) for regular (rectangular) rooms. Similarly, for irregular rooms using Eq. (2.71), we have: 𝑛(𝑓) =𝑑𝑁(𝑓)𝑑𝑓=4𝜋𝑉𝑐3𝑓2 + 3 𝜋 𝑉23𝑐2 𝑓 +12 𝑉138𝑐 (2.74) The modal count in a given frequency band, ∆𝑓, can also be calculated using this modal-density concept as: ∆𝑁 = 𝑛(𝑓) ∆𝑓 (2.75) Since the standing waves corresponding to each eigenfrequency are, in general, associated with some particular set of direction cosines, an increase in ∆𝑁 indicates an increase in the randomness of the directions of the associated waves. So this phenomenon is a good characterization tool for the measurement of the diffuseness of a sound field. Similarly, as explained earlier, Eq. (2.75) will give the theoretical prediction of modal density; actual prediction can be done by counting modes in a band using Eq. (2.33) and then dividing this number by the frequency bandwidth, ∆𝑓. In this study, we will always use the actual values for modal count and density, unless otherwise mentioned. 44 2.5.7 Modal Spacing Modal spacing, 𝛿𝑓, is the average frequency spacing between resonance peaks, and a measure of the spread (Fig. 2.5) of the modes as a function of frequency; it can be calculated using the concept of modal density as: 𝛿𝑓 =∆𝑓∆𝑁=1𝑛(𝑓) (2.76) In the case of the dominance of oblique modes, especially when wavelength, 𝜆 = 𝑐/𝑓, corresponding to driving frequency, 𝑓, is less than about one quarter of the smallest dimension of the room, the number of oblique modes in any frequency band is considerably larger than the number of tangential or axial modes [32], and the expression for modal spacing can be approximated as Eq. (2.77): 𝛿𝑓 =∆𝑓∆𝑁≈𝑐34𝜋𝑉𝑓2 (2.77) From the diffuse-field point of view, the modal spacing should be as low and uniform as possible. This approximation for modal spacing of Eq. (2.77) is valid at mid and high frequencies – in other words, when the field is sufficiently diffuse. This is the theoretical evaluation of modal spacing; calculating the spacing between adjacent resonance frequencies within frequency band, ∆𝑓, and averaging them, will give the actual average value of modal spacing, 𝛿𝑓, for the desired frequency band. 2.5.8 Modal Overlap and Modal-Overlap Factor Like the concept of modal count, modal overlap is also very important from a diffuse-field point of view, especially to determine the transition point between diffuse and non-diffuse conditions, as there exists a number of transition/cut-off-frequency definitions based on this modal-overlap concept. This term ‘modal overlap’ is quantified in terms of modal-overlap factor, 𝑀𝑓, and is defined as the ratio of the resonance-peak width, 𝛿𝑤, to the average modal spacing, 𝛿𝑓, as: 𝑀𝑓 =𝛿𝑤𝛿𝑓 (2.78) 45 The modal overlap is interpreted physically as the average number of modes excited by a driving frequency, under the assumption that each mode is excited only if the driving frequency is within a resonance-peak width centered at the normal frequency of the mode. This is another important tool to characterize a diffuse sound field. Mathematically, the modal overlap can be expressed as: 𝑀 = 𝛿𝑤 𝑛(𝑓) (2.79) Figure 2.5: Modal overlapping for overlap factor, Mf =3. As the driving frequency increases, the average frequency spacing decreases and the resonance peaks begin to overlap. Eventually, when the resonance peak-width is much larger than the average modal spacing, the response curve becomes fairly smooth. When the condition: 𝛿𝑤 ≫ 𝛿𝑓 (2.80) or, otherwise: 𝑀𝑓 =𝛿𝑤𝛿𝑓 ≫ 1 (2.81) is fulfilled, modal overlapping takes place (Fig. 2.5). Based on Eq. (2.81), it is possible to define a cut-off/transition frequency for a diffuse sound field. However, what should be the exact relation – i.e. by what extent the peak width should be greater than the modal spacing – remains undefined. That’s why a number of cut-off frequency definitions – i.e. Schroeder frequency limit [47], the Slingerland, Elfstorm and Grun limit [48], etc. – have been defined over the past years, based on different considerations. 46 2.5.9 Mode-Propagation Angle Part of the definition of a diffuse field tells us that there should be equal probability of sound waves approaching a receiver position from any direction – i.e. from any angle. Therefore, it is instructive to look at the range of angles covered by the plane waves associated with the local or normal modes. Studying the trajectories of planar waves associated with the normal modes will give only a qualitative measure of the modal sound field. Figure 2.6: Mode-propagation angle. To describe the modal field quantitatively, we need to take account of the propagation direction of different planar waves after reflection from each surface. For each normal mode, we can quantify the direction-of-propagation angles, 𝜃𝑥, 𝜃𝑦 and 𝜃𝑧, as defined relative to lines that are normal to the 𝑥, 𝑦 and 𝑧 axes, respectively (Fig. 2.6), using the calculated wavenumber vector, 𝑘𝑙𝑚𝑛, and its components, 𝑘𝑥𝑙, 𝑘𝑦𝑚 and 𝑘𝑧𝑛, as: 𝜃𝑥 = sin−1 (𝑘𝑥𝑙𝑘𝑙𝑚𝑛) 𝜃𝑦 = sin−1 (𝑘𝑦𝑚𝑘𝑙𝑚𝑛) 𝜃𝑧 = sin−1 (𝑘𝑧𝑛𝑘𝑙𝑚𝑛) (2.82a) (2.82b) (2.82c) 47 Then we can also quantify the propagation of each planar-wave consisting of the modes, using this 𝜃𝑥 , 𝜃𝑦 and 𝜃𝑧 definitions. If the angles associated with different planar waves cover almost the whole range of angles, then it can be considered as an auspicious sign of field diffuseness. Equivalent angles do not identify the cut-off frequency above which the field can be assumed to be diffuse; rather they illustrate one aspect of a diffuse sound field. 2.5.10 Interference and the Waterhouse Effect According to the definition of the ideal diffuse field (PWM), there would be random phase relations between waves passing through a particular position in space and, hence, there would be no interference between waves, as the field is unbounded. However, in practice, as we are approximating a diffuse field in a bounded space, there will be non-random phase relations between the incident wave and the reflected wave, and this will lead to interference, especially in the vicinity of the room surfaces. This phenomenon is known as the Waterhouse Effect [20]. Because of this interference effect, there will be non-uniform energy density throughout the space, which is a limitation on considering the field as diffuse. It has been found that there is a 3-dB difference in pressure level between points closer to a surface and points farther from that surface, which indicates that the energy density in front of a surface is twice that at a point far from that surface. Similarly, near to an edge the difference is 6 dB and in a corner this difference is 9 dB. This is why it is always recommended in reverberation-room-method standards not to do any measurements close to the room surfaces, edges or corners. Based on this observation, and also since we cannot avoid this interference effect, approximating a diffuse field in the central zone of the room is more reasonable than considering the whole space as diffuse [21]. 48 Chapter 3 Characterization of Diffuse Sound Fields and Finite-Element Modeling The aim of this chapter is to discuss ways that can be utilized to enhance sound-field diffuseness. At the same time, some physical/mathematical descriptors that can be used to characterize the degree of diffuseness of a diffuse field are also discussed. Finally, it discusses how to model diffuse fields using a finite-element model. 3.1 Different Ways to Enhance Sound-Field Diffuseness There are a number of factors that can be manipulated to control the degree of diffuseness of the sound field in a reverberation room that are mentioned in the literature [7]. Among them, some important factors used in this study to promote diffuseness in a reverberation room are discussed below. 3.1.1 Room Size Providing a large volume is the most straight-forward way of promoting field diffuseness in a reverberation room. As frequency increases, the number of modes in a given frequency band increases very rapidly and tends to fill the frequency spectrum; as well, their angles of incidence become more uniformly distributed. That means that the higher the frequency, the greater is the number of modes in any frequency band and, hence, a flatter frequency response and less spatial fluctuation in the pressure field, promoting a diffuse field in that frequency range. However, at low frequencies, the scenario is different, due to the lower number of modes in the frequency bands, which results in an irregular frequency response indicating insufficient field diffuseness. It would be possible to promote diffuseness at low frequencies by building an enormous room, as mentioned in the literature [7]. However, in that case the restrictive factors are the increasing absorption contributed by the air to the total room absorption, as well as the cost associated with 49 the construction. The first condition tends to violate one of the assumptions of diffuse-field formulae that there is negligible energy loss in the room. 3.1.2 Room Shape The shape of the room, along with its size, also plays an important role in promoting a diffuse sound field in a reverberation room. Reverberation rooms could be of different shapes: regular rectangular rooms with the longest dimension (length) in the 𝑥-coordinate direction, regular rectangular rooms with the longest dimension (height) in the vertical or 𝑧-coordinate direction, irregular, oblique-shaped rooms with splayed walls, etc. However, there has been a long tradition of using splayed-wall reverberation rooms, in which none of the two wall pairs is parallel to each other, promoting field diffuseness. Actually, splaying the walls does not help much, as it does not increase the number of modes, which is solely dependent on the volume of the room. Splays only shift the modes, as well as the angles of incidence of the modes, without increasing the number of modes. Splayed walls, however, help to reduce ‘flutter echoes’ and, therefore, prevent ‘two-dimensional reverberation’ which is likely to occur in the case of a highly absorptive sample applied to a single surface of the room, similar to the case of measuring the absorption coefficients of test samples in a reverberation room. 3.1.3 Room Proportion The room proportions – the distribution of the room dimensions in space – particularly for rectangular-shaped rooms, has a significant effect on the modal distribution of the room, which is directly linked to field diffuseness. The dimensions of well-designed rectangular reverberation rooms should be selected so that they are not related by whole numbers. If the dimensions of the room are related by a whole number (say, 𝐿𝑥 = 2𝐿𝑦 = 3𝐿𝑧) some of the natural frequencies will coincide – i.e. modes with different (𝑙,𝑚, 𝑛) values will have the same modal frequencies. As a result, the frequency response of the room will be more irregular. This phenomenon, called modal degeneracy [31], should be avoided. 50 Bolt [49] and Sepmeyer [50] have studied the modal distribution of various rectangular rooms, and found that certain room proportions favour a uniform distribution of modes along the frequency axis, which leads to a flatter frequency response/more uniform transfer function, as well as a uniform angular distribution of modes, in the room. However, these calculations are valid only for empty, rectangular rooms or for rooms with slight perturbations of the rectangular shape. 3.1.4 Hanging Diffusers/Volume Scatterers Sound scattering is a fundamental phenomenon encountered in room acoustics that plays a crucial role in defining the acoustic properties of spaces. It encompasses edge-diffraction effects, as well as the diffuse reflection of sound waves from not-perfectly-smooth/uniform surfaces. Arrays of hanging diffusers/scatterers, distributed randomly throughout a reverberation room, promote the spatial uniformity of the field – in particular, in the case of absorption measurement [51]. However, an unambiguous criterion concerning the appropriate size and number of such diffusers is required to obtain an optimal field diffuseness. It is common experience that the temporal decay rate increases at first with an increasing numbers of diffusers; finally, with a very high density of diffusers, the decay rate decreases again, as there is less energy absorption on the wall in the latter case and more reflection between the diffusers. Therefore, a compromise is required between the size of the diffusers and the wavelength at low frequencies, and between the number of panels and the degree of diffuseness required. Kuttruff [51] has derived an expression to calculate the number of diffusers required to achieve optimal field diffuseness as: 𝑁𝑑𝑖𝑓𝑓 =ℎ𝑉𝑄𝐻 (3.1) Here, 𝑄 is the diffuser cross-sectional area, 𝐻 is the longest room dimension and ℎ is a parameter having values defined by Kuttruff [51] as: 0.5 < ℎ < 2 (3.2) The diffuser cross-sectional area, Q, is defined as: 51 𝑄 =𝑆𝑠2 (3.3) Here, 𝑆𝑠 is the projected surface area. At low frequencies, 𝑆𝑠 is considered only for one side of the diffusers, while both sides are considered at higher frequencies. 3.1.5 Surface Diffusion Diffusing elements applied to the room surfaces reduce the energy reflected in specular reflection and ensure spatial scattering in many directions rather than just redirecting the sound in one direction that is different from the specular direction. Consequently, a number of diffusing surfaces have been developed over the past years, among them the Maximum Length Sequence Diffuser (MLSD), Quadratic Residue Diffuser (QRD), Modulated Quadratic Residue Diffuser (MQRD), etc. In this study, we will be using QRD to promote diffuse surface reflection. In QRD, also known as the Schroeder diffuser, sequences of arrays of wells of different depths are applied to achieve diffuseness across a wide frequency range. Quadratic residue sequences are given as [52]: 𝑆𝑛 = 𝑛2 mod 𝑁 (3.4) where 𝑁 is a prime number and 𝑛2 mod 𝑁 is the lowest non-negative remainder of modulo 𝑁. Such sequences are periodic with period 𝑁, and symmetric around 𝑛 = 0 and 𝑛 = (𝑁 − 1)/2. For instance, the quadratic residue sequence for 𝑁 = 17 is: 𝑆𝑛 = (0, 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, 2, 8, 16, 9, 4, 1… . ) (3.5) The QRD has a working frequency range determined by the depths and widths of the wells. The high-frequency limit is determined by the well width as: 𝑓𝑚𝑎𝑥 =𝑐2𝑤 (3.6) where 𝑤 is the well width. The low-frequency limit depends on the well depth and is defined as: 𝑓𝑚𝑖𝑛 =𝑐𝑆𝑛2𝑁𝑑𝑛 (3.7) 52 where 𝑑𝑛 is the nth well depth. This minimum frequency limit is also known as the design frequency. One point to be mentioned here is that the design frequency is not the lowest frequency at which surface produces more scattering than a plane surface; rather it is the frequency at which scattering can have evenly distributed energy-diffraction lobes. It has been observed that Schroeder diffusers scatter differently than a plane surface at an octave or two below the design frequency [52]. Figure 3.1: Lateral cross-section through a Schroeder quadratic residue diffuser with 𝑵 = 𝟕. In Figure 3.1, a cross-sectional view of a regular Schroeder QRD is presented, while in Figure 3.2, a modified version with the minimum surface area is presented. We will be using this modified version in our study, while modeling QRD to provide diffuse surface reflection, to minimize the surface absorption. Figure 3.2: Lateral cross-section through a Schroeder quadratic residue diffuser with 𝑵 = 𝟕. When designing a QRD, we need to avoid the critical frequency at which diffusers behave as a plane surface, which occurs when all the depths are an integer multiple of half a wavelength. To 53 avoid this, it is necessary to place the first critical frequency above the maximum frequency limit of the device given by Eq. (3.6), which is possible by selecting a prime number as follows: 𝑁 =𝑐2𝑤𝑓𝑚𝑖𝑛 (3.8) To obtain the optimal performance from a QRD, it is recommended to use a lower number of periods, with a large period width (𝑁𝑤).The former will ensure periodicity, but with the diffraction lobe not too narrow; the latter will ensure a large number of grating lobes, an indication of good diffuseness. The low-frequency efficiency of a Schroeder diffuser depends on the ratio of maximum sequence number (𝑆𝑚𝑎𝑥) to the prime number (𝑁); the lower is the ratio, the higher is the low-frequency efficiency of the device. 3.1.6 Low-Frequency Absorption As mentioned before, at low frequencies there exist a lower number of well-separated modes. In that frequency range, therefore, it is desirable to make the average modal width at least equal to the average modal spacing, in order that the few available modes cover the frequency range most uniformly. In the absence of low-frequency absorption, the modal width is very small and, therefore, the mode is poorly excited. If low-frequency absorption is provided, the modal-widths spread and tend to overlap the skirts of the adjacent modes, resulting in a relatively flatter frequency-response curve. To achieve the same objective, some literature [29] proposes absorbent corner treatments of rooms – placing tetrahedral blocks covered with absorbers at the corners, as mentioned before. 3.2 Descriptors to Characterize Sound-Field Diffuseness As mentioned before, there is no direct metric to evaluate the degree of field diffuseness. As a result, a variety of methods have been proposed in the literature, based on different studies [7, 53]. We consider the following descriptors in our study: 54 3.2.1 Cut-off/Transition Frequency A tentative criterion for the existence of sufficient field diffuseness is set in terms of the cut-off frequency or the transition frequency, which indicates the low-frequency limit of the diffuse field. In general, two types of definitions are used to determine the low-frequency limit: one is based on the modal-overlap-factor concept and the other on the modal count or the least permissible number of modes in a given frequency band. For the first case, the cut-off frequency is defined as [21]: 𝑓𝑐 = √𝑇60𝑀𝑓𝑐38.8𝜋𝑉 (3.9) For a modal-overlap factor of 𝑀𝑓 = 3, the corresponding limit based on Eq. (3.9) is known as the Schroeder cut-off frequency [47]: 𝑓𝑐 = 2000√𝑇60𝑉 (3.10) Here, 𝑉 is the volume of the reverberation room, and 𝑇60 is the reverberation time. This limit proposed by Schroeder is pure-tone-excitation-based, and is known to be very restrictive from an application point of view [8]. For the second case, 20 modes per band are commonly used; for an octave band, this number is obtained when the room volume is equal to the wavelength cubed [8], and the cut-off frequency is defined as: 𝑓𝑐 =𝑐√𝑉3 (3.11) On the other hand, for a third-octave band this number is obtained when the room volume is equal to four times the wavelength cubed [7], and the cut-off frequency is defined as: 𝑓𝑐 =𝑐√𝑉/43 (3.12) 55 The definitions provided in Eq. (3.3) and (3.4) are statistically-based, and are shown to be very useful from an operational point of view [8]. In a statistical sense, the number of modes in a given frequency bandwidth is dependent only on the width of the band, the frequency and the volume of the room [7]. 3.2.2 Directional Uniformity of Propagating Modes Modes with random phase relations should be incident from all possible directions at a given location in order to consider the field as diffuse. We consider a graphical representation of the mode-propagation angles as a function of frequency as another descriptor to characterize field diffuseness. The mode-propagation angles can be calculated using Eq. (2.81). However, it has to be noted that Eq. (2.81) can be used only for the case of vector representation of the modes – in other words, only applicable for the analytical/theoretical modal approach but not for the numerical approach. 3.2.3 Reverberant-Field Uniformity Spatial uniformity of the steady-state reverberant sound energy is one of the criteria that needs to be fulfilled for a field to be considered as diffuse. As we are approximating a diffuse field in a room that is enclosed by surfaces, achieving a field with uniform reverberant energy density is not possible due to the Waterhouse Effect [20], as mentioned before. There will be a spatial variation in sound pressure levels (SPL), with the highest variation near the walls and the least at the room center. In order to quantify the field variability as a measure of field uniformity, the standard deviation of the reverberant SPL is considered as the descriptor to characterize the degree of field diffuseness. The following expression is used to calculate the standard deviations: 𝜎𝐿𝑝 =√∑ (𝐿𝑝,𝑛 − ?̅?𝑝)2𝑁𝑟𝑒𝑐𝑛=1𝑁𝑟𝑒𝑐 − 1 (3.13) 56 Here, 𝑁𝑟𝑒𝑐 is the number of receiver positions, 𝐿𝑝,𝑛 is the SPL at the nth receiver location and ?̅?𝑝 is the reverberant SPL averaged over all receiver locations. According to ISO, a standard deviation of 1.5 dB for SPL is considered as sufficient for diffuse-field approximation. 3.2.4 Prediction of Sound-Pressure Level Knowing the power output of a source and the total absorption in a reverberant room, one can predict the average sound-pressure level in the room, under the assumption of a diffuse field, using the following expression [23]: 𝐿𝑝 = 𝐿𝑤 + 10 log10 (𝑞4𝜋𝑟2+4(1 − 𝛼)𝐴) dB (3.14) where 𝑞 is the source directivity factor with 𝐴sabine = 𝛼𝑆 and 𝐴Eyring = −𝑆 𝑙𝑛(1 − 𝛼) (3.15) disregarding the air-absorption factor. The prediction by Eq. (3.14) can be checked against the predicted average sound-pressure level. 3.2.5 Decay-Rate/ Reverberation-Time Uniformity Reverberation time, which is a measure of the ‘liveness’ of a room, is considered as the most important quantity for any reverberation-room prediction. This is because many other quantities – i.e. absorption coefficients, sound-power levels of sources, etc. – are calculated from the reverberation time. Since diffuse-field reverberation formulae depend only on the dimensions and the total absorption in the room, reverberation time should be the same at all positions in the room. As the Waterhouse Effect is always present, there will be a spatial variation in time-decay rates. Same as for the pressure-field uniformity, the standard deviation of the decay rate/reverberation time is considered as the descriptor to characterize the degree of field diffuseness. The following expression is used to calculate the standards deviations: 57 𝜎𝑇60 =√∑ (𝑇60,𝑛 − ?̅?60)2𝑁𝑟𝑒𝑐𝑛=1𝑁𝑟𝑒𝑐 − 1 (3.16) Here, 𝑁𝑟𝑒𝑐 is the number of receiver positions, 𝑇60,𝑛 is the RT at the nth receiver location, and ?̅?60 is the reverberation time (RT) averaged over all receiver locations. 3.2.6 Prediction of Decay Rate/Reverberation Time Knowing the total absorption in a reverberation room, one can predict the reverberation time in the room using Eq. (2.3) or (2.8), under the assumption of a diffuse field. This prediction can be checked against the predicted reverberation time averaged over a number of receiver locations using the following expression [55]: ?̅?60 =𝑁𝑟𝑒𝑐∑1𝑇60,𝑛𝑁𝑟𝑒𝑐𝑛=1 (3.19) This averaged value of reverberation time is used in Eq. (3.21) and Eq. (3.22) to calculate the power level of a source and the absorption coefficient of a test absorber, respectively. 3.2.7 Linearity of Temporal-Decay Curves Degree of linearity of temporal-decay curves can be a good indicator of field diffuseness. According to ASTM C423, both in the steady state and during the temporal decay, the decay curves of sound-pressure level versus time should be linear to conform to the assumption of the diffuse-field formulae. As discussed in a previous section, at low frequency decay curves show a monotonic curvature due to the dominant presence of highly-energetic axial and tangential modes. As frequency increases, decay curves become linear due to the dominant presence of less-energetic oblique modes. To measure the degree of nonlinearity of the decay curves – in other words, the curvature of the decay curves – quantitatively, correlation coefficient, 𝑟, (between the predicted decay curve and the best-fitted straight line) are used [44]. However, an effective measure of the curvature is obtained by means of magnifying the deviation from perfect correlation, e.g., according to the following expression: 58 Curvature, 𝜅 = 1000(1 − abs(𝑟)) (3.20) 3.2.8 Prediction of Source-Power Level Determination of the sound-power level of a source using the reverberation-room method is a steady-state approach, and involves exciting the room with an acoustic source, and measuring the mean spatially-averaged reverberant SPL; if the reverberation times are known, then the source sound-power level can be calculated as [54]: 𝐿𝑤 = 𝐿𝑝 − 10 log10(𝑇60) + 10 log10(𝑉) + 10 log10 (1 +𝑆𝜆8𝑉) − 13.9 dB (3.21) where 𝜆 is the wavelength of sound at the band center frequency. The constant 13.9 dB has been calculated for a pressure of one atmosphere and a temperature of 20 0C. For the approach of Eq. (3.21) to be valid, there should be at least 20 acoustic modes in the frequency band of interest [54], as required for the assumption of a diffuse field mentioned before. It is observed that the sound power emitted by the source at low frequencies varies substantially with the room and position of the sound source unless the room is very large and heavily damped [31]. This indicates that it is extremely important to average over many source positions if the sound power of a source with a significant content of pure tones is to be determined in a reverberation room. 3.2.9 Prediction of Absorption Coefficient As prescribed by ASTM C423, the absorption coefficient of a specimen can be measured using the reverberation-room method and Sabine’s formula as: 𝛼𝑟𝑒𝑣 =0.161𝑉𝑆𝑠𝑎𝑚(1𝑇60,2−1𝑇60,1) (3.22) Here, 𝑆𝑠𝑎𝑚is the specimen area and 𝑇60,1 and 𝑇60,2 are the reverberation times without and with the specimen, respectively. Degree of field diffuseness can be measured by comparing the 59 reverberation-room absorption coefficient, 𝛼𝑟𝑒𝑣, to either the statistical absorption coefficient calculated using an impedance-tube method or the theoretical random-incidence absorption coefficient calculated using the following equation: 𝛼𝑟𝑎𝑛 = ∫4𝑅𝑒(𝜁𝑛) cos(𝜃)(|𝜁𝑛| cos(𝜃))2 + 2𝑅𝑒(𝜁𝑛) cos(𝜃) + 1sin(2𝜃) 𝑑𝜃𝜋/20 (3.23) Here, 𝜁𝑛 is the normal specific impedance. It is assumed that incident sound waves having random phase relations are uniformly distributed over all possible directions. The random-incidence absorption coefficient should have a value less than unity. 3.3 Methodology: Modeling and Characterizing a Diffuse Sound Field In this section, a complete demonstration of the steps used in this study to model and characterizing a diffuse sound field in a reverberation room is presented using the concepts/definitions/terminologies discussed in the previous sections/chapters. The demonstration is given here for Case #1: the ISO 354 reverberation room with the longest x-dimension (see section 4.2.1.1 of Chapter 4 for details). The complete implementation for the given case study can be separated into the following two steps:  Finite-element modelling  Post-processing. 3.3.1 Finite-Element Modeling A numerical, FE-based, modal approach has been adopted in this study. It basically solves either the classical Helmholtz equation (frequency domain) or the wave equation (time domain) with impedance boundary conditions. A number of steps are involved in the modeling section, and these are described as follows: 60 First step: The very first step of modeling is the geometry creation. The geometry of the reverberation room modeled here for demonstration is shown in Figure 4.1. While creating the geometry, the minimum number of diffusers required to ensure optimum field diffuseness is calculated following the instructions given in section 3.1.4. The procedure is explained below. Calculation of number of diffusers: The minimum number of diffusers required to obtain optimum field diffuseness is calculated using Eq. (3.1), considering a value of h = 0.5 given in Eq. (3.2). For an acoustic diffuser having an area of 3 m2 (one side), the calculated minimum number of diffusers required to obtain optimal diffuseness at low frequency is 8. Figure 3.3: Domain discretization/meshing of the geometry. Second step: The second step of modeling is mesh creation/domain discretization. Solving acoustical problems numerically, while maintaining a desired accuracy level, is very challenging, as the finite-element model must discretize space into some number of elements per wavelength; large volumes involve a large number of elements and quickly become computationally expensive at high frequencies. In this study, we considered 3D, quadratic, tetrahedral elements to discretize our domains; 5 elements per wavelength were considered, satisfying the rule of thumb of 5 to 10 elements per wavelength to ensure sufficient accuracy [39]. The meshing of the geometry considering 5 elements per wavelength is shown in Figure 3.3. The mesh-convergence procedure is explained below. 61 Finite-element mesh convergence: As we are using the finite-element (FE) method, it has to be kept in mind that the accuracy of the solution is linked to the mesh size. Numerical solution always yields an approximate solution, due to the discontinuous representation of the continuous system which results from discretizing the domain into a number of finite subdomains. As mesh resolution increases – in other words, as mesh size decreases – the solution moves toward the exact solution for the equations being solved. However, since we are limited by finite computational resources and time, we have to rely on an approximation to the real solution. The goal of simulation is to minimize the difference (error) between the exact and the approximate solutions, and to ensure that the error is below some acceptable tolerance. Ensuring that the solution obtained has converged requires tracking the variation of a characteristic output parameter with mesh size, and determining at which mesh size the parameter has “converged” to the correct value. When the variation with mesh size almost reaches a steady state, the solution corresponding to that particular mesh size at that stage is considered as the converged solution. Acoustical problems require twelve degrees of freedom (DOF) per wavelength to accurately represent the acoustic wave. This requirement develops from the error associated with approximating a sine wave with a polynomial. Twelve DOFs provide a solution with less than 1% deviation from the approximated sine wave. Based on this required minimum number of DOF to represent a wave accurately, the required minimum number per wavelength of quadratic elements, the default element type used by COMSOL, is 5. Bibby [39] investigated the mesh convergence phenomenon by considering different numbers of elements per wavelength, ranging from 0.5 to 15, and concluded that 5 elements per wavelength is sufficient for the solution to converge with sufficient accuracy. Third step: The third step is physics setting. Here, the study was performed in frequency domain, which means we solve the Helmholtz equation in conjunction with the impedance boundary conditions. Because of the constraint on computational capacity, as well as our interest in the low-frequency range, we restricted our study to nine third-octave bands centered at 50, 63, 80, 100, 125, 160, 200, 250 and 315 Hz; in other words, to three octave bands centered at 63, 125 and 250 Hz. We also considered frequency- as well as angle-of-incidence-invariant impedance boundary conditions (discussed in a previous section). This is a valid assumption for surfaces with heavy, rigid construction [13] as is the case for reverberation-room surfaces. Here, we 62 assumed a specific surface impedance of 389, equivalent to an absorption coefficient of 0.02 calculated using Eq. (3.23), typical of concrete walls [54]. Final step: The final step is solving the problem. To perform the numerical simulation, a FE-based commercial software (COMSOL 4.3b) is used. A computer with an Intel i7 processor and 24 GB of memory was purpose-built for these simulations. Run times were typically under 24 to 48 hours to predict steady-state sound pressures for a frequency spectrum ranging from 44 to 355 Hz with a frequency resolution of 0.5 Hz, which is considered in this study, and 6 to 8 hours to predict the first eight-hundred room eigenfrequencies. However, run times are highly dependent on geometry, mesh and frequency resolution. At each frequency of interest, COMSOL solves the inhomogeneous Helmholtz equation, providing the acoustic-pressure solution for the entire acoustic domain. The broadband FR and eigenfrequency calculation procedure are explained below. Construction of broadband frequency response (FR): Eventually, a frequency sweep starting from the lowest (44 Hz) to the highest frequency of interest (355 Hz) with the above mentioned resolution of 0.5 Hz will generate the desired broadband frequency spectrum, known as the room frequency response (FR)/room transfer function. Computation of room eigenfrequencies: In the case of eigenfrequency calculation, COMSOL solves the homogeneous Helmholtz equation, and provides the desired number of eigenfrequencies as specified by the user. From the room eigenfrequencies and the pressure-field solution, and the fact that the solution is harmonic, all of the quantities of interest are calculated. 3.3.2 Post-Processing In this study, we used the steady-state, frequency-domain interface of the COMSOL acoustic module to simulate both the room eigenfrequencies and the pressure field at different receiver locations. Thus, our primary output from COMSOL modeling is the eigenfrequencies in Hz and 63 the sound pressures in Pascals at the receiver positions as a function of frequency, known as the frequency response (FR)/transfer function. However, our goal is to predict the modal compositions – i.e. number of modes, modal density, modal overlap, etc. – and reverberation times (RT) of the room. In order to do so, we need some further calculation over these frequency responses and eigenfrequencies. The complete post-processing task can be divided into two steps:  Post-processing with eigenfrequencies  Post-processing with frequency responses. 3.3.2.1 Post-Processing with Eigenfrequencies Eigenfrequencies can be predicted by both the theoretical and numerical modal approaches. However, the theoretical modal approach can only be used for an empty, rectangular room to resolve the eigenfrequencies, as well as the acoustic-pressure field. Therefore, this approach is not applicable in our current case, as the room is equipped with a number of diffusing panels and a mounting bed for the test absorber. For demonstration, for the given reverberation room but without diffusers, test absorber and mounting bed, the eigenfrequencies, along with the corresponding damping constants, calculated in the 100-Hz third-octave band, using both the theoretical and numerical modal approaches, are presented in Table 3.1. The theoretical eigenfrequencies, as well as the damping constants, are calculated by solving Eq. (2.30) and then using Eq. (2.33). To solve Eq. (2.30), a method called IN/GB is used; the MATLAB script is presented in Appendix A. A number of MATLAB scripts were also written to implement the theoretical modal approach; these are presented in Appendix B and Appendix C. On the other hand, numerical eigenfrequencies are obtained by the COMSOL eigenvalue solver which solves Eq. (2.58). It is evident from Table 3.1 that the theoretical results show a trend of increasing deviation from the numerical results as we move from the band center frequency towards the band edges. This is due to the driving-frequency-dependent characteristics of the theoretical approach, as mentioned before. 64 For our current case, we resolved the eigenfrequencies using the numerical approach. Then different modal compositions – i.e. number of modes, modal density, modal spacing, modal width, modal overlap, modal-overlap factor, etc., as well as the modal damping constants and reverberation times – are calculated in third-octave and octave bands for two conditions – ‘without’ and ‘with’ the test absorber. Table 3.1: Predicted modal frequencies and damping constants in the 100-Hz third-octave band. ′lmn′⏞ Mode index Eigenfrequenciy Damping constant ⏞ Theoretical Eigenfrequency Damping constant ⏞ Numerical (Hz) (1/s) (Hz) (1/s) '122' 91.14 1.11 91.14 1.01 '231' 91.94 1.10 91.94 1.01 '321' 92.62 1.09 92.62 1.01 '212' 96.02 0.88 96.02 0.84 '040' 97.08 0.69 97.08 0.67 '330' 100.34 0.80 100.34 0.80 '140' 100.38 0.80 100.38 0.80 '222' 101.32 1.00 101.32 1.01 '410' 102.24 0.62 102.24 0.63 '032' 103.90 0.85 103.90 0.88 '041' 105.24 0.84 105.24 0.88 '132' 107.00 0.95 107.00 1.01 '420' 107.23 0.75 107.23 0.80 '331' 108.26 0.94 108.26 1.01 '141' 108.30 0.94 108.30 1.01 '240' 109.71 0.73 109.71 0.80 '411' 110.02 0.77 110.02 0.84 '312' 111.74 0.76 111.74 0.84 Calculation of modal compositions: Modal compositions are calculated following the instructions discussed in sections 2.5.4 – 2.5.8. Modal compositions, like the number of modes are calculated by direct counting of modes in a given frequency band; dividing this number with the width of the band gives the modal density. Then the modal spacing, modal width, modal overlap and modal overlap factors are calculated as a function of third-octave and octave bands using Eq. (2.76), Eq. (2.68), Eq. (2.79) and Eq. (2.78), respectively. 65 Construction of modal-decay curves: Modal-decay curves are constructed following the instructions discussed in section 2.5.2. Calculation of modal reverberation-times: Modal reverberation times are calculated following the instructions discussed in section 2.5.1. At first, modal decay curves are constructed using Eq. (2.65) and, then, modal reverberation times are calculated corresponding to 60-dB decay. Figure 3.4: Predicted mode propagation angles as function of frequency. Mode-propagation angles: One noticeable aspect of the theoretical modal approach is that it predicts the room modes as vectors – i.e. Eq. (2.30a), (2.30b) and (2.30c) predict vector components in three coordinate directions, respectively; therefore, it provides the scope for calculating the mode propagation directions with the help of Eq. (2.81). This above-depicted (Figure 3.4) mode-propagation angles as a function of frequency was possible due to the modes vector form, which is only possible with the theoretical modal approach. However, the same is not possible with the numerical approach, which always gives the modes in scalar form. Though the theoretical approach is valid only for an empty, rectangular room, it can be used to get insights into the angular distribution of the modes in a non-empty, rectangular room. Figure 3.4 depicts the mode propagation angles as a function of frequency with the trend that, as the frequency increases, the modal-angular distribution become homogeneous. 1011021030102030405060708090Frequency (Hz)Angle ( 0 ) axial modes tangential modes oblique modes66 3.3.2.2 Post-Processing with Frequency Responses (FR) The procedure for calculating the broadband IR from the broadband FR is as follows: Construction of broadband impulse response (IR): The IR is a representation of the quantity of interest in the time domain (in our case acoustic pressure as a function of time). In order to generate the broadband room IR corresponding to its FR, we used the Discrete Inverse Fourier Transformation (DIFT) technique. In MATLAB, this was performed by using a default built-in function called ‘ifft’. While performing the DIFT using ‘ifft’, a Nyquist frequency of 355 Hz (maximum frequency of the broadband FR resulting from COMSOL) – in other words, a sampling frequency of 710 Hz (twice the Nyquist frequency) – is considered with a frequency resolution of 0.5 Hz (same as the FR resolution); zero padding is done for the part of the frequency spectrum ranging from 0 to 35.5 Hz, as we are not considering this frequency range in our study. The IR depicts the direct sound wave in the form of the impulse which reaches the receiver position first, followed by other reflected sound waves with gradually decaying magnitude, which (the reverberant sound) finally decays to zero. Calculation in frequency bands: Until now, the room FR and the corresponding IR, constructed using DIFT, have broadband frequency spectra ranging from zero up to the Nyquist frequency (355 Hz). However, we are interested to express different room-acoustical parameters in desired third-octave and octave bands, as mentioned in a previous section. This means that we have to filter both the broadband room FR and IR to get the FRs and IRs, respectively, in third-octave and octave bands. FRs as a function of third-octave and octave bands can be done very easily – simply, by sorting the frequency contents from the broadband FR into different frequency bands of interest. However, to construct IRs in third-octave and octave bands is not as easy as for the FRs. In order to do that, the broadband IR is passed through a band-pass filter. Here, a third-order Butterworth filter is used as the filter. The MATLAB scripts written to implement all these steps are presented in Appendices D and E. From the band FRs and IRs, the following room-acoustical parameters are calculated as a function of frequency band. Calculation of sound-pressure level (SPL): The SPLs are calculated from the FRs using Eq. (2.16). 67 Accuracy calculation of sound-pressure level (SPL): The accuracy of the predicted SPLs is checked against the actual (diffuse-field) values obtained from Eq. (3.14). The prediction accuracy is predicted using the following equation: Prediction error, 𝑢𝐿𝑝 =abs(Actual SPL − Predicted SPL)Actual SPL× 100 % (3.24) Calculation of SPL standard deviation: The standard deviations of the predicted SPLs are calculated using Eq. (3.13). Calculation of reverberation time (RT): One of the objectives of reverberation-room analysis is to predict the RT of the room – an important characteristic property of the room; calculation of many other quantities like, absorption, power level of a source, etc. depends on this quantity. Prediction of band RT from band FR involves a number of steps:  Constructing band Impulse Response (IR) from band Frequency Response (FR)  Constructing band Energy-Decay Curve (EDC) from band Impulse Response (IR)  Predicting band Reverberation Time (RT) from band Energy Decay Curve (EDC). Construction of band impulse response: The band IRs are constructed from the broadband IR using a band-pass filter, as mentioned in the previous section. Construction of band energy-decay curves: The EDC is a plot of the logarithm of the calculated band IR. It is also known as an integrated impulse decay curve, and is done using the Schroeder backward-integral formula given in Eq. (2.67). Prediction of band reverberation times: Theoretically, RT is defined as the time required for the EDC to drop by 60 dB from its steady state level when the source is cut off. The reverberation time was determined by means of a least square fit evaluation of the slope between -5 and -25 dB of the decay curves. 68 Accuracy calculation of reverberation time (RT): The accuracy of the predicted RTs is checked against the actual (diffuse-field) values obtained from Eq. (2.3) or (2.8). The prediction accuracy is predicted using the following equation: Prediction error, 𝑢𝑇60 =abs(Actual RT − Predicted RT)Actual RT× 100 % (3.25) Calculation of RT standard deviation: The standard deviations of the predicted RTs are calculated using Eq. (3.16). Calculation of curvatures for the temporal-decay curves: The curvatures of the constructed temporal-decay curves are calculated using Eq. (3.20). Calculation of sound-power level of a source: The sound-power levels of a source are predicted using Eq. (3.21), as recommended by ISO 3741 [6]. Accuracy calculation of sound-power level of a source: The accuracy of the predicted sound-power levels of a source are predicted using the following expression: Prediction error, 𝑢𝐿𝑤 = √(𝜕𝐿𝑤𝜕𝐿𝑝𝑢𝐿𝑝)2+ (𝜕𝐿𝑤𝜕𝑇60𝑢𝑇60)2 (3.26) Calculation of absorption coefficients of a test absorber: Calculation of the room-averaged absorption is done using Eq. (2.3). The average absorption coefficient of the room is calculated using Eq. (3.21). To calculate the absorption coefficient of a sample absorber, Eq. (3.16) is used. Accuracy calculation of absorption coefficients of a test absorber: The accuracy of the predicted room-averaged absorption coefficients, calculated using Eq. (2.3) and (2.8), are compared to the theoretical random absorption coefficient of 0.02. The accuracy of the predicted absorption coefficients of the test absorber are compared to the theoretical value of 0.76. 69 3.4 Prediction Demonstration for the Given Reverberation Room In this section, the predictions performed for the given reverberation room in both third-octave and octave bands, and ‘without’ and ‘with’ the test absorber, are presented with the help of the descriptors discussed above to characterize a diffuse sound field in this reverberation room. In the next chapter, we will follow exactly the same procedure while doing predictions in reverberation rooms of different configurations. 3.4.1 Third-Octave-Band Prediction without Test Absorber The results are discussed in terms of the descriptors described in Chapter 3 (see section 3.2) using the calculation procedure as described also in Chapter 3 (see section 3.3). Cut-off frequency: The predicted modal compositions shown in Table 3.2 reveal that a value for the minimum number of modes of around 20 (exact number is 18), which is the basis for the statistically-based cut-off-frequency definition, occurs from the 100-Hz third-octave band. However, the corresponding low-frequency limit is 103 Hz, which makes the 100-Hz third-octave disqualified as the first low-frequency prediction band. That means, according to the statistically-based cut-off frequency definition, this reverberation room can be used for prediction from the 125-Hz third-octave band. The modal density, modal spacing and modal-width values corresponding to this 125-Hz third-octave are 1.21, 0.84 and 0.35, respectively. On the other hand, the modal overlap and modal overlap factor values corresponding to the 125-Hz third-octave are 0.43 and 0.42, respectively; they are far below the limit 3, as required by the Schroder pure-tone-based cut-off-frequency definition. Taking a look at Table 3.2, it is clear that this expected value of 3 for the modal overlap does not happen in any of the third-octave bands considered here. The corresponding Schroeder low-frequency limit, considering a band-averaged value of 5.7 s for the RT, as shown in Table 3.4, is 392 Hz. 70 Table 3.2: Predicted third-octave-band modal compositions without test absorber. 50 63 80 100 125 160 200 250 315⏞ 𝑇ℎ𝑖𝑟𝑑−𝑜𝑐𝑡𝑎𝑣𝑒−𝑏𝑎𝑛𝑑 𝑐𝑒𝑛𝑡𝑒𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 (𝐻𝑧) Number of modes, 𝑁(𝑓) 4 5 11 18 35 61 116 219 419 Modal density, 𝑛(𝑓) 0.33 0.36 0.58 0.78 1.21 1.65 2.52 3.78 5.74 Modal spacing, 𝛿𝑓 3.09 3.35 1.63 1.25 0.84 0.61 0.40 0.27 0.17 Modal width, 𝛿𝑤 0.32 0.30 0.32 0.33 0.35 0.37 0.39 0.40 0.41 Modal overlap, 𝑀 0.11 0.11 0.18 0.26 0.43 0.60 0.99 1.51 2.33 Modal overlap factor, 𝑀𝑓 0.10 0.09 0.19 0.27 0.42 0.60 0.99 1.51 2.34 Spatial uniformity of SPL: From Table 3.3, the predicted standard deviations of SPL, a measure of spatial variation, reveal that in lower frequency bands – especially in the 50-, 63- and 80-Hz third-octaves – this variation is greater than 2 dB. On the other hand, a deviation of less than 2 dB occurs from the 100-Hz third-octave band with the maximum in the 125-Hz third-octave-band. According to the ISO standard, a maximum of 1.5 dB deviation is allowable, which is achieved in this case from the 160-Hz third-octave band. Table 3.3: Predicted third-octave-band SPL statistics without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 143.7 146.2 156.2 149.8 153.3 156.8 161.1 167.5 171.0 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 139.8 145.6 155.0 147.5 153.9 157.3 161.3 166.5 169.3 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 150.3 153.0 149.2 150.6 157.2 154.7 160.9 166.6 169.7 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 148.5 147.6 154.3 152.1 151.8 158.3 162.6 166.3 170.5 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 135.2 146.7 155.0 150.5 154.5 156.1 164.0 167.5 170.2 Predicted average SPL, ?̅?𝑝 (dB) 146.3 148.8 154.5 150.4 154.5 156.8 162.2 166.9 170.2 Standard deviation, 𝜎𝐿𝑝 (dB) 7.0 3.2 2.8 1.7 2.0 1.3 1.3 0.6 0.7 Expected Sabine SPL, 𝐿𝑝 (dB) 149.2 151.3 156.5 151.8 155.9 158.2 163.7 168.3 171.2 Sabine error (%) 1.9 1.7 1.3 1.0 0.9 0.9 0.9 0.8 0.6 Expected Eyring SPL, 𝐿𝑝 (dB) 149.2 151.3 156.5 151.8 155.9 158.2 163.6 168.3 171.1 Eyring error (%) 1.90 1.64 1.28 0.93 0.88 0.85 0.91 0.82 0.56 71 Table 3.4: Predicted third-octave-band RT statistics without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.22 7.41 7.13 6.80 6.45 6.22 5.82 5.66 5.54 Predicted RT @ Rec#1: 𝑇60 (s) 6.39 6.92 6.16 6.28 6.86 5.78 4.67 5.41 5.56 Predicted RT @ Rec#2: 𝑇60 (s) 6.09 6.32 5.96 6.65 6.40 4.89 5.23 4.94 5.44 Predicted RT @ Rec#3: 𝑇60 (s) 5.80 5.99 6.41 6.28 5.32 5.82 4.88 4.75 4.95 Predicted RT @ Rec#4: 𝑇60 (s) 5.99 6.06 6.35 6.16 6.89 5.39 5.32 4.97 5.29 Predicted RT @ Rec#5: 𝑇60 (s) 10.4 6.3 6.4 7.2 5.1 6.2 5.3 4.7 4.8 Predicted average RT: ?̅?60 (s) 6.61 6.31 6.25 6.50 6.01 5.58 5.07 4.93 5.20 Standard deviation, 𝜎𝑇60 (s) 1.97 0.37 0.19 0.44 0.86 0.49 0.29 0.29 0.32 Expected Sabine RT: 𝑇60 (s) 4.97 4.97 4.97 4.97 4.97 4.97 4.97 4.97 4.97 Sabine error (%) 33.0 26.9 25.8 30.7 20.9 12.2 2.0 0.8 4.5 Expected Eyring RT, 𝑇60 (s) 4.92 4.92 4.92 4.92 4.92 4.92 4.92 4.92 4.92 Eyring error (%) 34.4 28.2 27.1 32.0 22.1 13.3 3.0 0.2 5.6 Decay rate (dB/s) 9.07 9.51 9.60 9.23 9.98 10.8 11.8 12.2 11.5 Accuracy of the predicted SPLs: The accuracy of the predicted steady-state SPLs is checked against the Sabine and Eyring versions of the diffuse-field formulae as presented in Table 3.3. The errors associated with the predicted SPL is always less than 1.9% for both the Sabine and Eyring prediction cases, with the maximum error associated with the 50-Hz third-octave band. It can be seen from Table 3.3 that the errors associated with the Eyring version are less than with the Sabine version, the opposite scenario compared to the RT prediction case. Table 3.5: Predicted third-octave-band decay-curve curvature statistics without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 5.1 7.4 19.9 40.6 15.7 6.0 21.6 16.8 6.8 𝜅 @ Rec#2 6.2 6.5 11.0 12.1 13.6 13.7 17.8 4.4 6.5 𝜅 @ Rec#3 18.2 16.7 10.0 6.0 8.7 7.2 16.3 8.3 14.3 𝜅 @ Rec#4 15.9 12.3 13.9 14.2 18.2 19.5 7.0 12.1 3.0 𝜅 @ Rec#5 17.0 9.3 11.7 34.7 21.5 7.6 9.6 12.2 4.5 Average, 𝜅 12.5 10.5 13.3 21.6 15.5 10.8 14.5 10.8 7.0 Modal, 𝜅 0.06 0.01 0.02 0.03 0.03 0.04 0.04 0.03 0.02 72 Table 3.6: Predicted third-octave-band source-power level statistics without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 149.7 151.9 157.0 152.3 156.5 158.7 164.2 168.9 171.7 Sabine error (%) 21.8 18.6 18.0 20.6 15.1 9.5 2.0 1.1 3.8 Eyring error (%) 22.7 19.5 18.8 21.4 16.0 10.4 2.8 0.8 4.7 Predicted, ?̅?𝑤 (dB) (1 source) 146.7 148.9 154.0 149.3 153.5 155.7 161.2 165.9 168.7 Spatial uniformity of RT: From Table 3.4, it is clear that the standard deviations of RT, a measure of spatial variation, are always less than 0.5 s, except for the 50-Hz and 125-Hz third-octave bands. A high standard deviation in 50-Hz third-octave may be explainable by less frequential-averaging; however, a deviation of 0.88 s in the 125-Hz third-octave is not easily explained. Table 3.7: Predicted third-octave-band room absorption statistics without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 3.62 3.79 3.82 3.68 3.98 4.29 4.72 4.85 4.60 Sabine expected absorption, 𝐴 (m2) 4.81 4.81 4.81 4.81 4.81 4.81 4.81 4.81 4.81 Sabine error (%) 24.8 21.2 20.5 23.5 17.3 10.8 1.9 0.8 4.3 Eyring expected absorption, 𝐴 (m2) 4.86 4.86 4.86 4.86 4.86 4.86 4.86 4.86 4.86 Eyring error (%) 25.5 22.0 21.2 24.2 18.1 11.7 2.9 0.2 5.3 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 24.8 21.2 20.5 23.5 17.3 10.8 1.9 0.8 4.3 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 25.3 21.8 21.1 24.0 17.9 11.6 2.9 0.2 5.2 Accuracy of the predicted RTs: The accuracy of the predicted RTs is checked against the expected values of the RTs calculated using the Sabine and Eyring versions of diffuse-field formulae as presented in Table 3.4. The predicted RTs in different third-octave bands range from 6.61 to 4.93 s, with the maximum in the 50-Hz third-octave band and the minimum in the 250-Hz third-octave band; while the expected Sabine and Eyring RTs are 4.97 s and 4.92 s, respectively. In both the cases, the error is high in the lower frequency bands. This might be due 73 to the lower absorption than expected in that frequency range, which results in a higher predicted RT and, hence, a higher deviation from the expected RT. As the frequency increases, absorption also increases, which results in a lower RT and, therefore, a smaller deviation from the expected value. The error is less than 5% from the 200-Hz third-octave in both cases. It is also obvious from Table 3.4 that the errors associated with the Eyring version are higher than with the Sabine version. Linearity of temporal decay curves: From Table 3.5, the predicted curvatures of the energy-decay curves reveal that the decay of sounds occurs in a non-linear fashion; though a linear one is expected. The greater is the deviation of curvatures from unity, the greater is the significance of the non-linear decay effect. For all third-octave bands, these curvatures are less than 22, with a maximum of 21.6 occurring in the 100-Hz third-octave band. For the theoretical modal-decay curves, the curvatures are fairly close to unity, an indication of linear decay process. The modal-decay curves and the energy-decay curves predicted in the third-octave-bands are presented in Figure 3.5 and Figures 3.6 – 3.10, respectively. Accuracy of the predicted source-power levels: The predicted source-power levels and the corresponding errors in different third-octave bands are presented in Table 3.6. The source-power level is a derivative of the steady-state SPL and the RT. Therefore, the accuracy of the predicted source-power level completely depends on the accuracy of the predicted RT and SPL. The errors associated with the predicted source power levels are less than 3% from the 200-Hz third-octave band. The error is greater than 10% in the rest of the bands. In this case, the Eyring error is greater than the Sabine error. Accuracy of the predicted room absorptions: Table 3.7 shows that the predicted average room absorption is less than the expected values, and the difference gradually increases in the lower third-octave bands. It is seen from Table 3.7 that errors are less than 3% from the 200-Hz third- octave band, and greater than 10% in the rest of the bands. The same is true for the room absorption-coefficient prediction case, as the absorption-coefficient predictions depend on the accuracy of the absorption predictions. The room-absorption coefficient prediction statistics are also presented in Table 3.7. It is also notable that, the Eyring errors are greater than the Sabine errors. 74 Figure 3.5: Theoretical modal-decay curves and the corresponding RTs, in third-octave bands and without test absorber. 0 2 4 6-70-60-50-40-30-20-100Modal RT, T60 for the 50-Hz band Time (s)Level (dB)T60 = 7.22 s0 2 4 6 8-70-60-50-40-30-20-100Modal RT, T60 for the 63-Hz band Time (s)Level (dB)T60 = 7.41 s0 2 4 6-70-60-50-40-30-20-100Modal RT, T60 for the 80-Hz band Time (s)Level (dB)T60 = 7.13 s0 2 4 6-70-60-50-40-30-20-100Modal RT, T60 for the 100-Hz band Time (s)Level (dB)T60 = 6.8 s0 2 4 6-70-60-50-40-30-20-100Modal RT, T60 for the 125-Hz band Time (s)Level (dB)T60 = 6.45 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT, T60 for the 160-Hz band Time (s)Level (dB)T60 = 6.22 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT, T60 for the 200-Hz band Time (s)Level (dB)T60 = 5.82 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT, T60 for the 250-Hz band Time (s)Level (dB)T60 = 5.66 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT, T60 for the 315-Hz band Time (s)Level (dB)T60 = 5.54 s75 Figure 3.6: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #5. 0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 10.373 s0 1-20020Time (s)45 50 55-800-600-400-2000200Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.329 s0 1-20020Time (s)60 65 70-200002000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 6.4 s0 1-50050Time (s)70 80-50000500010000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 7.227 s0 1-40-2002040Time (s)90 100 110-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.082 s0 1-50050Time (s)120 130 140-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 6.176 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 5.322 s0 1-1000100Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.652 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 4.817 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)76 Figure 3.7: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #4. 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 5.989 s0 1-20020Time (s)45 50 55-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.061 s0 1-20020Time (s)60 65 70-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 6.353 s0 1-50050Time (s)70 800500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 6.162 s0 1-50050Time (s)90 100 110-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 6.889 s0 1-40-2002040Time (s)120 130 140-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 5.388 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 5.317 s0 1-2000200Time (s)180 200 220-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.965 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 5.288 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)77 Figure 3.8: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #3. 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 5.802 s0 1-20020Time (s)45 50 55-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.992 s0 1-40-2002040Time (s)60 65 70-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 6.41 s0 1-20020Time (s)70 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 6.276 s0 1-40-2002040Time (s)90 100 110-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.319 s0 1-50050Time (s)120 130 140-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 5.821 s0 1-50050Time (s)150 160 170-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 4.881 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.753 s0 1-2000200Time (s)240 260 280-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 4.954 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)78 Figure 3.9: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #2. 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 6.094 s0 1-10010Time (s)45 50 55-1500-1000-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.324 s0 1-20-10010Time (s)60 65 70-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 5.963 s0 1-50050Time (s)70 80-50000500010000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 6.654 s0 1-20020Time (s)90 100 110020004000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 6.4 s0 1-50050Time (s)120 130 140-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 4.893 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 5.23 s0 1-2000200Time (s)180 200 220-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.944 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 5.442 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)79 Figure 3.10: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and without test absorber, at receiver #1. 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 6.393 s0 1-10010Time (s)45 50 55020004000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 6.92 s0 1-40-2002040Time (s)60 65 70020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 6.158 s0 1-50050Time (s)70 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 6.278 s0 1-50050Time (s)90 100 110-4000-200002000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 6.864 s0 1-50050Time (s)120 130 140-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 5.78 s0 1-50050Time (s)150 160 170-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 4.672 s0 1-1000100200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.407 s0 1-400-2000200400Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 5.556 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)80 3.4.2 Third-Octave-Band Prediction with Test Absorber The results are discussed in terms of the descriptors described in Chapter 3 (see section 3.2) using the calculation procedure as described also in Chapter 3 (see section 3.3). Cut-off frequency: Analyzing the modal compositions as shown in Table 3.8 reveal that all the modal compositions, except the modal width and the modal overlap/modal overlap factor, are the same as in the ‘without sample absorber’ case. This is expected, as no change has been made to the room size and shape upon which the room modal composition depends. Therefore, there will be no change in the statistically-based low-frequency limit, as described in the previous ‘without sample absorber’ case. It is noticeable that there is a great increase in the modal-width values and, therefore, an increase in the modal-overlap/modal-overlap factors exceeding the minimum limit of 3 as prescribed by Schroeder. This increase in the modal-overlap factors may lead to the conclusion that the room sound field is more diffuse than it was before. Actually, this is not the case; this can be understood better if we take a look into the definition of the modal-overlap factor, which is the ratio of the modal width to the modal spacing. There should be a minimum modal spacing with a greater modal width in order to consider a field to be diffuse. In this current case, there is only a large increase in the modal widths, while the modal spacings are almost the same as in the ‘without sample absorber’ case (see Table 3.2). That means that the increase in the modal-overlap factor is solely due to the increased modal widths, which can be attributed to the increased absorption in the room caused by adding the test sample absorber. Table 3.8: Predicted third-octave-band modal compositions with test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 4 6 13 17 38 62 122 219 423 Modal density, 𝑛(𝑓) 0.33 0.43 0.68 0.74 1.31 1.68 2.65 3.78 5.79 Modal spacing, 𝛿𝑓 3.02 2.70 1.47 1.13 0.76 0.59 0.38 0.26 0.17 Modal width, 𝛿𝑤 0.91 6.50 6.50 3.30 4.11 3.26 3.97 2.93 3.21 Modal overlap, 𝑀 0.30 2.79 4.45 2.44 5.39 5.46 10.5 11.1 18.6 Modal overlap factor, 𝑀𝑓 0.30 2.41 4.43 2.91 5.41 5.48 10.6 11.1 18.6 81 Spatial uniformity of SPL: The standard deviations of the SPL for this ‘with absorber’ case, as a measure of the spatial variation are presented in Table 3.9. It is noticeable that the deviations are greater in the low-frequency bands, and are below the ISO-prescribed limit of 1.5 dB from the 160-Hz third-octave band. Overall, the deviations are greater than the values obtained in the ‘without absorber’ case. Table 3.9: Predicted third-octave-band SPL statistics with test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 140.0 142.7 152.1 148.1 152.9 153.7 157.1 163.9 166.5 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 135.2 139.4 149.3 144.2 149.9 152.2 155.6 162.0 164.1 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 145.7 145.9 146.1 147.0 148.5 150.5 158.5 161.3 165.2 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 143.2 143.0 149.6 149.7 148.6 152.6 161.2 162.4 164.4 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 131.9 142.6 150.4 146.2 152.6 153.0 158.6 162.0 164.7 Predicted average SPL, ?̅?𝑝 (dB) 141.6 143.2 149.9 147.4 150.9 152.5 158.6 162.4 165.1 Standard deviation, 𝜎𝐿𝑝 (dB) 6.26 2.39 2.23 2.08 2.15 1.20 2.13 0.98 0.93 Table 3.10: Predicted third-octave-band RT statistics with test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 3.12 4.11 3.42 4.69 3.16 2.73 2.81 2.49 2.22 Predicted RT @ Rec#1: 𝑇60 (s) 3.84 3.52 2.13 3.93 2.82 2.71 2.66 2.33 4.04 Predicted RT @ Rec#2: 𝑇60 (s) 3.28 2.95 2.62 3.38 2.79 2.53 2.81 1.98 1.98 Predicted RT @ Rec#3: 𝑇60 (s) 3.04 3.29 2.66 2.87 2.93 1.88 2.08 2.03 1.78 Predicted RT @ Rec#4: 𝑇60 (s) 3.61 3.93 3.09 3.90 3.33 2.67 2.56 2.24 3.07 Predicted RT @ Rec#5: 𝑇60 (s) 10.0 2.38 2.87 3.16 3.10 2.41 2.26 2.18 2.09 Predicted average RT: ?̅?60 (s) 3.93 3.12 2.63 3.40 2.98 2.40 2.44 2.14 2.36 Standard deviation, 𝜎𝑇60 (s) 3.11 0.60 0.36 0.47 0.22 0.34 0.30 0.15 0.98 Decay rate (dB/s) 15.3 19.2 22.8 17.7 20.1 25.0 24.6 28.0 25.4 Spatial uniformity of RT: The standard deviations of the RT for this ‘with absorber’ case, as a measure of the spatial variation are presented in Table 3.10. It is obvious that deviations are below 0.5 s in all third-octave bands, except the 50- and 315-Hz bands. Why a higher deviation 82 is happening in the 315-Hz band is not clear. It may be an anomaly like the 125-Hz band high RT deviation prediction made in the previous ‘without absorber’ case, or may not be. However, one thing is fairly clear – the deviations are greater than in the previous ‘without absorber’ case in almost all third-octave bands. It is also noticeable that the predicted average RTs are lower than in the ‘without absorber’ case, as is expected due to the increased absorption. Linearity of decay curves: The curvatures of the energy-decay curves as presented in Table 3.11 reveal that the curvatures are much greater than unity confirming the presence of the non-linear effect in the decay process. It is also noticeable that the curvatures are greater than those obtained in the ‘without absorber’ case. This means that the non-linear effect is much stronger in the ‘with absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Figure 3.11 and Figures 3.12 – 3.16, respectively. Table 3.11: Predicted third-octave-band decay-curve curvature statistics with test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 14.7 58.5 33.7 10.6 6.3 7.3 20.5 20.4 104 𝜅 @ Rec#2 29.3 7.7 45.8 33.1 10.5 12.8 11.2 9.0 7.0 𝜅 @ Rec#3 8.9 19.7 18.1 27.8 16.1 20.6 10.0 7.3 8.6 𝜅 @ Rec#4 3.9 17.2 25.0 11.4 16.5 42.0 88.3 12.7 37.2 𝜅 @ Rec#5 123 36.4 15.4 29.4 12.2 6.1 5.3 9.3 9.9 Average, 𝜅 36.0 27.9 27.6 22.5 12.3 17.8 27.1 11.7 33.3 Modal, 𝜅 1.24 0.28 1.45 2.40 1.36 1.94 8.39 4.06 3.92 Table 3.12: Predicted third-octave-band sample absorber absorption coefficient statistics of a test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.23 0.36 0.49 0.31 0.37 0.53 0.47 0.59 0.51 Sabine error (%) 70.0 52.8 35.9 59.1 50.7 30.6 38.2 23.0 32.5 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.20 0.30 0.39 0.27 0.31 0.41 0.37 0.44 0.40 Eyring error (%) 73.2 60.4 49.3 64.9 58.9 46.1 50.7 41.7 47.2 83 Figure 3.11: Theoretical modal-decay curves and the corresponding RTs, in third-octave bands with test absorber. Accuracy of the sample absorption coefficients: The accuracy of the absorption coefficients of an absorber sample predicted using the Sabine and Eyring versions of the diffuse-field formulae is checked against the theoretical random absorption-coefficient values predicted using Eq. (3.17); these are presented in Table 3.12. It is noticeable that predicted values deviate from the theoretical values to a great extent. The Sabine predictions show an error of around 30% from the 160-Hz third-octave band; in the rest of the low-frequency bands, the error is greater than 0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 3.12 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.11 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 3.42 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 4.69 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.16 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 2.73 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.81 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.49 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.22 s84 Figure 3.12: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #5. 30%. It is also evident that the Eyring prediction errors are higher than with the Sabine predictions. 0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 10.033 s0 1-20020Time (s)45 50 55-2000200Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 2.375 s0 1-20020Time (s)60 65 70-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 2.867 s0 1-50050Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 3.16 s0 1-40-2002040Time (s)90 100 110-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 3.096 s0 1-50050Time (s)120 130 140020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 2.407 s0 1-1000100Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.258 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.176 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 2.085 s0 1-2000200Time (s)300 350-50000500010000Frequency (Hz)Pressure (Pa)85 Figure 3.13: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #4. 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 3.614 s0 1-20020Time (s)45 50 55-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 3.932 s0 1-20020Time (s)60 65 70-1000010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 3.092 s0 1-50050Time (s)70 80-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 3.903 s0 1-50050Time (s)90 100 110-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.328 s0 1-20020Time (s)120 130 140-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 2.67 s0 1-50050Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.556 s0 1-2000200Time (s)180 200 220-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.24 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 3.066 s0 1-2000200Time (s)300 350-10000-500005000Frequency (Hz)Pressure (Pa)86 Figure 3.14: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #3. 0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 3.035 s0 1-20020Time (s)45 50 55-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 3.291 s0 1-20020Time (s)60 65 70-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 2.658 s0 1-40-20020Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 2.872 s0 1-20020Time (s)90 100 110010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.934 s0 1-2002040Time (s)120 130 140-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 1.881 s0 1-50050Time (s)150 160 170-10000100020003000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 2.082 s0 1-1000100Time (s)180 200 220-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.029 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 1.779 s0 1-400-2000200400Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)87 Figure 3.15: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #2. 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 3.276 s0 1-50510Time (s)45 50 55-600-400-2000200Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 2.954 s0 1-10010Time (s)60 65 70-1000-5000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 2.624 s0 1-50050Time (s)70 80020004000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 3.376 s0 1-20020Time (s)90 100 110-500050010001500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.787 s0 1-40-2002040Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.529 s0 1-50050Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.813 s0 1-1000100Time (s)180 200 220-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 1.978 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 1.983 s0 1-2000200Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)88 Figure 3.16: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands and with test absorber, at receiver #1. 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 3.836 s0 1-10010Time (s)45 50 5550010001500Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 3.523 s0 1-20020Time (s)60 65 70-500050010001500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 2.132 s0 1-50050Time (s)70 80-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 3.932 s0 1-40-2002040Time (s)90 100 110-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 2.815 s0 1-50050Time (s)120 130 140-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.709 s0 1-50050Time (s)150 160 170-6000-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.66 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.329 s0 1-400-2000200Time (s)240 260 280-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 4.037 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)89 3.4.3 Octave-Band Prediction without Test Absorber The results are discussed in terms of the descriptors described in Chapter 3 (see section 3.2) using the calculation procedure as described also in Chapter 3 (see section 3.3). Cut-off frequency: The modal compositions data shown in Table 3.13 implies that prediction can be done from the 63-Hz octave band, as it contains the expected number of modes of 20. However, the corresponding statistically-based cut-off frequency limit as 65 Hz, which means prediction cannot be done from the 63-Hz octave band but from the 125-Hz band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz band are 1.26, 0.78, 0.36 and 0.46, respectively. On the other hand, the pure-tone-based Schroeder limit corresponding to a modal overlap factor of 3 and a band-averaged RT of 5.8 s, as shown in Table 3.15, is 395 Hz. Spatial uniformity of SPL: The predicted SPL standard deviations are within 0.4 to 0.6 dB in all octave bands as shown in Table 3.14. Table 3.13: Predicted octave-band modal compositions without test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 20 112 756 Modal density, 𝑛(𝑓) 0.45 1.26 4.25 Modal spacing, 𝛿𝑓 2.12 0.78 0.24 Modal width, 𝛿𝑤 0.31 0.36 0.40 Modal overlap, 𝑀 0.14 0.45 1.71 Modal overlap factor, 𝑀𝑓 0.15 0.46 1.71 Accuracy of the predicted SPLs: The accuracy of the predicted steady-state SPLs is checked against the Sabine and Eyring versions of the diffuse-field as shown in Table 3.14. The predicted results show errors below 1% from the 125-Hz octave band, while the error is around 1.5% in the 63-Hz band for both the Sabine and Eyring cases. The predicted SPLs show greater deviation from the Sabine diffuse-field predictions than from the Eyring predictions. 90 Table 3.14: Predicted octave-band SPL statistics without test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 152.8 155.1 169.0 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 151.6 155.4 167.7 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 151.4 155.8 167.9 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 151.7 156.1 168.5 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 151.7 155.2 168.8 Predicted average SPL, ?̅?𝑝 (dB) 151.9 155.5 168.4 Standard deviation, 𝜎𝐿𝑝 (dB) 0.54 0.42 0.59 Expected Sabine SPL, 𝐿𝑝 (dB) 154.2 157.0 169.8 Sabine error (%) 1.50 0.97 0.78 Expected Eyring SPL, 𝐿𝑝 (dB) 154.1 157.0 169.7 Eyring error (%) 1.47 0.95 0.76 Table 3.15: Predicted octave-band RT without test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.23 6.44 5.63 Predicted RT @ Rec#1: 𝑇60 (s) 7.98 6.31 5.19 Predicted RT @ Rec#2: 𝑇60 (s) 6.22 5.73 5.17 Predicted RT @ Rec#3: 𝑇60 (s) 5.84 5.55 4.92 Predicted RT @ Rec#4: 𝑇60 (s) 7.03 5.92 5.12 Predicted RT @ Rec#5: 𝑇60 (s) 6.41 5.86 4.83 Predicted average RT: ?̅?60 (s) 6.62 5.86 5.04 Standard deviation, 𝜎𝑇60 (s) 0.84 0.28 0.16 Expected Sabine RT: 𝑇60 (s) 4.97 4.97 4.97 Sabine error (%) 33.1 17.9 1.41 Expected Eyring RT, 𝑇60 (s) 4.92 4.92 4.92 Eyring error (%) 34.5 19.1 2.43 Decay rate (dB/s) 9.07 10.2 11.9 91 Table 3.16: Predicted octave-band decay-curve curvature statistics without test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 6.85 5.46 3.25 𝜅 @ Rec#2 5.67 6.73 3.61 𝜅 @ Rec#3 16.4 3.63 7.76 𝜅 @ Rec#4 3.51 10.2 3.44 𝜅 @ Rec#5 7.69 3.29 3.68 Average, 𝜅 8.03 5.87 4.35 Modal, 𝜅 0.02 0.06 0.03 Spatial uniformity of RT: The predicted RT standard deviations are less than 0.3 s from the 125-Hz band; while 0.85 s in the 63-Hz octave band. These are shown in Table 3.15. Accuracy of the predicted RTs: The accuracy of the predicted RTs is checked against the Sabine and Eyring versions of the diffuse-field formulae, as shown in Table 3.15. The predicted results show errors below 20% from the 125-Hz band, while the errors are greater than 20% in the rest of the bands. The predicted results show greater deviation from the Eyring diffuse-field predictions than from the Sabine predictions. Linearity of decay curves: The curvatures are within 4 to 9 in all octave bands as shown in Table 3.16. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Figure 3.17 and Figures 3.18 – 3.22, respectively. Table 3.17: Predicted octave-band source-power level statistics without test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 154.7 157.6 170.3 Sabine error (%) 21.8 13.3 1.4 Eyring error (%) 22.7 14.2 2.2 Predicted, ?̅?𝑤 (dB) (1 source) 151.7 154.6 167.3 92 Table 3.18: Predicted octave-band room absorption statistics without test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 3.61 4.08 4.74 Sabine expected absorption, 𝐴 (m2) 4.81 4.81 4.81 Sabine error (%) 24.9 15.2 1.39 Eyring expected absorption, 𝐴 (m2) 4.86 4.86 4.86 Eyring error (%) 25.6 16.1 2.38 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 Sabine error (%) 24.9 15.2 1.39 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 25.4 15.9 2.35 Accuracy of the predicted source-power levels: The predicted source-power levels show errors below 15% from the 125-Hz band; the errors are greater than 15% in the rest of the bands as shown in Table 3.17. Accuracy of the predicted room absorptions: The predicted room absorptions, as well as the absorption coefficients, show errors below 16% from the 125-Hz band, while errors are greater than 16% in the rest of the bands. These are shown in Table 3.18. 93 Figure 3.17: Theoretical modal-decay curves and the corresponding RTs, in octave bands without test absorber. Figure 3.18: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #5. Figure 3.19: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #4. 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 7.23 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.44 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.63 s0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.405 s0 1-50050Time (s)60 80-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.863 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.829 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 7.028 s0 1-50050Time (s)60 800500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.918 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.121 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)94 Figure 3.20: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #3. Figure 3.21: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #2. Figure 3.22: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and without test absorber, at receiver #1. 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.841 s0 1-50050Time (s)60 80-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.547 s0 1-1000100Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.918 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.219 s0 1-50050Time (s)60 80-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.728 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.17 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 7.983 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 6.308 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.185 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)95 3.4.4 Octave-Band Prediction with Test Absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit. There is also no change in the modal compositions except the modal-width and, hence, the modal-overlap as shown in Table 3.19. The same reasoning is applicable here what is discussed in the ‘third-octave-band prediction section with absorber’ case. Table 3.19: Predicted octave-band modal compositions with test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 22 116 766 Modal density, 𝑛(𝑓) 0.50 1.30 4.30 Modal spacing, 𝛿𝑓 1.92 0.77 0.23 Modal width, 𝛿𝑤 5.67 3.58 3.24 Modal overlap, 𝑀 2.84 4.66 14.0 Modal overlap factor, 𝑀𝑓 2.95 4.64 14.0 Spatial uniformity of SPL: The predicted SPL deviations are less than 1.5 dB in all octave bands as shown in Table 3.20. However, overall deviations are higher than in the ‘without absorber’ case. Spatial uniformity of RT: The predicted RT deviations are less than 0.45 s from the 125-Hz band, while 1.47 s in the 63-Hz band, as shown in Table 3.21. However, overall deviations are greater than in the ‘without absorber’ case. Linearity of decay curves: The curvatures are less than 15 from the 125-Hz band, while 28.4 in the 63-Hz band, as shown in Table 3.22. However, overall curvatures are greater than in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Figure 3.23 and Figures 3.24 – 3.28, respectively. 96 Table 3.20: Predicted octave-band SPL statistics with test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 148.7 153.1 164.8 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 145.9 150.7 162.7 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 146.2 149.8 163.4 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 146.8 151.6 163.8 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 147.3 152.3 163.3 Predicted average SPL, ?̅?𝑝 (dB) 147.1 151.7 163.7 Standard deviation, 𝜎𝐿𝑝 (dB) 1.11 1.31 0.78 Accuracy of the sample absorption coefficients: The accuracy of the predicted absorption coefficients of an absorber sample is checked against the theoretical results as shown in Table 3.23. The Sabine predictions show errors of 26% in the 250-Hz band, while errors are greater than 50% in the rest of the bands. On the other hand, the Eyring predictions yield an error around 40% in the 250-Hz band, while errors are greater than 55% in the rest of the bands. Table 3.21: Predicted octave-band RT statistics with test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 3.79 4.03 2.50 Predicted RT @ Rec#1: 𝑇60 (s) 3.22 2.96 2.84 Predicted RT @ Rec#2: 𝑇60 (s) 3.09 2.69 2.02 Predicted RT @ Rec#3: 𝑇60 (s) 3.39 2.61 1.89 Predicted RT @ Rec#4: 𝑇60 (s) 6.33 3.54 2.54 Predicted RT @ Rec#5: 𝑇60 (s) 2.90 2.84 2.04 Predicted average RT: ?̅?60 (s) 3.49 2.90 2.21 Stnd. deviation, 𝜎𝑇60 (s) 1.47 0.37 0.41 Decay rate (dB/s) 17.2 20.7 27.1 97 Table 3.22: Predicted octave-band decay-curve curvature statistics with test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 68.6 3.8 31.4 𝜅 @ Rec#2 13.5 5.7 6.3 𝜅 @ Rec#3 12.1 3.1 3.5 𝜅 @ Rec#4 39.7 16.9 19.5 𝜅 @ Rec#5 8.4 7.3 1.8 Average, 𝜅 28.4 7.4 12.5 Modal, 𝜅 1.41 8.60 6.04 Table 3.23: Predicted octave-band sample absorber absorption coefficient statistics of a test absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.30 0.39 0.56 Sabine error (%) 60.6 49.1 26.0 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.26 0.32 0.43 Eyring error (%) 66.0 57.8 43.4 98 Figure 3.23: Theoretical modal-decay curves and the corresponding RTs, in octave bands with test absorber. Figure 3.24: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #5. Figure 3.25: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #4. 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 3.79 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 4.03 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.5 s0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 2.897 s0 1-40-2002040Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.843 s0 1-1000100Time (s)100 150-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.043 s0 1-5000500Time (s)200 300-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.334 s0 1-50050Time (s)60 80-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.538 s0 1-1000100Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.541 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)99 Figure 3.26: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #3. Figure 3.27: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #2. Figure 3.28: Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands and with test absorber, at receiver #1. 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 3.393 s0 1-40-2002040Time (s)60 80-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.607 s0 1-50050Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 1.891 s0 1-5000500Time (s)200 300-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 3.09 s0 1-50050Time (s)60 80020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.692 s0 1-50050Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.015 s0 1-400-2000200400Time (s)200 300-500005000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 3.224 s0 1-50050Time (s)60 80-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 2.963 s0 1-1000100Time (s)100 150-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.839 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)100 Chapter 4 Revisiting Existing Standards and Evaluating their Effectiveness In this chapter, at first, we will revisit existing standards regarding reverberation-room measurement using diffuse-field formulae. Then, we will evaluate their effectiveness at achieving a diffuse sound field in a reverberation room, and promoting the accurate measurement of acoustical quantities, using finite-element modeling as discussed in the previous chapter. We will also discuss the accuracy of RTs, steady-state SPLs, source sound-power levels, and the absorption coefficients of an absorbent specimen, determined by the reverberation-room standard methods, with respect to both the Sabine and Eyring versions of the diffuse-field formulae. 4.1 Revisiting Standards: The Reverberation-Room Method The key characteristics of a reverberation room, as prescribed by existing standards [3, 4, 5, 6] are: constructed of heavy materials – i.e. massive masonry, concrete, etc. – to obtain the least absorption at surfaces; room dimensions such that no two dimensions are equal, and that the largest to the smallest dimension ratio is greater than 2:1; and, a number of diffusers should be hung randomly in the room volume to ensure sufficient sound-field diffuseness. In addition to this, some standards go further and clearly specify the minimum room volume and set an absorption limit for creating a diffuse sound field. For example, ASTM E90 and ASTM C423 specify a minimum room volume of 82 m3 and 125 m3, respectively, while ISO 354 prescribes a minimum volume of 150 m3. Regarding absorption, ASTM E90 limits the absorption area A in a room of volume V according to the following equation: 𝐴 < 𝑉2 3⁄ m2 for 𝑓 < 2000 𝑉1 3⁄⁄ Hz and 𝐴 ≤𝑉233 m2 for 2000𝑉13< 𝑓 < 2000 Hz (4.1) ASTM C423 also discusses absorption limits, but in terms of the average surface-absorption coefficient 𝛼, as follows: 101 𝛼 = 0.05 for 250 < 𝑓 < 2500 Hz and 𝛼 = 0.1 for 𝑓 ≥ 2500 Hz (4.2) Regarding the room shape – i.e. whether it should be regular rectangular or irregularly shaped – transmission-loss measurement standards state that, in the case of a rectangular room, the room proportion, 41 3⁄ : 21 3⁄ : 1 – which is known as the Bolt ratio [49] – could be a choice that will ensure a good modal distribution in low-frequency bands. Regarding source and receiver numbers and their relative positions, ASTM C423 says that 1 source and 5 receivers are sufficient, with no two receivers, which should be 0.75 m away from the sample surface, closer than 1.5 m. ISO 354 says there should be at least 2 sources and 3 receivers, with no two receivers, which should be 1 m away from the sample and reflective surfaces and 2 m from the sound source, closer than 1.5 m. Sources should be positioned near the trihedral corners of the room. Both ASTM standards state that complying with recommendations does not guarantee sufficient field diffuseness, and recommend the use of diffusing panels. According to ASTM C423, diffusing panels should have a low absorption coefficient; areas of 3 m2 for one side are recommended. It has been shown that an optimal value for the number of panels to be used to obtain greater diffuseness can be found [51], and these values are used in this study. Regarding measurement of absorption coefficients of sample absorbers, both ASTM C423 and ISO 354 state that the sample should be placed asymmetrically – i.e. with none of the sample edges parallel to the room walls – and 0.75 m from any reflective surfaces (the room walls) except the one against which the sample is placed (usually the floor). Both standards also recommend a mounting air gap of 400 mm when testing a sample absorber. Regarding the sample size, ISO 354 sets a minimum of 10 m2, while ASTM C423 sets the value to 5.57 m2. ASTM C423 also suggests covering the edges of the absorber with hard materials to avoid the edge-diffraction effect. Regarding the low-frequency/cut-off limit of a diffuse sound field, ASTM E90 and ASTM C423 specify a limit of 125 Hz and 100 Hz, respectively, while ISO 354 does not prescribe any limit. By way of comparison, critical facility conditions and testing requirements for different standards are presented in Table 4.1. 102 Table 4.1: Comparison of ASTM E90, ASTM C423 and ISO 354 facility conditions. Property ASTM E90 ASTM C423 ISO 354 Minimum room volume (m3) 82 125 150 Room shape Rectangular; dimensions ratio: 41 3⁄ : 21 3⁄ : 1 Maximum allowable absorption (m2)  𝑉2 3⁄ m2 for 𝑓 <2000 𝑉1 3⁄⁄ Hz  𝑉233 m2 for 2000𝑉13<𝑓 < 2000 Hz Maximum allowable absorption coefficient  0.05 for 250 <𝑓 < 2500 Hz  0.1 for 𝑓 ≥2500 Hz Minimum number of sources 1 2 Source placement Near room trihedral corners Minimum number of receivers 5 3 Receive placement 1.5 m apart, 0.75 m from sample surface 1.5 m apart, 2 m from sound source, 1 m from reflective surface and sample Diffusers Should have low absorption coefficient; areas of 3 m2 for one side Minimum sample size (m2) 5.57 10 Sample placement and orientation Asymmetric; 0.75 m from a reflective surface Asymmetric; 0.75 m from a reflective surface Sample mounting air gap (mm) 400 400 Cut-off limit (Hz) 125 100 103 4.2 Numerical Simulation: The Reverberation Room We simulate three different room volumes prescribed by three different standards – i.e. ISO 354, ASTM C423 and ASTM E90. For each of the volumes, we consider four different room shapes:  Case #1: Regular rectangular-shaped, with the longest x-dimension  Case #2: Regular rectangular-shaped, with the longest z-dimension  Case #3: Irregular oblique-shaped, with the longest z-dimension  Case #4: Irregular oblique-shaped, with the longest x-dimension. For all four cases, predictions were performed with 2 sources and 5 receiver positions placed accordingly as prescribed by standards, and for the following conditions:  Third-octave-band without absorber  Third-octave-band with absorber  Octave-band without absorber  Octave-band with absorber. In all cases, the contribution towards achieving a diffuse field was evaluated by predicting the parameters/descriptors described in the previous chapter (see section 3.2) using the calculation procedure also described in the previous chapter (see section 3.3). 4.2.1 ISO 354 Reverberation Room 4.2.1.1 Case #1: Regular Rectangular-Shaped with the Longest x-Dimension We consider a rectangular-shaped reverberation room of dimensions: 𝐿𝑥 = 6.71 m, 𝐿𝑦 =5.3 m and 𝐿𝑧 = 4.22 m. The Bolt room ratio is used to select the room dimensions. The room has a volume of 150 m3, and is equipped with 8 diffusing panels (the diffuser number calculation procedure is given in section 3.1.4), each with an average surface area of 3 m2 (one side) hanging from the ceiling, and a mounting bed of 400-mm height for the testing of the sample absorbers placed against the floor, maintaining standards prescriptions (see section 4.1). 104 Figure 4.1: Rectangular ISO 354 reverberation-room setup with the longest x-dimension. As a test absorber, we consider a sample of 5-cm thickness and surface area of 10.71 m2 having a specific acoustic impedance of 4.3-0.6i, which corresponds to a theoretical random-incidence absorption coefficient of 0.76 as calculated using Eq. (3.17) (see section 3.2.6). It is also ensured that the edges of the absorbers are not exposed to the sound field, to avoid the effect of edge-diffraction. The configuration of the reverberation room is shown in Figure 4.1. 4.2.1.1.1 Third-Octave-Band Prediction A detailed analysis for Case #1: the rectangular-shaped ISO 354 reverberation room with the longest x-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in the previous chapter (see sections 3.4.1 and 3.4.2, respectively). The results presented in sections 3.4.1 and 3.4.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in third-octave bands, the effectiveness of the rectangular-shaped ISO 354 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency is 103 Hz, which suggests that prediction can be done accurately from the 125-Hz band. The modal density, modal spacing and modal-105 width corresponding to this band are 1.21, 0.84 and 0.35, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency is 392 Hz.  The predicted SPL standard deviations are smaller than 1.5 dB from the 160-Hz third-octave band, while the deviations are greater than 2 dB in the rest of the bands – i.e. at < 160 Hz.  The predicted steady-state SPLs show errors below 1% from the 100-Hz third-octave band; the errors are greater than 1% in the rest of the bands, with the maximum of 1.9% in the 50-Hz band.  The predicted RT standard deviations are vary from 0.2 to 0.5 s from the 63-Hz band, with the exception 0.88 s in the 125-Hz band.  The predicted RTs show errors below 15% from the 160-Hz third-octave band; the errors are greater than 20% in the rest of the bands.  The predicted curvatures for the energy-decay curves are between 5 and 15 for all bands, with the exception of 21.6 in the 100-Hz band.  The predicted source-power levels show errors below 10% from the 160-Hz band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 12% from the 160-Hz band; the errors are greater than 15% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors below or around 30% from the 200-Hz band, while the predicted Eyring absorption coefficients show errors around 40% in the same bands. The errors are greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is apparent from the above discussion that prediction can be done accurately from the 160-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 15, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption-coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and the higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the 106 same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.1.1.2 Octave-Band Prediction A detailed analysis for Case #1: the rectangular-shaped ISO 354 reverberation room with the longest x-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in the previous chapter (see sections 3.4.3 and 3.4.4, respectively). The results presented in sections 3.4.3 and 3.4.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of rectangular-shaped ISO 354 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 65 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 1.26, 0.78, 0.36 and 0.46, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency limit is 395 Hz.  The predicted SPL standard deviations are within 0.4 to 0.6 dB for all octave bands.  The predicted steady-state SPLs show errors below 1% from the 125-Hz octave band, while the error is 1.5% in the 63-Hz band.  The predicted RT standard deviations are less than 0.3 s from the 125-Hz octave band.  The predicted RTs show errors below 20% from the 125-Hz octave band.  The predicted curvatures for the energy-decay curves are within 4 to 9 for all octave bands.  The predicted source-power levels show errors below 15% from the 125-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 16% from the 125-Hz band; the errors are greater than 20% in the rest of the bands. 107  The predicted Sabine absorption coefficients of the sample absorber show errors around 26% in the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 40% in the same band. The errors are greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is evident from the above discussion that prediction can be done accurately from the 125-Hz octave band with a lower RT deviation than 0.3 s, a lower SPL deviation than 0.6 dB, a lower curvature than 9, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 20% and 1%, respectively. However, the absorption coefficient prediction case ‘with absorber’ show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.1.2 Case #2: Regular Rectangular-Shaped with the Longest z-Dimension We consider a rectangular-shaped reverberation room of dimensions: 𝐿𝑥 = 4.22 m, 𝐿𝑦 =5.3 m and 𝐿𝑧 = 6.71 m. The room has the same volume (150 m3) and testing arrangements as described in Case #1. The area of the test absorber used, having the same thickness and absorption properties discussed in Case #1, is 8.647 m2. The configuration of the reverberation room is shown in Figure 4.2. 108 Figure 4.2: Rectangular ISO 354 reverberation-room setup with the longest z-dimension. 4.2.1.2.1 Third-Octave-Band Prediction A detailed analysis for Case #2: the rectangular-shaped ISO 354 reverberation room with the longest z-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices H.1 and H.2, respectively. The results presented in appendices H.1 and H.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in third-octave bands, the effectiveness of rectangular-shaped ISO 354 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 103 Hz, which suggests that prediction can be done accurately from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 1.24, 0.8, 0.37 and 0.46, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency limit is 390 Hz.  The predicted SPL standard deviations are less than, or around, 1.5 dB from the 80-Hz third-octave band, while the deviations are greater than 3 dB in the rest of the bands. 109  The predicted steady-state SPLs show errors below 1% from the 125-Hz third-octave band.  The predicted RT standard deviations are less than 0.4 s from the 80-Hz band.  The predicted RTs show errors below 20% from the 160-Hz third-octave band, while the errors are greater than 30% in the rest of the bands.  The predicted curvatures for the energy-decay curves are within 7 to 11 from the 100-Hz third-octave band.  The predicted source-power levels show errors below or around 15% from the 160-Hz third-octave band, while the errors are greater than 20% in the rest of the bands.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 15% from the 160-Hz band, while the errors are greater than 20% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 30% from the 160-Hz band, while the predicted Eyring absorption coefficients show errors around 45% in the same bands. Therefore, it is evident from the above discussion that prediction can be done accurately from the 160-Hz third-octave band with a lower RT deviation than 0.4 s, a lower SPL deviation than 1.5 dB, a lower curvature than 11, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 20% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.1.2.2 Octave-Band Prediction A detailed analysis for Case #2: the rectangular-shaped ISO 354 reverberation room with the longest z-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices H.3 and H.4, respectively. The results presented in appendices H.3 and H.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the 110 calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the rectangular-shaped ISO 354 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 65 Hz. On the other hand, the pure-tone-based Schroeder cut-off frequency limit is 400 Hz.  The predicted SPL standard deviations are less than 0.85 dB in all octave bands.  The predicted steady-state SPLs show errors below 1% from the 125-Hz octave band.  The predicted RT standard deviations are less than 0.2 s from the 125-Hz band.  The predicted RTs show errors below 25% from the 125-Hz octave band.  The predicted curvatures for the energy-decay curves are within 2 to 6 in all octave bands.  The predicted source-power levels show errors below 20% from the 125-Hz octave band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 20% from the 125-Hz band.  The predicted Sabine absorption coefficients of the sample absorber show errors around 24% in the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 40% in the same band. Therefore, it is evident from the above discussion that prediction can be done accurately from the 125-Hz octave band with a lower RT deviation than 0.2 s, a lower SPL deviation than 0.85 dB, a lower curvature than 6, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 25% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 111 4.2.1.3 Case #3: Irregular Oblique-Shaped with the Longest z-Dimension Figure 4.3: Oblique ISO 354 reverberation-room setup with the longest z-dimension. We consider a reverberation room of oblique shaped with the longest z-dimension. The room has the same volume (150 m3) and testing arrangements as described in Case #1 and Case #2. The area of the test absorber used, having the same thickness and absorption properties discussed in Case #1 and Case #2, is 9.46 m2. The configuration of the reverberation room is shown in Figure 4.3. 4.2.1.3.1 Third-Octave-Band Prediction A detailed analysis for Case #3: the oblique-shaped ISO 354 reverberation room with the longest z-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices J.1 and J.2, respectively. The results presented in appendices J.1 and J.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in third-octave bands, the effectiveness of oblique-shaped ISO 354 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows: 112  The statistically-based cut-off frequency limit is 103 Hz, which suggests that prediction can be done from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 1.17, 0.82, 0.37 and 0.45, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency limit is 390 Hz.  The predicted SPL standard deviations are less than 0.9 dB from the 160-Hz third-octave band, while the deviations are greater than 1.5 dB in the rest of the bands.  The predicted steady-state SPLs show errors below or around 1% from the 125-Hz third-octave band.  The predicted RT standard deviations are within 0.15 to 0.52 s from the 100-Hz band.  The predicted RTs show errors below 15% from the 125-Hz third-octave band.  The predicted curvatures for the energy decay curves are within 6 to 15 from the 125-Hz band.  The predicted source-power levels show errors below or around 15% from the 100-Hz third-octave band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 15% from the 100-Hz band, while the errors are greater than 20% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 30% from the 200-Hz band, while the predicted Eyring absorption coefficients show errors around 40% in the same bands. The errors are greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is evident from the above discussion that prediction can be done accurately from the 125-Hz third-octave band with a lower RT deviation than 0.52 s, a lower SPL deviation than 1.5 dB, a lower curvature than 15, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 113 4.2.1.3.2 Octave-Band Prediction A detailed analysis for Case #3: the oblique-shaped ISO 354 reverberation room with the longest z-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices J.3 and J.4, respectively. The results presented in appendices J.3 and J.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of oblique-shaped ISO 354 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 65 Hz. On the other hand, the pure-tone-based Schroeder cut-off frequency limit is 406 Hz.  The predicted SPL standard deviations are less than 1.5 dB in all octave bands.  The predicted steady-state SPLs show errors below or around 1% from the 125-Hz octave-band.  The predicted RT standard deviations are less than 0.2 s from the 125-Hz band.  The predicted RTs show errors below or around 15% from the 125-Hz octave-band.  The predicted curvatures for the energy-decay curves are within 1 to 6 in all octave bands.  The predicted source-power levels show errors below 15% from the 125-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 15% from the 125-Hz band; the errors are greater than 15% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 20% in the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 40% in the same band. The errors are greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is evident from the above discussion that prediction can be done accurately from the 125-Hz octave band with a lower RT deviation 0.2 s, a lower SPL deviation than 0.5 dB, a lower 114 curvature than 6, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.1.4 Case #4: Irregular Oblique-Shaped with the Longest x-Dimension Figure 4.4: Oblique ISO 354 reverberation-room setup with the longest x-dimension. We consider a reverberation room of oblique-shaped with the longest x-dimension. The room has the same volume (150 m3) and testing arrangements as described in Case #1, Case #2 and Case #3. The area of the test absorber used, having the same thickness and absorption properties discussed in Case #1, Case #2 and Case #3, is 10.71 m2. The configuration of the reverberation room is shown in Figure 4.4. 115 4.2.1.4.1 Third-Octave-Band Prediction A detailed analysis for Case #4: the oblique-shaped ISO 354 reverberation room with the longest x-dimension prediction in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices L.1 and L.2, respectively. The results presented in appendices L.1 and L.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in third-octave bands, the effectiveness of the ISO 354 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 103 Hz, which suggests that prediction can be done from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 1.14, 0.87, 0.36 and 0.41, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency limit is 395 Hz.  The predicted SPL standard deviations are less than 1.5 dB from the 160-Hz third-octave band, while the deviations are greater than 1.5 dB in the rest of the bands.  The predicted steady-state SPLs show errors below or around 1% from the 125-Hz third-octave band.  The predicted RT standard deviations are less than 0.5 s from the 125-Hz band.  The predicted RTs show errors around 10% from the 125-Hz third-octave band.  The predicted curvatures for the energy-decay curves are within 4 to 18 from the 125-Hz band.  The predicted source-power levels show errors below or around 10% from the 100-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors around 15% from the 100-Hz band, while the errors are greater than 20% in the rest of the bands. 116  The predicted Sabine absorption coefficients of the sample absorber show errors around 40% from the 160-Hz band, while the predicted Eyring absorption coefficients errors around 50% in the same bands. Therefore, it is evident from the above discussion that prediction can be done from the 125-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 2 dB, a lower curvature than 18, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 10% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.1.4.4 Octave-Band Prediction A detailed analysis for Case #4: the oblique-shaped ISO 354 reverberation room with the longest x-dimension prediction in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices L.3 and L.4, respectively. The results presented in appendices L.3 and L.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the ISO 354 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency is 65 Hz. On the other hand, the pure-tone-based Schroeder cut-off frequency is 406 Hz.  The predicted SPL standard deviations are less than 1.5 dB in all octave bands.  The predicted steady-state SPLs show errors below or around 1% from the 125-Hz octave band.  The predicted RT standard deviations are less than 0.35 s from the 125-Hz band. 117  The predicted RTs show errors below or around 15% from the 125-Hz octave band.  The predicted curvatures for the energy-decay curves are within 3 to 12 in all octave bands.  The predicted source-power levels show errors below 10% from the 125-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 10% from the 125-Hz band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 30% from the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 45% in the same bands. Therefore, it is evident from the above discussion that prediction can be done accurately from the 125-Hz octave band with a lower RT deviation than 0.35 s, a lower SPL deviation than 1.5 dB, a lower curvature than 7, and predicted RTs and SPLs that conform to the expected diffuse field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.1.5 Discussion In this section, based on the predictions done for all the ISO 354 reverberation rooms, we will summarize the results in terms of the descriptors as follows: Cut-off frequency: For all of the ISO 354 reverberation-room cases, the required minimum number of modes around 20, as suggested by the statistically-based cut-off-frequency definition, occurs from the 100-Hz third-octave band, as shown in Figure 4.5. However, the corresponding low-frequency limit calculated using Eq. (3.12) is 103 Hz, which suggests prediction should only be done from the 125-Hz third-octave band. The same is true for the octave-band prediction, as shown in Figure 4.6. The 63-Hz octave band contains the required minimum modes of 20; however, the limit calculated using Eq. (3.11) is 65 Hz, which means prediction should only be 118 Figure 4.5: Distribution of number of modes in third-octave bands for ISO 354 reverberation rooms. Figure 4.6: Distribution of number of modes in octave bands for ISO 354 reverberation rooms. done from the 125-Hz octave band. On the other hand, for both the third-octave and octave-band predictions, the Schroeder cut-off limit is around 400 Hz. Spatial uniformity of SPL: The SPL deviations, as a measure of the spatial variation, in third-octave bands for different ISO reverberation rooms are shown in Figure 4.7. It is noticeable that, the deviations are less than 1.5 dB from the 160-Hz third-octave band for all reverberation room types. In particular, the rectangular room with the longest z-dimension shows deviations less than 1.5 dB from the 80-Hz band. Both of the oblique-shaped rooms show a relatively smaller deviations compared to the other room types, disregarding some exceptions in the 100- and 125-Hz bands. In the case of octave-band prediction, the deviations are smaller than 1.5 dB in all octave bands for all reverberation-room types, as shown in Figure 4.8. In particular, the rectangular room with the longest z-dimension, and the rectangular room with the longest x-dimension, show deviations smaller than 1 dB from the 63-Hz octave-band. 119 Figure 4.7: SPL standard deviation in third-octave bands for ISO 354 reverberation rooms. Figure 4.8: SPL standard deviation in octave bands for ISO 354 reverberation rooms. Accuracy of the predicted SPLs: The predicted SPLs and the corresponding SPL-prediction errors, with respect to the Sabine diffuse-field steady-state SPL formula, in third-octave bands are shown in Figure 4.9 and Figure 4.10, respectively. It is obvious that for all types of reverberation rooms, the SPLs can be predicted assuming a diffuse sound field with an error less than, or around, 1.2% from the 80-Hz third-octave band. Figure 4.9: Predicted SPL in third-octave bands for ISO 354 reverberation rooms. 50 63 80 100 125 160 200 250 315020406080100120140160180200Third-octave center frequency (Hz)L p (dB) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension120 Figure 4.10: SPL-prediction error in third-octave bands for ISO 354 reverberation rooms. On the other hand, the octave-band prediction of the SPLs and the corresponding SPL-prediction errors are shown in Figure 4.11 and Figure 4.12, respectively. It is obvious that for all reverberation-room types, the errors are less than 1% from the 125-Hz octave band. Figure 4.11: Predicted SPL in octave bands for ISO 354 reverberation rooms. Figure 4.12: SPL-prediction error in different octave-bands for ISO 354 reverberation rooms. Spatial uniformity of RT: The RT deviations, as a measure of spatial variation, in third-octave bands for different ISO reverberation rooms are shown in Figure 4.13. It is noticeable that the deviations are less than 0.5 s from the 160-Hz third-octave band for all reverberation-room types. In particular, the oblique room with the longest x-dimension shows relatively lower deviations in 50 63 80 100 125 160 200 250 31500.511.522.5Third-octave center frequency (Hz)SPL prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension63 125 250050100150200Octave-band center frequency (Hz)L p (dB) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension63 125 25000.511.52Octave-band center frequency (Hz)SPL prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension121 the low-frequency bands – i.e. 50, 63, 80 and 100 Hz – compared to the other room types, disregarding some exceptions in the case of one or two bands. Both of the oblique-shaped rooms show deviations below 0.5 s from the 125-Hz third-octave band. In the case of octave-band prediction, the SPL deviations are smaller than 0.5 s from the 125-Hz octave band for all reverberation rooms as shown in Figure 4.14. It is also noticeable that the deviations are smaller in octave bands than in third-octave bands. Figure 4.13: RT standard deviation in third-octave bands for ISO 354 reverberation rooms. Figure 4.14: RT standard deviation in octave bands for ISO 354 reverberation rooms. Accuracy of the predicted RTs: The predicted RTs and the corresponding RT-prediction errors, with respect to the Sabine diffuse-field reverberation-time formula, in third-octave bands are shown in Figure 4.15 and Figure 4.16, respectively. It is obvious that, for the oblique room with the longest x-dimension, the RTs can be predicted assuming a diffuse sound field with an error less than, or around, 15% from the 100-Hz third-octave band. 50 63 80 100 125 160 200 250 31500.511.5Third-octave center frequency (Hz)T60 (s) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension63 125 25000.20.40.60.811.2Octave-band center frequency (Hz)T60 (s) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension122 Figure 4.15: Predicted RT in third-octave-bands for ISO 354 reverberation rooms. Figure 4.16: RT-prediction error in third-octave bands for ISO 354 reverberation rooms. On the other hand, the octave-band prediction of the RTs and the corresponding RT-prediction errors are shown in Figure 4.17 and Figure 4.18, respectively. It is obvious that, for the oblique room with the longest x-dimension, the errors are less than 12% from the 125-Hz octave band. Figure 4.17: Predicted RT in octave bands for ISO 354 reverberation rooms. 50 63 80 100 125 160 200 250 3150246810Third-octave center frequency (Hz)T60 (s) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension50 63 80 100 125 160 200 250 315010203040Third-octave center frequency (Hz)RT prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension63 125 2500246810Octave-band center frequency (Hz)T60 (s) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension123 Figure 4.18: RT-prediction error in octave bands for ISO 354 reverberation rooms. Linearity of temporal decay curves: The curvatures, as a measure of the degree of non-linearity of the temporal energy-decay curves, in third-octave bands for different ISO reverberation rooms are shown in Figure 4.19. It is noticeable that the curvatures are less than, or around, 15 from the 100-Hz third-octave band for all reverberation-room types, disregarding some exceptions for the rectangular reverberation room with the longest x-dimension. In particular, both the rectangular room with the longest z-dimension, and the oblique room with the longest x-dimension, show relatively lower non-linearity compared to the other room types, especially in the low-frequency bands. In the case of octave-band prediction, the curvatures are less than 12 in all octave bands for all reverberation-room types as shown in Figure 4.20. For the rectangular room with the longest z-dimension and the oblique room with the longest z-dimension show values less than 8 in all octave bands. Figure 4.19: Curvature of decay-curve in third-octave bands for ISO 354 reverberation rooms. 63 125 25001020304050Octave-band center frequency (Hz)RT prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension50 63 80 100 125 160 200 250 3150510152025Third-octave center frequency (Hz)Curvature,  Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension124 Figure 4.20: Curvature of decay-curve in octave bands for ISO 354 reverberation rooms. Accuracy of the predicted source-power levels: The predicted source-power levels and the corresponding source-power-level prediction errors, with respect to the Sabine diffuse-field prediction, in third-octave-bands are shown in Figure 4.21 and Figure 4.22, respectively. It is obvious that, for the oblique room with the longest x-dimension, the source-power levels can be predicted assuming a diffuse sound field with an error less than, or around, 12% from the 100-Hz third-octave band. Figure 4.21: Predicted source-power level in third-octaves for ISO 354 reverberation rooms. Figure 4.22: Power-level prediction error in third-octaves for ISO 354 reverberation rooms. 63 125 25002468101214Octave-band center frequency (Hz)Curvature,  Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension50 63 80 100 125 160 200 250 315050100150200Third-octave center frequency (Hz)Lw (dB) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension50 63 80 100 125 160 200 250 3150510152025Third-octave center frequency (Hz)Lw prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension125 On the other hand, the octave-band prediction of source-power levels and the corresponding source-power-level prediction errors are shown in Figure 4.23 and Figure 4.24, respectively. It is obvious that, for the oblique room with the longest x-dimension, the errors are less than 10% from the 125-Hz octave band. Figure 4.23: Predicted source-power level in octave-bands for ISO 354 reverberation rooms. Figure 4.24: Power-level prediction error in octave bands for ISO 354 reverberation rooms. Accuracy of the room-averaged absorption coefficients: The Sabine predictions of room-averaged absorption coefficients and the corresponding prediction errors done in the ‘without absorber’ case, in third-octave bands, are shown in Figure 4.25 and Figure 26. It is obvious that at higher frequency bands, the errors are lower than in the low-frequency bands. The same is also true for octave-band prediction of room-averaged absorption coefficients and the corresponding prediction-errors, as presented in Figure 4.27 and Figure 4.28. 63 125 250050100150200Octave-band center frequency (Hz)Lw (dB) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension63 125 2500510152025Octave-band center frequency (Hz)Lw prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension126 Figure 4.25: Sabine-predicted room-averaged absorption coefficients in third-octave bands for ISO 354 reverberation rooms. Figure 4.26: Sabine-prediction error for the room-averaged absorption coefficients in third-octave bands for ISO 354 reverberation rooms. Figure 4.27: Sabine-predicted room-averaged absorption coefficients in octave bands for ISO 354 reverberation rooms. Figure 4.28: Sabine-prediction error for room-averaged absorption coefficients in octave bands for ISO 354 reverberation rooms. 50 63 80 100 125 160 200 250 31501020304050Third-octave center frequency (Hz)Sabine  prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension63 125 25001020304050Octave-band center frequency (Hz)Sabine  prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension127 Accuracy of the sample absorption coefficients: The predicted absorption coefficients of a sample absorber calculated using the Sabine diffuse-field formula, and the corresponding prediction errors, with respect to the theoretical random absorption coefficients, in third-octave bands, are shown in Figure 4.29 and 4.30, respectively. It is noticeable that the errors are greater (compared to the theoretical value of 0.76) in almost all third-octave bands – in particular, the errors are excessive below the 160-Hz third-octave band. However, it seems the oblique room with the longest x-dimension gives less error compared to the other reverberation rooms. Figure 4.29: Sabine-predicted absorption coefficients of a sample absorber in third-octave bands for ISO 354 reverberation rooms. Figure 4.30: Sabine-prediction error for the absorption coefficients of sample absorber in third-octave bands for ISO 354 reverberation rooms. The predicted absorption coefficients of a sample absorber calculated using the Eyring diffuse-field formula, and the corresponding prediction-error with respect to the theoretical random absorption coefficients, in third-octave bands, are shown in Figure 4.31 and 4.32, respectively. It is noticeable that the prediction-errors are higher (compared to the theoretical value of 0.76) in almost all third-octave bands –they are even greater than the Sabine predictions. However, it seems the oblique room with the longest x-dimension gives less error compared to the other reverberation rooms. 50 63 80 100 125 160 200 250 315020406080100Third-octave center frequency (Hz)Sabine  prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension128 Figure 4.31: Eyring-predicted absorption coefficients of a sample absorber in third-octave bands for ISO 354 reverberation rooms. Figure 4.32: Eyring-prediction error for the absorption coefficients of sample absorber in third-octave bands for ISO 354 reverberation rooms. On the other hand, the octave-band prediction of absorption coefficients of a sample absorber calculated using the Sabine formula, and the corresponding prediction-errors are shown in Figure 4.33 and Figure 4.34, respectively. It is obvious that, the oblique room with the longest x-dimension performs better than the other reverberation rooms. Figure 4.33: Sabine-predicted absorption coefficients of a sample absorber in octave bands for ISO 354 reverberation rooms. 50 63 80 100 125 160 200 250 315020406080100Third-octave center frequency (Hz)Eyring  prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension129 Figure 4.34: Sabine-prediction error for the absorption coefficients of sample absorber in octave-bands for ISO 354 reverberation rooms. The octave-band prediction of absorption coefficients of a sample absorber calculated using the Eyring formula, and the corresponding prediction-errors are shown in Figure 4.35 and Figure 4.36, respectively. It is obvious that, the oblique room with the longest x-dimension performs better than the other reverberation rooms. However, the errors are greater than in the Sabine predictions. Figure 4.35: Eyring-predicted absorption coefficients of a sample absorber in octave bands for ISO 354 reverberation rooms. Figure 4.36: Eyring-prediction error for the absorption coefficients of sample absorber in octave bands for ISO 354 reverberation rooms. 63 125 250020406080100Octave-band center frequency (Hz)Sabine  prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension63 125 250020406080100Octave-band center frequency (Hz)Eyring  prediction error (%) Oblique with longest x-dimensionOblique with longest z-dimensionrectangular with longest z-dimensionrectangular with longest x-dimension130 The large errors associated with the absorption-coefficient predictions imply that the sound field is not sufficiently diffuse. Comparing the test absorber absorption coefficients to the room-averaged absorption coefficients makes it clear that, there is less error in the room-averaged absorption coefficients than those predicted for a test absorber in the ‘with absorber’ case. It can be inferred from the above discussion that the presence of the test absorber makes the field less diffuse than it was without the test absorber. This can also be understood from the higher RT and SPL standard-deviation predictions, and the higher curvatures for the decay curves, which are mentioned in the previous sections. Based on the discussion presented above, it can be concluded that the oblique-shaped ISO 354 reverberation room with the longest x-dimension performs better than the other reverberation rooms. In the case of third-octave-band predictions, prediction can be done accurately from the 100-Hz third-octave band; it will yield an RT-prediction error of less than 15%, an SPL-prediction error of less than 1%, an RT standard deviation of less than 0.5 s, an SPL standard deviation of less than 1.5 dB, a curvature of the temporal decay curve of less than 15, a source-power-level prediction error of less than 12%, and relatively better predictions of the absorption coefficients than the other rooms. On the other hand, in the case of octave-band predictions, prediction can be done accurately from the 125-Hz octave band; it will yield an RT-prediction error of less than 12%, an SPL-prediction error of less than 1%, an RT standard deviation of less than 0.5 s, an SPL standard deviation of less than 1.5 dB, a curvature of the temporal decay curve of less than 12, a source-power-level prediction error of less than 10%, and relatively better predictions of the absorption coefficients than the other rooms. The deviations become smaller in the higher frequency bands. It is also evident that prediction in octave bands gives relatively better predictions than in the third-octave bands. 131 4.2.2 ASTM C423 Reverberation Room 4.2.2.1 Case #1: Regular Rectangular-Shaped with the Longest x-Dimension We consider a rectangular-shaped reverberation room of dimensions: 𝐿𝑥 = 6.32 m, 𝐿𝑦 =4.99 m and 𝐿𝑧 = 3.98 m. The Bolt room ratio is used to select the room dimensions. The room has a volume of 125 m3, and is equipped with 7 diffusing panels, each with an average surface area of 3 m2 (one side) hanging from the ceiling, and a mounting bed of 400-mm height for the testing of the sample absorbers placed against the floor, maintaining standards prescriptions (see section 4.1). As a test absorber, we consider a sample of 5-cm thickness and surface area of 7.534 m2 having a specific acoustic impedance of 4.3-0.6i, which corresponds to a theoretical random-incidence absorption coefficient of 0.76 as calculated using Eq. (3.17) (see section 3.2.6). It is also ensured that the edges of the absorbers are not exposed to the sound field, to avoid the effect of edge-diffraction. The configuration of the reverberation room is shown in Figure 4.37. Figure 4.37: Rectangular ASTM C423 reverberation-room setup with the longest x-dimension. 132 4.2.2.1.1 Third-Octave-Band Prediction A detailed analysis for Case #1: the rectangular-shaped ASTM C423 reverberation room with the longest x-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices L.1 and L.2, respectively. The results presented in appendices L.1 and L.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in third-octave bands, the effectiveness of the rectangular-shaped ASTM C423 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 108 Hz, which suggests that prediction can be done accurately from the 125-Hz band. The modal density, modal spacing and modal-width corresponding to this band are 0.97, 1.02 and 0.36, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency is 411 Hz.  The predicted SPL standard deviations are lower than 1.5 dB from the 160-Hz third-octave band, while the deviations are greater than 2 dB in the rest of the bands.  The predicted steady-state SPLs show errors below 1% from the 100-Hz third-octave band, while the errors are greater than 1% in the rest of the bands.  The predicted RT standard deviations are lower than 0.5 s from the 63-Hz third-octave band.  The predicted RTs show errors below 10% from the 160-Hz third-octave band, while the errors are greater than 15% in the rest of the bands.  The predicted curvatures for the energy-decay curves are within 4 to 23 in all third-octave bands, with the exception of 23 in the 125-Hz band.  The predicted source-power levels show errors below 10% from the 160-Hz band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 10% from the 160-Hz band, while the errors are greater than 15% in the rest of the bands. 133  The predicted Sabine absorption coefficients of the sample absorber show errors below or around 25% from the 160-Hz band, while the predicted Eyring absorption coefficients show errors around 40% in the same bands. The error is greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is apparent from the above discussion that prediction can be done accurately from the 160-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 23, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 10% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.2.1.2 Octave-Band Prediction A detailed analysis for Case #1: rectangular-shaped ASTM C423 reverberation room with the longest x-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices L.3 and L.4, respectively. The results presented in appendices L.3 and L.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the rectangular-shaped ASTM C423 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 68 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 1.08, 0.92, 0.37 and 0.4, 134 respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency is 430 Hz.  The predicted SPL standard deviations are lower than 1.5 dB from the 125-Hz octave band.  The predicted steady-state SPLs show errors below 1% from the 125-Hz octave band, while the error is greater than 1% in the 63-Hz band.  The predicted RT standard deviations are smaller than 0.5 s from the 125-Hz octave band.  The predicted RTs show errors below 15% from the 125-Hz octave band.  The predicted curvatures for the energy-decay curves are within 2 to 11 in all octave bands.  The predicted source-power levels show errors below 12% from the 125-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 13% from the 125-Hz band;, while the errors are greater than 20% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 20% in the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 40% in the same band. The error is greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is evident from the above discussion that prediction can be done accurately from the 125-Hz octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 11, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case ‘with absorber’ show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 135 4.2.2.2 Case #2: Regular Rectangular-Shaped with the Longest z-Dimension Figure 4.38: Rectangular ASTM C423 reverberation-room setup with the longest z-dimension. We consider a rectangular-shaped reverberation room of dimensions: 𝐿𝑥 = 3.98 m, 𝐿𝑦 =4.99 m and 𝐿𝑧 = 6.32 m. The room has the same volume (125 m3) and testing arrangements as described in Case #1. The area of the test absorber used, having the same thickness and absorption properties discussed in Case #1, is 6.214 m2. The configuration of the reverberation room is shown in Figure 4.38. 4.2.2.2.1 Third-Octave-Band Prediction A detailed analysis for Case #2: the rectangular-shaped ASTM C423 reverberation room with the longest z-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices N.1 and N.2, respectively. The results presented in appendices N.1 and N.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. 136 Based on the results presented in third-octave bands, the effectiveness of the rectangular-shaped ASTM C423 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 108 Hz, which suggests that prediction can be done accurately from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 1.07, 0.92, 0.39 and 0.42, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency is 423 Hz.  The predicted SPL standard deviations are smaller than, or around, 1.5 dB from the 125-Hz third-octave band.  The predicted steady-state SPLs show errors below 1% from the 125-Hz third-octave band.  The predicted RT standard deviations are less than 0.5 s in all third-octave bands.  The predicted RTs show errors below 15% from the 160-Hz third-octave band, while the errors are greater than 20% in the rest of the bands.  The predicted curvatures for the energy-decay curves are within 5 to 17 in all third-octave bands.  The predicted source-power levels show errors less than, or around, 13% from the 125-Hz third-octave band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 15% from the 125-Hz band, while the errors are greater than 20% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 40% from the 200-Hz band, while the predicted Eyring absorption coefficients show errors around 50% in the same bands. Therefore, it is evident from the above discussion that prediction can be done accurately from the 160-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 17, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient 137 prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.2.2.2 Octave-Band Prediction A detailed analysis for Case #2: the rectangular-shaped ASTM C423 reverberation room with the longest z-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices N.3 and N.4, respectively. The results presented in appendices N.3 and N.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the rectangular-shaped ASTM C423 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 68 Hz. On the other hand, the pure-tone-based Schroeder cut-off frequency limit is 430 Hz.  The predicted SPL standard deviations are smaller than 1.5 dB in all octave bands.  The predicted steady-state SPLs show errors below 1% from the 125-Hz octave band.  The predicted RT standard deviations are less than 0.3 s in all octave bands.  The predicted RTs show errors below 12% from the 250-Hz octave band.  The predicted curvatures for the energy-decay curves are within 3 to 5 in all octave bands.  The predicted source-power levels show errors below 10% from the 250-Hz octave band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 10% from the 250-Hz band. 138  The predicted Sabine absorption coefficients of the sample absorber show errors around 30% in the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 45% in the same band. Therefore, it is evident from the above discussion that prediction can be done accurately from the 250-Hz octave band with a lower RT deviation than 0.3 s, a lower SPL deviation than 1.5 dB, a lower curvature than 5, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 12% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.2.3 Case #3: Irregular Oblique-Shaped with the Longest z-Dimension Figure 4.39: Oblique ASTM C423 reverberation-room setup with the longest z-dimension. We consider a reverberation room of oblique shaped with the longest z-dimension. The room has the same volume (150 m3) and testing arrangements as described in Case #1 and Case #2. The 139 area of the test absorber used, having the same thickness and absorption properties discussed in Case #1 and Case #2, is 6.484 m2. The configuration of the reverberation room is shown in Figure 4.39. 4.2.2.3.1 Third-Octave-Band Prediction A detailed analysis for Case #3: the oblique-shaped ASTM C423 reverberation room with the longest z-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices P.1 and P.2, respectively. The results presented in appendices P.1 and P.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in third-octave bands, the effectiveness of the oblique-shaped ASTM C423 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency is 108 Hz, which suggests that prediction can be done from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 0.97, 1.02, 0.37 and 0.36, respectively. On the other hand, the pure-tone-based Schroeder cut-off limit is 423 Hz.  The predicted SPL standard deviations are less than 1.5 dB from the 160-Hz third-octave band, while the deviations are greater than 1.5 dB in the rest of the bands.  The predicted steady-state SPLs show errors below or around 1% from the 125-Hz third-octave band.  The predicted RT standard deviations are smaller than 0.5 s from the 100-Hz third-octave band.  The predicted RTs show errors below 10% from the 200-Hz third-octave band.  The predicted curvatures for the energy decay curves are within 5 to 17 from the 100-Hz third-octave band. 140  The predicted source-power levels show errors below or around 10% from the 200-Hz third-octave band.  On the other hand, the predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 10% from the 200-Hz third-octave band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 35% from the 160-Hz band, while the predicted Eyring absorption coefficients show errors around 50% in the same bands. The errors are greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is evident from the above discussion that prediction can be done accurately from the 160-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 17, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.2.3.2 Octave-Band Prediction A detailed analysis for Case #3: the oblique-shaped ASTM C423 reverberation room with the longest z-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices P.3 and P.4, respectively. The results presented in appendices P.3 and P.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the oblique-shaped ASTM C423 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows: 141  The statistically-based cut-off frequency limit is 68 Hz. On the other hand, the pure-tone-based Schroeder cut-off frequency limit is 452 Hz.  The predicted SPL standard deviations are smaller than 1.5 dB from the 125-Hz octave band.  The predicted steady-state SPLs show errors below or around 1% from the 125-Hz octave-band.  The predicted RT standard deviations are less than 0.3 s from the 125-Hz octave band.  The predicted RTs show errors below or around 10% from the 250-Hz band.  The predicted curvatures for the energy-decay curves are within 2 to 5 from the 125-Hz octave-band.  The predicted source-power levels show errors below 10% from the 250-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 10% from the 250-Hz band; the errors are greater than 15% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 20% in the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 40% in the same band. The errors are greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is evident from the above discussion that prediction can be done accurately from the 125-Hz octave band with a lower RT deviation 0.3 s, a lower SPL deviation than 1.5 dB, a lower curvature than 5, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 142 4.2.2.4 Case #4: Irregular Oblique-Shaped with the Longest x-Dimension Figure 4.40: Oblique ASTM C423 reverberation-room setup with the longest x-dimension. We consider a reverberation room of oblique-shaped with the longest x-dimension. The room has the same volume (125 m3) and testing arrangements as described in Case #1, Case #2 and Case #3. The area of the test absorber used, having the same thickness and absorption properties discussed in Case #1, Case #2 and Case #3, is 7.534 m2. The configuration of the reverberation room is shown in Figure 4.40. 4.2.2.4.1 Third-Octave-Band Prediction A detailed analysis for Case #4: the oblique-shaped ASTM C423 reverberation room with the longest x-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices R.1 and R.2, respectively. The results presented in appendices R.1 and R.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. 143 Based on the results presented in third-octave bands, the effectiveness of the ASTM C423 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency is 108 Hz, which suggests that prediction can be done from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 0.97, 1.05, 0.37 and 0.36, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency l is 452 Hz.  The predicted SPL standard deviations are less than 1.5 dB from the 160-Hz third-octave band, while the deviations are greater than 1.5 dB in the rest of the bands.  The predicted steady-state SPLs show errors below or around 1% from the 160-Hz third-octave band.  The predicted RT standard deviations are less than 0.5 s from the 63-Hz third-octave band.  The predicted RTs show errors around 10% from the 125-Hz third-octave band.  The predicted curvatures for the energy-decay curves are within 5 to 13 from the 63-Hz third-octave band.  The predicted source-power levels show errors below or around 10% from the 125-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors around 10% from the 125-Hz band, while the errors are greater than 20% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 40% from the 160-Hz band, while the predicted Eyring absorption coefficients show errors around 50% in the same bands. Therefore, it is evident from the above discussion that prediction can be done from the 125-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.9 dB, a lower curvature than 13, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 10% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, 144 which is apparent from the higher RT and SPL deviations, and higher. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.2.4.2 Octave-Band Prediction A detailed analysis for Case #4: the oblique-shaped ASTM C423 reverberation room with the longest x-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices P.3 and P.4, respectively. The results presented in appendices P.3 and P.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the ASTM C423 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency is 68 Hz. On the other hand, the pure-tone-based Schroeder cut-off frequency is 430 Hz.  The predicted SPL standard deviations are less than 1.5 dB from the 125-Hz octave band.  The predicted steady-state SPLs show errors below or around 1.1% in all octave bands.  The predicted RT standard deviations are less than 0.3 s from the 125-Hz band.  The predicted RTs show errors below or around 13% from the 125-Hz octave band.  The predicted curvatures for the energy-decay curves are within 2 to 6 from the 125-Hz octave band.  The predicted source-power levels show errors below 10% from the 125-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 10% from the 125-Hz band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 50% from the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 58% in the same bands. 145 Therefore, it is evident from the above discussion that prediction can be done accurately from the 125-Hz octave band with a lower RT deviation than 0.3 s, a lower SPL deviation than 1.5 dB, a lower curvature than 6, and predicted RTs and SPLs that conform to the expected diffuse field values with an error of 13% and 1.1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.2.5 Discussion In this section, based on the predictions done for all the ASTM C423 reverberation rooms, we will summarize the results in terms of the descriptors as follows: Cut-off frequency: For all of the ASTM C423 reverberation-room cases, the required minimum number of modes of 20, as suggested by the statistically-based cut-off-frequency definition, occurs from the 125-Hz third-octave band, as shown in Figure 4.41. At the same time, the corresponding low-frequency limit calculated using Eq. (3.12) is 108 Hz, which also suggests that prediction can be done from the 125-Hz third-octave band. The same is true for the octave-band prediction, as shown in Figure 4.42. The 125-Hz octave band contains the required minimum modes of 20; the corresponding limit calculated using Eq. (3.11) is 68 Hz, which Figure 4.41: Distribution of mode number in third-octave bands for ASTM C423 reverberation rooms. 146 Figure 4.42: Distribution of mode number in octave bands for ASTM C423 reverberation rooms. means prediction can be done from the 125-Hz octave band. On the other hand, for both the third-octave and octave-band predictions, the Schroeder cut-off limit is around 425 Hz. Spatial uniformity of SPL: The SPL deviations, as a measure of spatial variation, in third-octave bands for different ASTM C423 reverberation rooms are shown in Figure 4.43. It is noticeable that, the deviations are less than 1.5 dB from the 160-Hz third-octave band for all reverberation room types. In particular, the oblique room with the longest x-dimension shows deviations less than, or around, 1.5 dB from the 100-Hz third-octave band, and the rectangular room with the longest z-dimension shows deviations less than 1.5 dB in all third-octave bands. In the case of octave-band prediction, the deviations are smaller than 1.5 dB from the 125-Hz octave band for all reverberation-room types, as shown in Figure 4.44. In particular, the rectangular room with the longest z-dimension, and the oblique room with the longest z-dimension, show deviations smaller than 1.5 dB in all octave bands. Figure 4.43: SPL standard deviation in third-octave bands for ASTM C423 reverberation rooms. 147 Figure 4.44: SPL standard deviation in octave bands for ASTM C423 reverberation rooms. Accuracy of the predicted SPLs: The predicted SPLs and the corresponding SPL-prediction errors, with respect to the Sabine diffuse-field steady-state SPL formula, in third-octave bands are shown in Figure 4.45 and Figure 4.46, respectively. It is obvious that for all types of reverberation rooms, the SPLs can be predicted assuming a diffuse sound field with an error less than, or around, 1.2% from the 100-Hz third-octave band. Figure 4.45: Predicted SPL in third-octave bands for ASTM C423 reverberation rooms. Figure 4.46: SPL-prediction error in third-octave bands for ASTM C423 reverberation rooms. 148 On the other hand, the octave-band prediction of the SPLs and the corresponding SPL-prediction errors are shown in Figure 4.47 and Figure 4.48, respectively. It is obvious that for all reverberation-room types, the errors are less than 1.3% in all octave bands. Figure 4.47: Predicted SPL in octave bands for ASTM C423 reverberation rooms. Figure 4.48: SPL-prediction error in octave-bands for ASTM C423 reverberation rooms. Spatial uniformity of RT: The RT deviations, as a measure of spatial variation, in third-octave bands for different ASTM C423 reverberation rooms are shown in Figure 4.49. It is noticeable that the deviations are less than, or around 0.6 s from the 63-Hz third-octave band for all reverberation-room types disregarding some exceptions for the rectangular reverberation room with the longest x-dimension. In particular, the oblique room with the longest x-dimension shows relatively lower deviations from the 63-Hz band than the other room types. The rectangular-shaped room with the longest z-dimension shows smaller deviations below 0.5 s in all third-octave bands disregarding the exception in the 315-Hz band. In the case of octave-band prediction, the SPL deviations are smaller than 0.5 s from the 125-Hz octave band for all reverberation rooms as shown in Figure 4.50. In particular, the rectangular-shaped room with the longest z-dimension shows smaller deviations below 0.5 s in all octave bands. 149 Figure 4.49: RT standard deviation in third-octave bands for ASTM C423 reverberation rooms. Figure 4.50: RT standard deviation in octave bands for ASTM C423 reverberation rooms. Accuracy of the predicted RTs: The predicted RTs and the corresponding RT-prediction errors, with respect to the Sabine diffuse-field reverberation-time formula, in third-octave bands are shown in Figure 4.51 and Figure 4.52, respectively. It is obvious that, for the oblique room with the longest x-dimension, the RTs can be predicted assuming a diffuse sound field with an error less than, or around, 15% from the 125-Hz third-octave band. Figure 4.51: Predicted RT in third-octave-bands for ASTM C423 reverberation rooms. 150 Figure 4.52: RT-prediction error in third-octave bands for ASTM C423 reverberation rooms. On the other hand, the octave-band prediction of the RTs and the corresponding RT-prediction errors are shown in Figure 4.53 and Figure 4.54, respectively. It is obvious that, for the oblique room with the longest x-dimension, the errors are less than, or around, 10% from the 125-Hz octave band. Figure 4.53: Predicted RT in octave bands for ASTM C423 reverberation rooms. Figure 4.54: RT-prediction error in octave bands for ASTM C423 reverberation rooms. 151 Linearity of temporal decay curves: The curvatures, as a measure of the degree of non-linearity of the temporal energy-decay curves, in third-octave bands for different ASTM C423 reverberation rooms are shown in Figure 4.55. It is noticeable that both the rectangular room with the longest z-dimension, and the oblique room with the longest x-dimension, show relatively lower non-linearity compared to the other room types. In particular, the curvatures are less than, or around, 10 from the 63-Hz third-octave band for the oblique room with the longest x-dimension. In the case of octave-band prediction, the curvatures are less than 10 in all octave bands for the oblique room with the longest x-dimension, as shown in Figure 4.56. For the rectangular room with the longest z-dimension, curvatures are less than 5 in all octave bands. Figure 4.55: Curvature of decay-curve in third-octave bands for ASTM C423 reverberation rooms. Figure 4.56: Curvature of decay-curve in octave bands for ASTM C423 reverberation rooms. Accuracy of the predicted source-power levels: The predicted source-power levels and the corresponding source-power-level prediction errors, with respect to the Sabine diffuse-field 152 prediction, in third-octave-bands are shown in Figure 4.57 and Figure 4.58, respectively. It is obvious that, for the oblique room with the longest x-dimension, the source-power levels can be predicted assuming a diffuse sound field with an error less than, or around, 10% from the 125-Hz third-octave band. Figure 4.57: Predicted source-power level in third-octaves for ASTM C423 reverberation rooms. Figure 4.58: Power-level prediction error in third-octaves for ASTM C423 reverberation rooms. On the other hand, the octave-band prediction of source-power levels and the corresponding source-power-level prediction errors are shown in Figure 4.59 and Figure 4.60, respectively. It is obvious that, for the oblique room with the longest x-dimension, the errors are less than 10% from the 125-Hz octave band. Figure 4.59: Predicted source-power level in octave-bands for ASTM C423 reverberation rooms. 153 Figure 4.60: Power-level prediction error in octave bands for ASTM C423 reverberation rooms. Accuracy of the room-averaged absorption coefficients: The Sabine prediction of room-averaged absorption coefficients and the corresponding prediction-errors done in the ‘without absorber’ case, in third-octave bands, are shown in Figure 4.61 and Figure 4.62. It is evident that at higher frequency bands, the errors are lower than in the low-frequency bands. It is obvious that, for the oblique room with the longest x-dimension, the errors are less than 10% from the 125-Hz octave band. The same is also true for octave-band prediction of room-averaged absorption coefficients and the corresponding prediction-errors, as presented in Figure 4.63 and Figure 4.64. Figure 4.61: Sabine-predicted room-averaged absorption coefficients in third-octave bands for ASTM C423 reverberation rooms. 154 Figure 4.62: Sabine-prediction error for the room-averaged absorption coefficients in third-octave bands for ASTM C423 reverberation rooms. Figure 4.63: Sabine-predicted room-averaged absorption coefficients in octave bands for ASTM C423 reverberation rooms. Figure 4.64: Sabine-prediction error for room-averaged absorption coefficients in octave bands for ASTM C423 reverberation rooms. Accuracy of the sample absorption coefficients: The predicted absorption coefficients of a sample absorber calculated using the Sabine diffuse-field formula, and the corresponding 155 prediction errors, with respect to the theoretical random absorption coefficients, in third-octave bands, are shown in Figure 4.65 and 4.66, respectively. It is noticeable that the errors are greater (compared to the theoretical value of 0.76) in almost all third-octave bands – in particular, the errors are excessive below the 160-Hz third-octave band. Figure 4.65: Sabine-predicted absorption coefficients of a sample absorber in third-octave bands for ASTM C423 reverberation rooms. Figure 4.66: Sabine-prediction error for the absorption coefficients of sample absorber in third-octave bands for ASTM C423 reverberation rooms. The predicted absorption coefficients of a sample absorber calculated using the Eyring diffuse-field formula, and the corresponding prediction-error with respect to the theoretical random absorption coefficients, in third-octave bands, are shown in Figure 4.67 and 4.68, respectively. It is noticeable that the prediction-errors are higher (compared to the theoretical value of 0.76) in almost all third-octave bands –they are even greater than the Sabine predictions. 156 Figure 4.67: Eyring-predicted absorption coefficients of a sample absorber in third-octave bands for ASTM C423 reverberation rooms. Figure 4.68: Eyring-prediction error for the absorption coefficients of sample absorber in third-octave bands for ASTM C423 reverberation rooms. On the other hand, the octave-band prediction of absorption coefficients of a sample absorber calculated using the Sabine formula, and the corresponding prediction-errors are shown in Figure 4.69 and Figure 4.70, respectively. Figure 4.69: Sabine-predicted absorption coefficients of a sample absorber in octave bands for ASTM C423 reverberation rooms. 157 Figure 4.70: Sabine-prediction error for the absorption coefficients of sample absorber in octave-bands for ASTM C423 reverberation rooms. The octave-band prediction of absorption coefficients of a sample absorber calculated using the Eyring formula, and the corresponding prediction-errors are shown in Figure 4.71 and Figure 4.72, respectively. It is obvious that, the errors are greater than in the Sabine predictions. Figure 4.71: Eyring-predicted absorption coefficients of a sample absorber in octave bands for ASTM C423 reverberation rooms. Figure 4.72: Eyring-prediction error for the absorption coefficients of sample absorber in octave bands for ASTM C423 reverberation rooms. 158 The large errors associated with the absorption-coefficient predictions imply that the sound field is not sufficiently diffuse. Comparing the test absorber absorption coefficients to the room-averaged absorption coefficients makes it clear that, there is less error in the room-averaged absorption coefficients than those predicted for a test absorber in the ‘with absorber’ case. It can be inferred from the above discussion that the presence of the test absorber makes the field less diffuse than it was without the test absorber. This can also be understood from the higher RT and SPL standard-deviation predictions, and the higher curvatures for the decay curves, which are mentioned in the previous sections. Based on the discussion presented above, it can be concluded that the oblique-shaped ASTM C423 reverberation room with the longest x-dimension performs better than the other reverberation rooms. In the case of third-octave-band predictions, prediction can be done accurately from the 125-Hz third-octave band; it will yield an RT-prediction error of less than, or around, 15%, an SPL-prediction error of less than 1.2%, an RT standard deviation of less than 0.5 s, an SPL standard deviation of less than 1.5 dB, a curvature of the temporal decay curve of less than 12, a source-power-level prediction error of less than, or around, 10%, and relatively better predictions of the absorption coefficients than the other rooms. On the other hand, in the case of octave-band predictions, prediction can be done accurately from the 125-Hz octave band; it will yield an RT-prediction error of less than 10%, an SPL-prediction error of less than 1.2%, an RT standard deviation of less than 0.5 s, an SPL standard deviation of less than 1.5 dB, a curvature of the temporal decay curve of less than 5, a source-power-level prediction error of less than 10%, and relatively better predictions of the absorption coefficients than the other rooms. The deviations become smaller in the higher frequency bands. It is also evident that prediction in octave bands gives relatively better predictions than in the third-octave bands. 159 4.2.3 ASTM E90 Reverberation Room 4.2.3.1 Case #1: Regular Rectangular-Shaped with the Longest x-Dimension We consider a rectangular-shaped reverberation room of dimensions: 𝐿𝑥 = 5.48 m, 𝐿𝑦 =4.35 m and 𝐿𝑧 = 3.44 m. The Bolt room ratio is used to select the room dimensions. The room has a volume of 82 m3, and is equipped with 5 diffusing panels, each with an average surface area of 3 m2 (one side) hanging from the ceiling, and a mounting bed of 400-mm height for the testing of the sample absorbers placed against the floor, maintaining standards prescriptions (see section 4.1). As a test absorber, we consider a sample of 5-cm thickness and surface area of 6.259 m2 having a specific acoustic impedance of 4.3-0.6i, which corresponds to a theoretical random-incidence absorption coefficient of 0.76 as calculated using Eq. (3.17) (see section 3.2.6). It is also ensured that the edges of the absorbers are not exposed to the sound field, to avoid the effect of edge-diffraction. The configuration of the reverberation room is shown in Figure 4.73. Figure 4.73: Rectangular ASTM E90 reverberation-room setup with the longest x-dimension. 160 4.2.3.1.1 Third-Octave-Band Prediction A detailed analysis for Case #1: the rectangular-shaped ASTM E90 reverberation room with the longest x-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices T.1 and T.2, respectively. The results presented in appendices T.1 and T.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in third-octave bands, the effectiveness of the rectangular-shaped ASTM E90 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 125 Hz, which suggests that prediction can be done accurately from the 160-Hz third-octave band. The modal density, modal spacing and modal-width corresponding to this band are 1, 0.95, 0.43 and 0.46, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency is 498 Hz.  The predicted SPL standard deviations are lower than 1.5 dB from the 160-Hz third-octave band, while the deviations are greater than 2 dB in the rest of the bands.  The predicted steady-state SPLs show errors below 1% from the 125-Hz third-octave band, while the errors are greater than 1% in the rest of the bands.  The predicted RT standard deviations are lower than 0.5 s from the 125-Hz third-octave band.  The predicted RTs show errors below 15% from the 160-Hz third-octave band, while the errors are greater than 15% in the rest of the bands.  The predicted curvatures for the energy-decay curves are within 4 to 13 in all third-octave bands.  The predicted source-power levels show errors below 15% from the 160-Hz band, while the errors are greater than 15% in the rest of the bands. 161  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 15% from the 160-Hz band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors below or around 50% from the 160-Hz band, while the predicted Eyring absorption coefficients show errors around 55% in the same bands. The error is greater than 60% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is apparent from the above discussion that prediction can be done accurately from the 160-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 13, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.3.1.2 Octave-Band Prediction A detailed analysis for Case #1: the rectangular-shaped ASTM E90 reverberation room with the longest x-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices T.3 and T.4, respectively. The results presented in appendices T.3 and T.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the rectangular-shaped ASTM E90 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows: 162  The statistically-based cut-off frequency limit is 78 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 0.76, 1.32, 0.42 and 0.32, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency is 517 Hz.  The predicted SPL standard deviations are lower than 1.5 dB from the 125-Hz octave band.  The predicted steady-state SPLs show errors below 1% from the 125-Hz octave band, while the error is greater than 1% in the 63-Hz band.  The predicted RT standard deviations are smaller than 0.5 s from the 125-Hz octave band.  The predicted RTs show errors below 10% from the 250-Hz octave band.  The predicted curvatures for the energy-decay curves are within 3 to 6 in all octave bands.  The predicted source-power levels show errors below 10% from the 250-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 10% from the 250-Hz band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 35% in the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 50% in the same band. The errors are greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is evident from the above discussion that prediction can be done accurately from the 125-Hz octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 6, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 20% and 1%, respectively. However, the absorption coefficient prediction case ‘with absorber’ show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 163 4.2.3.2 Case #2: Regular Rectangular-Shaped with the Longest z-Dimension Figure 4.74: Rectangular ASTM E90 reverberation-room setup with the longest z-dimension. We consider a rectangular-shaped reverberation room of dimensions: 𝐿𝑥 = 3.44 m, 𝐿𝑦 =4.35 m and 𝐿𝑧 = 5.48 m. The room has the same volume (82 m3) and testing arrangements as described in Case #1. The area of the test absorber used, having the same thickness and absorption properties discussed in Case #1, is 5.075 m2. The configuration of the reverberation room is shown in Figure 4.74. 4.2.3.2.1 Third-Octave-Band Prediction A detailed analysis for Case #2: the rectangular-shaped ASTM E90 reverberation room with the longest z-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices V.1 and V.2, respectively. The results presented in appendices V.1 and V.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. 164 Based on the results presented in third-octave bands, the effectiveness of the rectangular-shaped ASTM E90 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 125 Hz, which suggests that prediction can be done accurately from the 160-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 1, 0.97, 0.44 and 0.46, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency is 527 Hz.  The predicted SPL standard deviations are smaller than 1.5 dB from the 160-Hz third-octave band.  The predicted steady-state SPLs show errors below 1% from the 80-Hz third-octave band.  The predicted RT standard deviations are less than 0.5 s from the 100-Hz third-octave band.  The predicted RTs show errors below 15% from the 250-Hz third-octave band, while the errors are greater than 20% in the rest of the bands.  The predicted curvatures for the energy-decay curves are within 4 to 10 from the 100-Hz third-octave band.  The predicted source-power levels show errors less than, or around, 15% from the 250-Hz third-octave band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 15% from the 250-Hz band, while the errors are greater than 20% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 45% from the 200-Hz band, while the predicted Eyring absorption coefficients show errors around 50% in the same bands. Therefore, it is evident from the above discussion that prediction can be done accurately from the 250-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 10, and predicted RTs and SPLs that conform to the expected diffuse-165 field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.3.2.2 Octave-Band Prediction A detailed analysis for Case #2: the rectangular-shaped ASTM E90 reverberation room with the longest z-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices V.3 and V.4, respectively. The results presented in appendices V.3 and V.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the rectangular-shaped ASTM E90 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency limit is 78 Hz. On the other hand, the pure-tone-based Schroeder cut-off frequency limit is 558 Hz.  The predicted SPL standard deviations are smaller than 1.5 dB in all octave bands.  The predicted steady-state SPLs show errors below 1% in all octave bands.  The predicted RT standard deviations are less than 0.5 s from the 125-Hz octave band.  The predicted RTs show errors below 15% from the 250-Hz octave band.  The predicted curvatures for the energy-decay curves are within 2 to 3 from the 125-Hz octave band.  The predicted source-power levels show errors below 15% from the 250-Hz octave band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 15% from the 250-Hz band. 166  The predicted Sabine absorption coefficients of the sample absorber show errors around 40% in the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 50% in the same band. Therefore, it is evident from the above discussion that prediction can be done accurately from the 250-Hz octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 3, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.3.3 Case #3: Irregular Oblique-Shaped with the Longest z-Dimension Figure 4.75: Oblique ASTM E90 reverberation-room setup with the longest z-dimension. We consider a reverberation room of oblique shaped with the longest z-dimension. The room has the same volume (82 m3) and testing arrangements as described in Case #1 and Case #2. The 167 area of the test absorber used, having the same thickness and absorption properties discussed in Case #1 and Case #2, is 5.008 m2. The configuration of the reverberation room is shown in Figure 4.75. 4.2.3.3.1 Third-Octave-Band Prediction A detailed analysis for Case #3: the oblique-shaped ASTM E90 reverberation room with the longest z-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices X.1 and X.2, respectively. The results presented in appendices X.1 and X.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in third-octave bands, the effectiveness of the oblique-shaped ASTM E90 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency is 125 Hz, which suggests that prediction can be done from the 160-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 1, 0.97, 0.43 and 0.44, respectively. On the other hand, the pure-tone-based Schroeder cut-off limit is 536 Hz.  The predicted SPL standard deviations are less than 1.5 dB from the 160-Hz third-octave band, while the deviations are greater than 1.5 dB in the rest of the bands.  The predicted steady-state SPLs show errors below 1% from the 125-Hz third-octave band.  The predicted RT standard deviations are smaller than 0.5 s from the 80-Hz third-octave band.  The predicted RTs show errors below 15% from the 200-Hz third-octave band.  The predicted curvatures for the energy decay curves are within 6 to 11 in all third-octave bands. 168  The predicted source-power levels show errors below or around 15% from the 200-Hz third-octave band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 15% from the 200-Hz third-octave band, while the errors are greater than 15% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 50% from the 160-Hz band, while the predicted Eyring absorption coefficients show errors around 55% in the same bands. The errors are greater than 60% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is evident from the above discussion that prediction can be done accurately from the 200-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 11, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.3.3.2 Octave-Band Prediction A detailed analysis for Case #3: the oblique-shaped ASTM E90 reverberation room with the longest z-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices X.3 and X.4, respectively. The results presented in appendices X.3 and X.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the oblique-shaped ASTM E90 reverberation room with the longest z-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows: 169  The statistically-based cut-off frequency is 78 Hz. On the other hand, the pure-tone-based Schroeder cut-off frequency is 536 Hz.  The predicted SPL standard deviations are smaller than 1 dB from the 125-Hz octave band.  The predicted steady-state SPLs show errors below or around 1% from the 125-Hz octave-band.  The predicted RT standard deviations are less than 0.6 s in all octave bands.  The predicted RTs show errors below or around 20% from the 125-Hz band.  The predicted curvatures for the energy-decay curves are within 2 to 4 in all octave bands.  The predicted source-power levels show errors below 18% from the 125-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 18% from the 125-Hz band; the errors are greater than 20% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 45% in the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 55% in the same band. The errors are greater than 50% in the rest of the bands for both Sabine and Eyring prediction cases. Therefore, it is evident from the above discussion that prediction can be done accurately from the 160-Hz octave band with a lower RT deviation 0.6 s, a lower SPL deviation than 1.5 dB, a lower curvature than 4, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 15% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 170 4.2.3.4 Case #4: Irregular Oblique-Shaped with the Longest x-Dimension Figure 4.76: Oblique ASTM E90 reverberation-room setup with the longest x-dimension. We consider a reverberation room of oblique-shaped with the longest x-dimension. The room has the same volume (82 m3) and testing arrangements as described in Case #1, Case #2 and Case #3. The area of the test absorber used, having the same thickness and absorption properties discussed in Case #1, Case #2 and Case #3, is 6.259 m2. The configuration of the reverberation room is shown in Figure 4.76. 4.2.3.4.1 Third-Octave-Band Prediction A detailed analysis for Case #4: the oblique-shaped ASTM E90 reverberation room with the longest x-dimension in third-octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices Z.1 and Z.2, respectively. The results presented in appendices Z.1 and Z.2 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. 171 Based on the results presented in third-octave bands, the effectiveness of the ASTM E90 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency is 125 Hz, which suggests that prediction can be done from the 160-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to this band are 1.51, 0.67, 0.39 and 0.58, respectively. On the other hand, the pure-tone-based Schroeder cut-off frequency is 531 Hz.  The predicted SPL standard deviations are less than 1.5 dB from the 100-Hz third-octave band, while the deviations are greater than 1.5 dB in the rest of the bands.  The predicted steady-state SPLs show errors below or around 1% from the 80-Hz third-octave band.  The predicted RT standard deviations are less than 0.5 s from the 125-Hz third-octave band with the exception in the 200-Hz band.  The predicted RTs show errors around 25% from the 200-Hz third-octave band.  The predicted curvatures for the energy-decay curves are within 10 to 20 from the 125-Hz third-octave band.  The predicted source-power levels show errors below or around 25% from the 200-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors around 25% from the 160-Hz band, while the errors are greater than 25% in the rest of the bands.  The predicted Sabine absorption coefficients of the sample absorber show errors around 40% from the 125-Hz band, while the predicted Eyring absorption coefficients show errors around 55% in the same bands. Therefore, it is evident from the above discussion that prediction can be done from the 200-Hz third-octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 20, and predicted RTs and SPLs that conform to the expected diffuse-field values with an error of 25% and 1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, 172 which is apparent from the higher RT and SPL deviations, and higher. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.3.4.2 Octave-Band Prediction A detailed analysis for Case #4: the oblique-shaped ASTM E90 reverberation room with the longest x-dimension in octave bands and ‘without’ and ‘with’ sample absorber is presented in appendices Z.3 and Z.4, respectively. The results presented in appendices Z.3 and Z.4 are discussed in terms of the descriptors described in the previous chapter (see section 3.2) using the calculation procedure as described also the previous chapter (see section 3.3). In this section, only the prediction summary is presented. Based on the results presented in octave bands, the effectiveness of the ASTM E90 reverberation room with the longest x-dimension at achieving sound-field diffuseness and the accurate prediction of relevant acoustical quantities can be summarized as follows:  The statistically-based cut-off frequency is 78 Hz. On the other hand, the pure-tone-based Schroeder cut-off frequency is 517 Hz.  The predicted SPL standard deviations are less than 1.5 dB from the 125-Hz octave band.  The predicted steady-state SPLs show errors below or around 1% from the 125-Hz octave band.  The predicted RT standard deviations are less than 0.5 s in all octave bands.  The predicted RTs show errors below or around 25% from the 250-Hz octave band.  The predicted curvatures for the energy-decay curves are within 5 to 10 in all octave bands.  The predicted source-power levels show errors below 25% from the 250-Hz band.  The predicted Sabine and Eyring room-averaged absorptions, as well as the absorption coefficients, show errors below 20% from the 125-Hz band, while the errors are greater than 15% in the rest of the bands. 173  The predicted Sabine absorption coefficients of the sample absorber show errors around 60% from the 250-Hz band, while the predicted Eyring absorption coefficients show errors around 65% in the same bands. Therefore, it is evident from the above discussion that prediction can be done accurately from the 250-Hz octave band with a lower RT deviation than 0.5 s, a lower SPL deviation than 1.5 dB, a lower curvature than 10, and predicted RTs and SPLs that conform to the expected diffuse field values with an error of 25% and 1.1%, respectively. However, the absorption coefficient prediction case of a sample absorber show less field diffuseness compared to the ‘without absorber’ case, which is apparent from the higher RT and SPL deviations, and higher curvatures. Consequently, the Sabine and Eyring absorption coefficients predicted in the same bands show a larger error compared to the other prediction cases – i.e. source-power-level prediction. 4.2.3.5 Discussion In this section, based on the predictions done for all the ASTM E90 reverberation rooms, we will summarize the results in terms of the descriptors as follows: Cut-off frequency: For all of the ASTM E90 reverberation-room cases, the required minimum number of modes around 20, as suggested by the statistically-based cut-off-frequency definition, occurs from the 125-Hz third-octave band, as shown in Figure 4.77. However, the corresponding low-frequency limit calculated using Eq. (3.12) is 125 Hz, which suggests prediction should only be done from the 160-Hz third-octave band. The same is true for the octave-band prediction, as shown in Figure 4.78. The 125-Hz octave band contains the required minimum modes of 20, and the limit calculated using Eq. (3.11) is 78 Hz, which also suggests that prediction can be done from the 125-Hz octave band. On the other hand, for both the third-octave and octave-band predictions, the Schroeder cut-off limit is greater than 500 Hz. 174 Figure 4.77: Distribution of mode number in third-octave bands for ASTM E90 reverberation rooms. Figure 4.78: Distribution of mode number in octave bands for ASTM E90 reverberation rooms. Spatial uniformity of SPL: The SPL deviations, as a measure of spatial variation, in third-octave bands for different ASTM E90 reverberation rooms are shown in Figure 4.79. It is noticeable that, the deviations are less than, or around, 1.5 dB from the 100-Hz third-octave band for all reverberation room types with the exception for the rectangular room with the longest x-dimension. In particular, the oblique room with the longest x-dimension shows deviations less than 1.5 dB from the 100-Hz band. Both of the oblique-shaped rooms show a relatively smaller deviations compared to the other room types, disregarding some exceptions. In the case of octave-band prediction, the deviations are smaller than 1.5 dB from the 125-Hz octave band for all reverberation-room types, as shown in Figure 4.80. 175 Figure 4.79: SPL standard deviation in third-octave bands for ASTM E90 reverberation rooms. Figure 4.80: SPL standard deviation in octave bands for ASTM E90 reverberation rooms. Accuracy of the predicted SPLs: The predicted SPLs and the corresponding SPL-prediction errors, with respect to the Sabine diffuse-field steady-state SPL formula, in third-octave bands are shown in Figure 4.81 and Figure 4.82, respectively. It is obvious that for all types of reverberation rooms, the SPLs can be predicted assuming a diffuse sound field with errors less than, 1% from the 125-Hz third-octave band. In particular, the oblique room with the longest x-dimension shows errors less than 1% from the 80-Hz band. 176 Figure 4.81: Predicted SPL in third-octave bands for ASTM E90 reverberation rooms. Figure 4.82: SPL-prediction error in third-octave bands for ASTM E90 reverberation rooms. On the other hand, the octave-band prediction of the SPLs and the corresponding SPL-prediction errors are shown in Figure 4.83 and Figure 4.84, respectively. It is obvious that for all reverberation-room types, the errors are less than 1% from the 125-Hz octave band. Figure 4.83: Predicted SPL in octave bands for ASTM E90 reverberation rooms. 177 Figure 4.84: SPL-prediction error in different octave-bands for ASTM E90 reverberation rooms. Spatial uniformity of RT: The RT deviations, as a measure of spatial variation, in third-octave bands for different ASTM E90 reverberation rooms are shown in Figure 4.85. It is noticeable that the deviations are less than 0.5 s from the 125-Hz third-octave band for all reverberation-room types disregarding some exceptions. In particular, the oblique room with the longest x-dimension shows deviations below 0.5 s from the 63-Hz third-octave band. In the case of octave-band prediction, the SPL deviations are smaller than 0.5 s from the 125-Hz octave band for all reverberation rooms, as shown in Figure 4.86. It is also noticeable that the deviations are smaller in octave bands than in third-octave bands. Figure 4.85: RT standard deviation in third-octave bands for ASTM E90 reverberation rooms. 178 Figure 4.86: RT standard deviation in octave bands for ASTM E90 reverberation rooms. Accuracy of the predicted RTs: The predicted RTs and the corresponding RT-prediction errors, with respect to the Sabine diffuse-field reverberation-time formula, in third-octave bands are shown in Figure 4.87 and Figure 4.88, respectively. It is obvious that, for all of the room types, the RTs can be predicted assuming a diffuse sound field with an error less than, or around, 25% from the 160-Hz third-octave band. Figure 4.87: Predicted RT in third-octave-bands for ASTM E90 reverberation rooms. Figure 4.88: RT-prediction error in third-octave bands for ASTM E90 reverberation rooms. 179 On the other hand, the octave-band prediction of the RTs and the corresponding RT-prediction errors are shown in Figure 4.89 and Figure 4.90, respectively. It is obvious that, for all of the room types, the errors are less than, or around 25% from the 125-Hz octave band. Figure 4.89: Predicted RT in octave bands for ASTM E90 reverberation rooms. Figure 4.90: RT-prediction error in octave bands for ASTM E90 reverberation rooms. Linearity of temporal decay curves: The curvatures, as a measure of the degree of non-linearity of the temporal energy-decay curves, in third-octave bands for different ASTM E90 reverberation rooms are shown in Figure 4.91. It is noticeable that the curvatures are less than, or around, 10 from the 100-Hz third-octave band for all reverberation-room types, disregarding some exceptions for the oblique reverberation room with the longest x-dimension. In particular, the oblique room with the longest z-dimension shows relatively lower non-linearity compared to the other room types. In the case of octave-band prediction, the curvatures are less than 8 from the 125-Hz octave band for all reverberation-room types, as shown in Figure 4.92. 180 Figure 4.91: Curvature of decay-curve in third-octave bands for ASTM E90 reverberation rooms. Figure 4.92: Curvature of decay-curve in octave bands for ASTM E90 reverberation rooms. Accuracy of the predicted source-power levels: The predicted source-power levels and the corresponding source-power-level prediction errors, with respect to the Sabine diffuse-field prediction, in third-octave-bands are shown in Figure 4.93 and Figure 4.94, respectively. It is obvious that, for the oblique room with the longest z-dimension, the source-power levels can be predicted assuming a diffuse sound field with an error less than, or around, 16% from the 100-Hz third-octave band. 181 Figure 4.93: Predicted source-power level in third-octaves for ASTM E90 reverberation rooms. Figure 4.94: Power-level prediction error in third-octaves for ASTM E90 reverberation rooms. On the other hand, the octave-band prediction of source-power levels and the corresponding source-power-level prediction errors are shown in Figure 4.95 and Figure 4.96, respectively. It is obvious that, for the oblique room with the longest z-dimension, the errors are less than 16% from the 125-Hz octave band. Figure 4.95: Predicted source-power level in octave-bands for ASTM E90 reverberation rooms. 182 Figure 4.96: Power-level prediction error in octave bands for ASTM E90 reverberation rooms. Accuracy of the room-averaged absorption coefficients: The Sabine prediction of room-averaged absorption coefficients and the corresponding prediction-errors done in the ‘without absorber’ case, in third-octave bands, are shown in Figure 4.97 and Figure 98. It is obvious that at higher frequency bands, the errors are lower than in the low-frequency bands. The same is also true for octave-band prediction of room-averaged absorption coefficients and the corresponding prediction-errors, as presented in Figure 4.99 and Figure 4.100. Figure 4.97: Sabine-predicted room-averaged absorption coefficients in third-octave bands for ASTM E90 reverberation rooms. 183 Figure 4.98: Sabine-prediction error for the room-averaged absorption coefficients in third-octave bands for ASTM E90 reverberation rooms. Figure 4.99: Sabine-predicted room-averaged absorption coefficients in octave bands for ASTM E90 reverberation rooms. Figure 4.100: Sabine-prediction error for room-averaged absorption coefficients in octave bands for ASTM E90 reverberation rooms. 184 Accuracy of the sample absorption coefficients: The predicted absorption coefficients of a sample absorber calculated using the Sabine diffuse-field formula, and the corresponding prediction errors, with respect to the theoretical random absorption coefficients, in third-octave bands, are shown in Figure 4.101 and 4.102, respectively. It is noticeable that the errors are greater (compared to the theoretical value of 0.76) in almost all third-octave bands – in particular, the errors are excessive below the 200-Hz third-octave band. However, it seems the oblique room with the longest x-dimension gives less error compared to the other reverberation rooms. Figure 4.101: Sabine-predicted absorption coefficients of a sample absorber in third-octave bands for ASTM E90 reverberation rooms. Figure 4.102: Sabine-prediction error for the absorption coefficients of sample absorber in third-octave bands for ASTM E90 reverberation rooms. The predicted absorption coefficients of a sample absorber calculated using the Eyring diffuse-field formula, and the corresponding prediction-error with respect to the theoretical random absorption coefficients, in third-octave bands, are shown in Figure 4.103 and 4.104, respectively. 185 It is noticeable that the prediction-errors are higher (compared to the theoretical value of 0.76) in almost all third-octave bands –they are even greater than the Sabine predictions. However, it seems the oblique room with the longest x-dimension gives less error compared to the other reverberation rooms. Figure 4.103: Eyring-predicted absorption coefficients of a sample absorber in third-octave bands for ASTM E90 reverberation rooms. Figure 4.104: Eyring-prediction error for the absorption coefficients of sample absorber in third-octave bands for ASTM E90 reverberation rooms. On the other hand, the octave-band prediction of absorption coefficients of a sample absorber calculated using the Sabine formula, and the corresponding prediction-errors are shown in Figure 4.105 and Figure 4.106, respectively. It is obvious that, the errors are excessive below the 125-Hz band for all reverberation-room types. 186 Figure 4.105: Sabine-predicted absorption coefficients of a sample absorber in octave bands for ASTM E90 reverberation rooms. Figure 4.106: Sabine-prediction error for the absorption coefficients of sample absorber in octave-bands for ASTM E90 reverberation rooms. The octave-band prediction of absorption coefficients of a sample absorber calculated using the Eyring formula, and the corresponding prediction-errors are shown in Figure 4.105 and Figure 4.106, respectively. It is obvious that the errors are greater than in the Sabine predictions. Figure 4.107: Eyring-predicted absorption coefficients of a sample absorber in octave bands for ASTM E90 reverberation rooms. 187 Figure 4.108: Eyring-prediction error for the absorption coefficients of sample absorber in octave bands for ASTM E90 reverberation rooms. The large errors associated with the absorption-coefficient predictions imply that the sound field is not sufficiently diffuse. Comparing the test absorber absorption coefficients to the room-averaged absorption coefficients makes it clear that, there is less error in the room-averaged absorption coefficients than those predicted for a test absorber in the ‘with absorber’ case. It can be inferred from the above discussion that the presence of the test absorber makes the field less diffuse than it was without the test absorber. This can also be understood from the higher RT and SPL standard-deviation predictions, and the higher curvatures for the decay curves, which are mentioned in the previous sections. Based on the discussion presented above, it can be concluded that the oblique-shaped ASTM E90 reverberation rooms with the longest z-dimension performs better than the other reverberation rooms. In the case of third-octave-band predictions, prediction can be done accurately from the 160-Hz third-octave band; it will yield an RT-prediction error of less than 25%, an SPL-prediction error of less than 1%, an RT standard deviation of less than 0.5 s, an SPL standard deviation of less than 1.5 dB, a curvature of the temporal decay curve of less than 10, a source-power-level prediction error of less than 16%, and relatively better predictions of the absorption coefficients than the other rooms. On the other hand, in the case of octave-band predictions, prediction can be done accurately from the 125-Hz octave band; it will yield an RT-prediction error of less than 25%, an SPL-prediction error of less than 1%, an RT standard deviation of less than 0.5 s, an SPL standard deviation of less than 1.5 dB, a curvature of the temporal decay curve of less than 8, a source-power-level prediction error of less than 16%, and relatively better predictions of the absorption coefficients than the other rooms. The deviations 188 become smaller in the higher frequency bands. It is also evident that prediction in octave bands gives relatively better predictions than in third-octave bands. 4.3 Summary Based on the above discussion, the best possible performance statistics by the ISO 354 reverberation room with the longest x-dimension is presented in Table 4.2 in terms of different quantitative parameters/descriptors. The same is also done for the ASTM C423 reverberation room with the longest x-dimension and ASTM E90 reverberation room with the longest z-dimension, as presented in Table 4.3 and Table 4.4 respectively. Finally, it can be concluded the following points: Table 4.2: Performance summary for the ISO 354 reverberation room with the longest x-dimension. Third-octave-band prediction Octave-band prediction Cut-off frequency (Hz) 100 125 SPL deviation (dB) 1.5 1.5 SPL accuracy (%) 1 1 RT deviation (s) 0.5 0.5 RT accuracy (%) 15 12 Curvature of energy-decay curve 15 12 Source-power-level accuracy (%) 12 10 Absorption-coefficient accuracy (%) 45 45 Table 4.3: Performance summary for the ASTM C423 reverberation room with the longest x-dimension. Third-octave-band prediction Octave-band prediction Cut-off frequency (Hz) 125 125 SPL deviation (dB) 1.5 1.5 SPL accuracy (%) 1.2 1.2 RT deviation (s) 0.5 0.5 RT accuracy (%) 15 10 Curvature of energy-decay curve 12 5 Source-power-level accuracy (%) 10 10 Absorption-coefficient accuracy (%) 48 55 189 Table 4.4: Performance summary for the ASTM E90 reverberation room with the longest z-dimension. Third-octave-band prediction Octave-band prediction Cut-off frequency (Hz) 160 125 SPL deviation (dB) 1.5 1.5 SPL accuracy (%) 1 1 RT deviation (s) 0.5 0.5 RT accuracy (%) 25 25 Curvature of energy-decay curve 10 8 Source-power-level accuracy (%) 16 16 Absorption-coefficient accuracy (%) 40 60  The ISO 354 reverberation rooms of volume 150 m3 give better field diffuseness/prediction accuracy than the two other room volumes (ASTM C423 of 125 m3 and ASTM E90 of 82 m3) for different room shapes.  Comparing results for different reverberation-room shapes make it clear that the oblique-shaped room with the longest x-dimension/shortest vertical dimension gives better prediction accuracy than the other room shapes for different room volumes.  Overall, the ISO 354 reverberation room with the longest x-dimension/shortest vertical dimension yields better field diffuseness and, hence, better prediction accuracy than the other reverberation rooms of different shapes and volumes.  Predicted SPLs show less uncertainty to the expected diffuse-field values.  Predicted RTs show greater uncertainty to the expected diffuse-field values than the SPL predictions.  Sound-field diffuseness is high in the ‘without absorber’ case than in the ‘with absorber’ case.  Octave-band prediction accuracy is slightly higher than in the third-octave-band prediction.  Sabine’s diffuse-field formula yield better prediction accuracy than the Eyring formula.  Analyzing all of the prediction cases, it is obvious that the Schroder cut-off limit is very restrictive, while statistically-based cut-off limits are helpful from the application point of view. 190  It is also notable that the minimum cut-off limits prescribed by standards – i.e. 125 Hz (160-Hz third-octave band) according to ASTM E90, 100 Hz (125-Hz third-octave band) according to ASTM C423, etc. – do not conform to our findings. At the same time, standards do not mention about the accuracy of the room-acoustical parameters that can be achieved at their prescribed cut-off limits. According to our findings, predictions can be done from the 160-Hz third-octave band for ASTM E90, 125-Hz third-octave band for ASTM C423 and 100-Hz third-octave band for ISO 354 – with the given room shape and with the expected accuracy of different room-acoustical parameters as presented in Table 4.4, Table 4.3 and Table 4.2, respectively. Based on this conclusion, in the next chapter, we will continue our study from the oblique-shaped ISO 354 reverberation room with the longest x-dimension. We will add some features – i.e. absorbent corner treatments, diffuse surface reflection, etc. – to improve the field diffuseness. 191 Chapter 5 Alternative Reverberation-Room Configurations and their Effectiveness In this chapter, based on the conclusions made in the previous chapter, we will model some other reverberation-room configurations, and then compare these results to those obtained in the previous chapter. In the previous chapter (Chapter 4), it was concluded that reverberation rooms of volume 150 m3, oblique-shaped with the typical dimensional orientation (shortest dimension in the vertical direction/longest x-dimension), and equipped with a number of diffusing panels hanging from the ceilings, give better sound-field diffuseness than the other reverberation-room types of different volumes and shapes. In this chapter, we start from these findings, and extend our study to the following cases:  Oblique-shaped room of volume 150 m3 (empty)  Oblique-shaped room of volume 150 m3 (empty) with absorbent corner treatments  Oblique-shaped room of volume 150 m3 with absorbent corner treatments and diffusers  Oblique-shaped room of volume 150 m3 (empty) with diffuse surface reflection  Oblique-shaped room of volume 150 m3 with diffuse surface reflection and diffusers. In Chapter 4, it had been observed that in the case of octave-band prediction, there was not much accuracy relative to third-octave prediction. That’s why, in this chapter, we will make prediction only in the third-octave bands. At the same time, as we are dealing with the same volume (150 m3) as for ISO reverberation rooms, we will not predict the modal compositions, which will be the same as predicted in Chapter 4 for ISO reverberation rooms. In Chapter 4, it was also observed that the prediction accuracy with the Sabine’s diffuse field formula is better than with the Eyring’s formula. Because of this, in this chapter we will use only Sabine’s formula to make predictions. In all prediction cases, 5 receivers and 1 source have been used. All other testing arrangements, as well as the absorption properties, are kept the same as in Chapter 4. 192 5.1 Oblique-Shaped, Empty Reverberation Room In last chapter, it was concluded that the oblique-shaped room with the shortest vertical dimension/ longest x-dimension equipped with diffusers yields better field diffuseness. Here, we will do a bit modification by considering only the empty room, but no diffusers, to check the effect on the field diffuseness. The reverberation room is shown in Figure 5.1. The predictions made in third-octave bands using this room are presented in Tables 5.1 and 5.2 for the cases ‘without’ and ‘with’ the test absorber, respectively. Figure 5.1: Oblique-shaped, empty reverberation-room setup. From Table 5.1, it can be observed that the SPL deviations are less than 1.5 dB from the 160-Hz third-octave band, and the SPL-prediction errors are smaller than 1.2% from the same band. On the other hand, the RT deviations are smaller than 0.75 s from the 80-Hz band, and the RT-prediction errors are smaller than 10% from the same band, disregarding the exception of the 160-Hz band. The source-power-level prediction errors are smaller than 10% from the 80-Hz band. The same is also true for the room-averaged absorption coefficients, disregarding the exception for the 160-Hz band. 193 Table 5.1: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty), without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 137.3 141.3 143.6 149.9 150.8 157.1 161.3 163.6 167.2 Standard deviation, 𝜎𝐿𝑝 (dB) 6.82 1.52 3.62 5.23 1.89 0.76 0.75 1.33 0.50 Sabine error (%) 1.7 1.6 1.5 1.4 1.3 1.2 1.0 0.8 0.5 Predicted average RT: ?̅?60 (s) 7.97 7.13 6.74 6.30 6.08 5.70 5.87 5.85 6.31 Standard deviation, 𝜎𝑇60 (s) 1.45 0.06 0.73 0.53 0.15 0.73 0.55 0.73 0.31 Sabine error (%) 23 10 4 3 6 12 9 10 3 Average, 𝜅 4.5 7.3 17.5 11.5 11.5 12.7 6.2 13.3 7.5 Predicted, ?̅?𝑤 (dB) 139.4 143.4 145.5 151.7 152.4 158.7 162.6 164.7 167.8 Sabine error (%) 13 6 3 2 5 9 7 7 2 Average absorption, 𝐴 (m2) 3.07 3.43 3.63 3.88 4.02 4.29 4.17 4.18 3.88 Sabine error (%) 19 9 4 3 7 14 10 11 3 Room absorption coeff, 𝛼𝑟𝑒𝑣 0.016 0.018 0.019 0.021 0.021 0.023 0.022 0.022 0.021 Sabine error (%) 19 9 4 3 7 14 10 11 3 From Table 5.2, it can be observed that the SPL deviations, as well as the RT deviations, are greater than in the ‘without test absorber’ case. The curvatures for the energy-decay curves also show the same trend. Table 5.2: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty), with test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 132.4 134.6 139.0 146.0 146.4 152.6 157.0 158.7 162.7 Standard deviation, 𝜎𝐿𝑝 (dB) 3.92 1.80 4.19 3.64 2.37 0.87 1.72 1.69 1.24 Predicted average RT: ?̅?60 (s) 3.81 3.15 3.58 3.55 2.68 2.71 2.91 2.72 2.94 Standard deviation, 𝜎𝑇60 (s) 0.77 0.52 0.84 0.94 0.65 0.23 0.21 0.24 0.41 Average, 𝜅 99.4 89.7 27.8 27.9 14.0 12.5 9.2 17.9 99.4 Specimen absorption coeff, 𝛼𝑟𝑒𝑣 0.31 0.40 0.30 0.28 0.47 0.44 0.39 0.45 0.41 Sabine error (%) 59 47 61 63 38 42 48 41 46 194 5.2 Oblique-Shaped, Empty Reverberation Room with Absorbent Corner Treatments Here, following the suggestion of Cop et. al. [29], we will do another modification of the reverberation room by placing tetrahedral blocks at all of the four floor-adjacent corners and covering them with the same test absorber that is mounted in the mounting bed for testing, but no diffusers, to observe its effect on field diffuseness. All these treatments are performed with a hope to damping the low-frequency modes – i.e. axial and tangential modes. The reverberation room is shown in Figure 5.2. The predictions made in third-octave bands for this room are presented in Tables 5.3 and 5.4 for the cases ‘without’ and ‘with’ the test absorber, respectively. Figure 5.2: Oblique-shaped, empty reverberation-room setup with absorbent corner treatments. From Table 5.3, it is clear that the SPL deviations and the SPL-prediction errors are smaller than 1.5 dB and 1.5%, respectively, from the 125-Hz band. The RT deviations and the RT-prediction errors are smaller than 0.5 s and 10%, respectively, from the 63-Hz band, disregarding some exceptions. The source-power prediction errors are around 10% from the 63-Hz band, disregarding the exception for the 160-, 200- and 250-Hz bands. The same is almost true for the room-averaged absorption-coefficient prediction accuracy. 195 Table 5.3: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty), with absorbent corner treatments, without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 136.2 138.5 141.5 146.5 150.0 154.9 159.9 162.3 165.9 Standard deviation, 𝜎𝐿𝑝 (dB) 6.90 1.08 3.92 4.64 1.05 0.92 0.70 1.14 0.66 Sabine error (%) 1.9 2.1 1.8 1.4 1.4 1.5 1.2 1.0 0.6 Predicted average RT: ?̅?60 (s) 6.03 4.93 4.79 5.03 4.61 4.10 4.28 4.36 4.93 Standard deviation, 𝜎𝑇60 (s) 1.84 0.48 0.81 1.02 0.40 0.15 0.13 0.15 0.23 Sabine error (%) 18 4 7 2 10 20 16 15 4 Average, 𝜅 44 17 19 10 13 7 5 5 9 Predicted, ?̅?𝑤 (dB) 139.4 142.1 144.8 149.2 152.8 157.9 162.4 164.5 167.5 Sabine error (%) 13 4 6 2 10 21 17 15 3 Average absorption, 𝐴 (m2) 3.98 4.87 5.01 4.77 5.21 5.85 5.60 5.50 4.87 Sabine error (%) 15 4 7 2 11 25 20 17 4 Room absorption coeff, 𝛼𝑟𝑒𝑣 0.021 0.026 0.027 0.025 0.028 0.031 0.030 0.029 0.026 Sabine error (%) 15 4 7 2 11 25 20 17 4 From Table 5.4, it can be said that the prediction accuracy of the absorption coefficients for the test absorber increased a bit in most of the third-octave bands compared to the previous ‘without absorbent corner treatment’ case. Table 5.4: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty), with absorbent corner treatments, with test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 131.9 134.0 137.3 144.3 146.0 150.6 156.0 157.8 162.0 Standard deviation, 𝜎𝐿𝑝 (dB) 4.25 1.45 4.57 2.69 1.69 0.85 1.10 1.71 1.42 Predicted average RT: ?̅?60 (s) 4.08 2.85 2.88 2.24 2.94 2.05 2.04 2.21 2.96 Standard deviation, 𝜎𝑇60 (s) 1.32 0.82 1.34 0.25 0.31 0.16 0.15 0.29 0.39 Average, 𝜅 216 138 70 39 33 30 26 14 74 Specimen absorption coeff, 𝛼𝑟𝑒𝑣 0.18 0.35 0.32 0.58 0.29 0.57 0.60 0.52 0.32 Sabine error (%) 76 54 57 24 62 25 21 31 58 196 5.3 Oblique-Shaped Reverberation Room with Diffusers and Absorbent Corner Treatments In this case, we integrated both absorbent corner treatments and diffusers to the reverberation room. The reverberation room is shown in Figure 5.3. The predictions made in third-octave bands for this room are presented in Tables 5.5 and 5.6 for the cases ‘without’ and ‘with’ the test absorber, respectively. Figure 5.3: Oblique-shaped reverberation-room setup with diffusers and absorbent corner treatments. From Table 5.5, it is observed that the SPL deviations and the SPL-prediction errors are smaller than 1.5 dB and 1.6%, respectively, from the 125-Hz third-octave band. The RT deviations and the RT-prediction errors are smaller than 0.42 s and 10%, respectively, from the 125-Hz band. The same is also true for the room–averaged absorption-coefficient prediction accuracy. The absorption coefficients of the test absorber are presented in Table 5.6. It is noticeable that there is a slight improvement in the prediction accuracy than in the previous ‘with corner treatments but no diffuser’ case. 197 Table 5.5: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 with diffusers and absorbent corner treatments, without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 137.0 139.6 141.4 146.5 149.7 155.1 159.2 162.0 164.2 Standard deviation, 𝜎𝐿𝑝 (dB) 3.65 3.24 2.01 3.21 1.39 1.24 1.33 0.46 0.94 Sabine error (%) 2.6 2.2 1.8 1.3 1.6 1.1 1.0 1.0 0.4 Predicted average RT: ?̅?60 (s) 4.41 4.32 4.34 4.59 3.83 4.05 3.90 3.72 4.45 Standard deviation, 𝜎𝑇60 (s) 0.66 0.52 0.66 0.87 0.18 0.10 0.31 0.24 0.42 Sabine error (%) 9 7 7 14 5 1 3 8 10 Average, 𝜅 34 4 8 19 13 8 7 6 10 Predicted, ?̅?𝑤 (dB) 142.2 144.3 145.6 150.0 153.6 158.4 162.4 165.2 166.4 Sabine error (%) 9 7 8 13 6 1 4 9 10 Average absorption, 𝐴 (m2) 5.38 5.48 5.47 5.16 6.20 5.85 6.08 6.38 5.33 Sabine error (%) 8 7 7 12 5 1 3 9 9 Room absorption coeff, 𝛼𝑟𝑒𝑣 0.022 0.022 0.022 0.021 0.025 0.024 0.025 0.026 0.022 Sabine error (%) 8 7 7 12 5 1 3 9 9 Table 5.6: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 with diffusers and absorbent corner treatments, with test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 131.7 134.0 138.3 144.4 144.6 150.5 156.2 158.2 161.1 Standard deviation, 𝜎𝐿𝑝 (dB) 2.44 3.59 3.95 4.45 1.89 2.78 1.86 0.42 0.97 Predicted average RT: ?̅?60 (s) 3.74 2.55 2.44 2.40 2.26 1.96 2.01 1.87 2.48 Standard deviation, 𝜎𝑇60 (s) 2.71 1.66 1.71 0.83 0.75 0.42 0.28 0.22 0.94 Average, 𝜅 126 149 93 16 42 30 35 30 68 Specimen absorption coeff, 𝛼𝑟𝑒𝑣 0.09 0.37 0.42 0.46 0.42 0.61 0.56 0.62 0.41 Sabine error (%) 88 51 45 39 45 20 26 18 46 198 5.4 Oblique-Shaped, Empty Reverberation Room with Diffuse Surface Reflection In this section, we modify the reverberation room by installing QRDs to provide diffuse surface reflection, as diffuse reflection helps to enhance field diffuseness, and considering no diffusers. The reverberation room is shown in Figure 5.4. The predictions made in third-octave bands for this room are presented in Tables 5.7 and 5.8 for the cases ‘without’ and ‘with’ the test absorber, respectively. Figure 5.4: Oblique-shaped, empty reverberation-room setup with diffuse surface reflection. To achieve diffuse reflection from the surfaces, QRDs are installed on all surfaces other than the floor. An array of 17×17 wells of varying depths is used as a period, and only one period is installed per surface to obtain diffuse reflection. Though, it is recommended to have a number of periods with large period width to achieve optimal diffuseness, due to computational restrictions we used only one period, instead of a series, with greater period width. It can be seen from Table 5.7 that the SPL deviations are smaller than 1.5 dB from the 160-Hz band, and the SPL-prediction errors are smaller than 1.1% from the 125-Hz band. The RT-prediction errors are greater than those obtained in the previous cases. The same is also true for the source-power level, as well as for the room-averaged absorption coefficient, prediction accuracy. 199 Table 5.7: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty) with diffuse surface reflection, without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 141.0 133.8 146.7 145.5 149.3 152.6 157.6 162.4 164.3 Standard deviation, 𝜎𝐿𝑝 (dB) 0.96 1.57 6.84 1.12 3.12 0.80 0.94 1.61 0.58 Sabine error (%) 1.3 1.5 1.3 1.4 0.9 1.1 0.7 0.4 0.1 Predicted average RT: ?̅?60 (s) 6.76 5.54 4.80 4.09 4.29 3.60 3.75 3.88 4.30 Standard deviation, 𝜎𝑇60 (s) 2.30 1.33 0.23 0.48 0.76 0.36 0.14 0.32 0.66 Sabine error (%) 146 102 75 49 56 31 36 41 56 Average, 𝜅 9.7 45.1 3.0 33.0 30.3 13.5 9.9 4.6 16.4 Predicted, ?̅?𝑤 (dB) 146.1 139.1 151.9 150.9 154.0 157.5 161.9 166.3 167.4 Sabine error (%) 94 80 68 52 57 37 42 46 57 Average absorption, 𝐴 (m2) 3.66 4.46 5.15 6.06 5.77 6.88 6.60 6.38 5.76 Sabine error (%) 59 50 43 33 36 24 27 29 36 Room absorption coeff, 𝛼𝑟𝑒𝑣 0.008 0.010 0.011 0.013 0.013 0.015 0.015 0.014 0.013 Sabine error (%) 59 50 43 33 36 24 27 29 36 The prediction errors for the absorption coefficients of the test absorber are presented in Table 5.8. It is obvious that the prediction accuracy is not as good as those obtained in the previous cases. Table 5.8: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 (empty) with diffuse surface reflection, with test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 139.1 131.5 141.6 141.4 147.0 151.4 154.1 158.7 161.0 Standard deviation, 𝜎𝐿𝑝 (dB) 2.24 3.02 6.45 1.90 2.66 2.11 1.33 1.26 1.32 Predicted average RT: ?̅?60 (s) 5.47 4.68 2.19 2.50 2.87 2.07 2.08 2.31 2.63 Standard deviation, 𝜎𝑇60 (s) 4.54 3.17 0.16 0.62 0.38 0.18 0.24 0.17 1.11 Average, 𝜅 50.9 136.8 50.0 54.2 44.3 11.4 6.2 11.2 62.7 Specimen absorption coeff, 𝛼𝑟𝑒𝑣 0.11 0.11 0.82 0.51 0.38 0.67 0.70 0.57 0.48 Sabine error (%) 85 86 7 33 51 12 8 24 37 200 5.5 Oblique-Shaped Reverberation Room with Diffusers and Diffuse Surface Reflection Here, we modify the reverberation room by considering both diffusers and QRD to observe their effect on field diffuseness. In The reverberation room is shown in Figure 5.5. The predictions made in third-octave bands using this room are presented in Tables 5.9 and 5.10 for the cases ‘without’ and ‘with’ the test absorber, respectively. Figure 5.5: Oblique-shaped reverberation-room setup with diffusers and diffuse surface reflection. The same QRD configurations are used here as used in the previous case. Taking a look into the RT-prediction accuracy, as well as the source-power level and room-averaged absorption-coefficient prediction accuracy, it is quite obvious that the prediction accuracy is poorer than the previous prediction cases. The SPL deviations are also higher than in the previous cases. The prediction accuracy of the absorption coefficients of a test absorber, as presented in Table 5.10, also reveals that they are not good as that obtained in the ‘without diffuse surface reflection’ case. The higher curvatures of the temporal decay curves also reveal the non-diffuse nature of the sound field. 201 Table 5.9: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 with diffusers and diffuse surface reflection, without test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 137.5 133.5 147.3 145.1 150.0 153.3 158.5 162.6 164.4 Standard deviation, 𝜎𝐿𝑝 (dB) 4.18 2.38 8.26 2.28 2.97 2.59 1.16 1.53 0.88 Sabine error (%) 1.6 0.5 1.0 1.7 1.1 1.0 0.6 0.4 0.2 Predicted average RT: ?̅?60 (s) 6.12 7.35 5.16 3.63 3.87 3.54 3.57 3.56 4.18 Standard deviation, 𝜎𝑇60 (s) 2.25 4.04 0.22 0.51 0.57 0.60 0.22 0.29 0.96 Sabine error (%) 149 200 110 48 58 44 46 45 70 Average, 𝜅 44.9 105.0 2.0 14.3 36.8 18.2 7.8 7.6 21.9 Predicted, ?̅?𝑤 (dB) 143.3 137.9 152.5 151.2 155.3 158.5 163.3 167.0 167.7 Sabine error (%) 106 118 93 57 65 54 55 55 73 Average absorption, 𝐴 (m2) 4.01 3.34 4.75 6.75 6.34 6.93 6.86 6.89 5.86 Sabine error (%) 60 67 52 32 37 31 31 31 41 Room absorption coeff, 𝛼𝑟𝑒𝑣 0.008 0.007 0.009 0.013 0.013 0.014 0.014 0.014 0.012 Sabine error (%) 60 67 52 32 37 31 31 31 41 Table 5.10: Third-octave-band predictions for an oblique-shaped reverberation room of volume 150 m3 with diffusers and diffuse surface reflection, with test absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted average SPL, ?̅?𝑝 (dB) 134.8 131.1 143.1 140.9 148.0 151.8 155.4 159.9 161.0 Standard deviation, 𝜎𝐿𝑝 (dB) 4.61 2.38 8.27 2.82 4.10 2.11 2.05 1.24 1.11 Predicted average RT: ?̅?60 (s) 4.31 4.68 2.63 2.72 2.47 1.95 1.91 2.16 2.61 Standard deviation, 𝜎𝑇60 (s) 5.89 4.72 0.92 1.10 0.45 0.34 0.12 0.21 0.93 Average, 𝜅 110 113 14 84 34 28 15 13 47 Specimen absorption coeff, 𝛼𝑟𝑒𝑣 0.22 0.25 0.61 0.30 0.47 0.74 0.79 0.59 0.47 Sabine error (%) 71 67 20 61 38 2 4 23 39 202 5.6 Discussion The SPL deviations predicted in third-octave bands for different reverberation-room configurations are presented in Figure 5.6. It is obvious that the deviations are smaller than the ISO-prescribed limit of 1.5 dB from the 160-Hz third-octave bands for all of the reverberation-room configurations except those equipped with the QRDs. Overall, it is also noticeable that the room equipped with diffusing panels hanging from the ceiling and absorbent corner treatments yields a smaller band-averaged deviation than the other room configurations. Figure 5.6: SPL deviations in third-octave bands for different reverberation-room configurations. The SPL-prediction errors in third-octaves for different reverberation-room configurations are presented in Figure 5.7. It is obvious that the errors are smaller than 1% from the 160-Hz third-octave bands for all of the reverberation-room configurations. Figure 5.7: SPL-prediction errors in third-octave bands for different reverberation-room configurations. 203 The RT deviations predicted in third-octave bands for different reverberation-room configurations are presented in Figure 5.8. It is obvious that the deviations are smaller than 0.5 s from the 160-Hz third-octave band for the room equipped with either diffusers or absorbent corner treatments, or both. However, the same is not true for the empty room and the room equipped with QRDs. Figure 5.8: RT deviations in third-octave bands for different reverberation-room configurations. The RT-prediction errors in third-octaves for different reverberation-room configurations are presented in Figure 5.9. It is obvious that the errors are smaller than 10% from the 125-Hz third-octave bands for the room equipped with diffusers and absorbent corner treatments. However, the same is not true for the room equipped with QRDs; rather they show greater errors than the other room configurations. Figure 5.9: RT-prediction errors in third-octave bands for different reverberation-room configurations. 204 The curvatures for the temporal-decay curves in third-octave bands for different reverberation-room configurations are presented in Figure 5.10. It is noticeable that the curvatures are smaller than 10 for the room equipped with diffusers and absorbent corner treatments from the 125-Hz third-octave band. The curvatures are a bit higher for the room equipped with QRDs. Figure 5.10: Curvatures of temporal-decay curves in third-octave bands for different reverberation-room configurations. The source-power-level prediction errors in third-octave bands for different reverberation-room configurations are presented in Figure 5.11. It is noticeable that the errors are smaller than 10% from the 125-Hz band for the empty room and the room with diffusers and absorbent corner treatments. On the other hand, prediction errors are greater for the rooms equipped with QRDs. It can also be observed that the prediction errors are much smaller for the empty reverberation room (no diffusers and absorbent corner treatments). Figure 5.11: Source-power-level prediction errors in third-octave bands for different reverberation-room configurations. 205 The room-averaged absorption coefficients predicted without the test absorber in third-octave bands for different reverberation-room configurations are presented in Figure 5.12. It is obvious that the predicted values are closer to the expected value of 0.02 - 0.025 for the room with both diffusers and absorbent corner treatments. The rooms equipped with QRDs yield less accurate results, as is clear from Figure 5.12. Figure 5.12: Room-averaged absorption coefficients in third-octave bands for different reverberation-room configurations. The predicted absorption coefficients of a test absorber in third-octave bands for different reverberation-room configurations are presented in Figure 5.13. It is noticeable that the predicted values are closer to the expected value of 0.76 for the room with absorbent corner treatments and the room with both diffusers and absorbent corner treatments. Though the prediction accuracies in some frequency bands for the rooms equipped with QRDs are good, overall band-averaged accuracy is better for the above-mentioned two other room configurations. Figure 5.13: Test absorber absorption coefficients in third-octave bands for different reverberation-room configurations. 206 It is evident that the accuracies of the room-averaged absorption coefficients predicted without the test absorber is better than the accuracy of the absorption coefficients predicted for the test absorber. This is because of the reduced field diffuseness which arises from the presence of the highly-absorptive test absorber. The greater RT and SPL deviations and the higher curvatures predicted in the presence of the test absorber, as presented earlier of this chapter, also support this claim. Based on the results discussed above in terms of different descriptors, it is clear that reverberation rooms equipped either with diffusers or absorbent corner treatments, or both, yield better field diffuseness than that with QRDs. It was also observed that the empty room with no diffusers and absorbent corner treatments, yields good prediction accuracy for the source-power level. On the other hand, the prediction accuracy of the absorption coefficients of the test absorber is better for the room equipped with both diffusers and absorbent corner treatments than for the other room configurations. In all of the cases, the rooms equipped with QRD yields less accurate results; this can be attributed to the increased surface absorption, which results from a significant increase in surface area (due to the installation of QRD). 207 Chapter 6 Conclusion 6.1 Contributions The main objective of this work was to investigate the optimal design of a reverberation-room configuration and testing setup to achieving a diffuse sound field in order to ensure the applicability of diffuse-field theories and, hence, sufficient prediction accuracy using the reverberation-room methods. Based on these objectives, the main contributions/accomplishments of this work can be summarized as: In our work, we started by delving into the concept of the diffuse sound field, and the theories that evolved based on this concept – in particular, the Sabine and Eyring formulae. The diffuse-sound-field concept was discussed with respect to both the theoretical and application points of view; then a mathematical formulation of the Sabine and Eyring theories that is based on this concept was presented. Next we studied techniques that can be used to predict the sound field in an enclosure like a reverberation room. Out of a number of available techniques, the phase-based wave approach (modal approach) was adopted in this study, due to a number of advantages over other techniques, and these were also discussed. Complete mathematical formulations for both of the two forms of the modal approach – theoretical/analytical form and numerical form – were discussed in great detail. While developing the mathematical formulations, a frequency-invariant, finite, real/complex-valued impedance was applied to the surfaces as boundary conditions to account for the real-life effect. At the same time, the concepts of locally-reacting and extended-reaction surfaces were discussed, and why the locally-reacting concept was used in this study. Then the concepts of modal count, modal density, modal spacing, modal width, modal overlapping, modal damping-constant/reverberation-time, modal propagation, modal decay, energy decay, etc. were discussed in detail; these are essential to understand the characteristics of a sound field in a reverberation room from a diffuse-field point of view. 208 Because of the lack of clear and direct ways to characterize a diffuse field, a number of quantitative parameters – like cut-off/transition-frequency definition to quantify the frequential limit of a diffuse sound field, reverberant SPL deviations to quantify the spatial variation of sound energy, decay-rate/reverberation-time deviations to quantify the constancy of energy decay, curvatures of temporal-decay curves to check the non-linearity of decay process, accuracy of the predicted steady-state SPLs and reverberation times with respect to the diffuse-field theories to check the quality of the field diffuseness and, finally, the prediction accuracy of source-power levels and absorption coefficients – were considered as descriptors in this study to characterize the degree of sound-field diffuseness in a reverberation room. In this study, a number of reverberation-room configurations and test setups were modeled with the help of the numerical finite-element-based modal approach, in order to identify the optimal configuration. While modeling reverberation rooms numerically, the QRD feature, which is numerically very expensive to model, was integrated to account for the effect of diffuse surface reflection. 6.2 Summary of Results We started by exploring three standards regarding reverberation-room methods. Regarding the commissioning of a reverberation room, three standards prescribe three different room volumes, and specify that they should be constructed of hard materials; however, no instruction was provided about the room shape. On the other hand, regarding the test setups, all standards prescribed almost the same idea – an optimal number of diffusing panels, a number of sources and receivers, asymmetric positioning of the test absorber, etc. Based on the standards prescriptions, we selected three room volumes – namely, 150 m3 from ISO 354, 125 m3 from ASTM C423 and 82 m3 from ASTM E90, and four room shapes for each of the volumes. A total of twelve room configurations were considered; each of them was equipped with a number of diffusing panels hanging from the ceilings. In all of the cases, predictions were performed with 2 sources and 5 receivers in both octave and third-octave bands, 209 and ‘without’ and ‘with’ the test absorber for each of the cases. The results can be summarize into the following point:  ISO 354 reverberation room: Among four ISO 354 reverberation rooms, the oblique-shaped room with the longest x-dimension yields better diffuseness than the other rooms.  ASTM C423 reverberation room: Among four ASTM C423 reverberation rooms, the oblique-shaped room with the longest x-dimension yields better diffuseness than the other rooms.  ASTM E90 reverberation room: Among four ASTM E90 reverberation rooms, the oblique-shaped room with the longest z-dimension yields better diffuseness than the other rooms. After comparing all the reverberation-room performance, it was noticed that the ISO reverberation rooms – in particular, the oblique-shaped with the longest x-dimension – yield better sound field diffuseness and, hence, improved prediction accuracy, than the other reverberation rooms of different volumes and shapes. It can be attributed to the greater room volume of 150 m3, as no other factors are involved here that affect the prediction accuracy. Based on the above findings, we extended this room configuration to a number of conditions – i.e. empty, empty with absorbent corner treatments, diffusers and absorbent corner treatments, empty and diffuse surface reflection, diffusers and diffuse surface reflection, etc. The findings can be summarized into the following points:  The reverberation room equipped with either diffusers or absorbent corner treatments, or both, yielded better results than the rooms with diffuse surface reflection. For absorption-coefficient prediction of a test absorber, it seems that the room with both diffusers and absorbent corner treatments yields better prediction accuracy, while the empty room with no diffusers and absorbent corner treatments yielded better accuracy for the source-power level prediction.  The poor prediction accuracy with the room equipped with diffuse surface reflection is due to the increased surface absorption caused by the installation of the QRDs, which 210 leads to a great increase in the surface area and, hence, greater absorption. Higher surface absorption affects the energy-decay process and induces a greater uncertainty in the predicted reverberation times and, consequently, affects the prediction accuracy of other acoustical parameters. Ultimately, we tried to rewrite a standard, which was one of our objectives, based on our findings. Due to our time, as well as computational, restrictions, it was not possible to explore all of the features regarding the reverberation room methods. However, a detailed exploration had been made over the size and shape of a reverberation room to achieve the better sound-field diffuseness. Based on our findings, we can make a number of recommendations to write a new standard. First, regarding room size: room of volume of 150 m3 is good one to achieve better field diffuseness. Second, regarding room shape: oblique-shaped room with typical dimensional orientation (longest x-dimension/shortest vertical dimension) is good one to achieve better field diffuseness. Third, regarding diffusing panels: room equipped with an optimum number of diffusers is good one to achieve better field diffuseness. Fourth, regarding absorbent corner treatments: room provided with low-frequency absorbers at the floor-adjacent corners seem to yield better field diffuseness. Fifth, room equipped with both diffusers and absorbent corner treatments seems to yield better field diffuseness for the prediction of surface-absorption coefficients for the above mentioned room size and shape. Sixth, room equipped with neither diffusers nor absorbent corner treatments (empty) seems to yield better field diffuseness for the prediction of source-power levels for the above mentioned room size and shape. 211 6.3 Future Recommendations Achieving a diffuse sound field in a reverberation room is an open-ended quest. There are lot of factors that need to be considered while commissioning a reverberation room to obtain sufficient field diffuseness. Because of both time and computational restrictions, we were not able to explore all the features of a reverberation room regarding its commissioning and test-setup arrangements to get the best features of a reverberation room. 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Cazzolato, Acoustic Analysis Using MATLAB and ANSYS, CRC Press, Florida, 2015. 217 Appendix: A Eigenvalue calculation using IN/GB Method function Krawczyk(f,X,L,k,Z1,Z2) %'Krawczyk.m' Finds all solutions(complex wave number) to nonlinear % eigenvalue equation for a specified coordinate direction. % Krawczyk(f,X,L,k,Z1,Z2) % % Uses the Krawczyk operator with generalised bisection to find bounds on all % solutions to a nonlinear system, given by the function f, in the box X. % % INPUT: f function of nonlinear system % X initial box % L dimension of the room % k driving wave number % Z1 specific impedance of wall 1(complex number) % Z2 specific impedance of wall 2(complex number) % OUTPUT: Wnum matrix of eigenvalues(wave number in the specified direction) % (complex number) % Eigenvalue calculation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% global i Wnum; n = length(X);% length of given interval fX=feval(f,X,L,k,Z1,Z2);%function evaluation of 'nleqn'at interval if max(mig(fX))==0 %checking that initial interval encloses the soution x = mid(X);%midpoint of given interval fx = feval(f,intval(x),L,k,Z1,Z2);%function evaluation of'nleqn' %at midpoint FX=feval(f,gradientinit(X),L,k,Z1,Z2);%gradient evaluation of'nleqn' J=FX.dx;% Jacobian matrix C=(mid(J)); % midpoint matrix of Jacobian/preconditioning matrix if det(C)==0 %checking that initial interval doesn't cause the %determinant of preconditioning matrix to be zero %Bisection of X %Coordinate direction selection M=rad(X).*max(abss(J))'; 218 [a,b]=max(M); X1=X; X2=X; X1(b) = infsup(inf(X(b)),mid(X(b)));% bisected first half Krawczyk('nleqn',X1,L,k,Z1,Z2); %calling the function X2(b) = infsup(mid(X(b)),sup(X(b)));% bisected 2nd half Krawczyk('nleqn',X2,L,k,Z1,Z2); %calling the function else K = x-C\fx+(eye(n)-C\J)*(X-x);%Krawczyk operator Xnew = intersect(K,X); %computed new interval if ~isempty(Xnew) %Checking when Xnew is not empty if ~in(K,X) %checking when Kis not in X %bisect Xnew %gradient evaluation of'nleqn' FX=feval(f,gradientinit(Xnew),L,k,Z1,Z2); J=FX.dx;% Jacobian matrix %Coordinate direction selection M=rad(Xnew).*max(abss(J))'; [a,b]=max(M); Xnew1=Xnew; Xnew2=Xnew; % bisected 1st half and calling function Xnew1(b) = infsup(inf(Xnew(b)),mid(Xnew(b))); Krawczyk('nleqn',Xnew1,L,k,Z1,Z2); % bisected 2nd half and calling function Xnew2(b) = infsup(mid(Xnew(b)),sup(Xnew(b))); Krawczyk('nleqn',Xnew2,L,k,Z1,Z2); else if rad(Xnew)<1e-6%PRINTING OUTPUTS i=i+1; Wnum(i)=complex(mid(Xnew(1)),mid(Xnew(2))); else %calling the function Krawczyk('nleqn',Xnew,L,k,Z1,Z2); end 219 end %end of ~in(K,X) end %end of ~isempty(Xnew) end %end of det(C)==0 end %end of max(mig(fX))==0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Subroutines used function [f]=nleqn(x,l,k,zeta1,zeta2) z1=1/zeta1;%characteristic admittance of wall 1 z2=1/zeta2;%characteristic admittance of wall 2 rz1=real(z1);iz1=imag(z1);rz2=real(z2);iz2=imag(z2); f=[(x(1)-rz1*k)*(x(1)-rz2*k)-(x(2)-iz1*k)*(x(2)-iz2*k)-exp(-2*x(2)*l)*... ( cos(2*x(1)*l)*((x(1)+rz1*k)*(x(1)+rz2*k)-... (x(2)+iz1*k)*(x(2)+iz2*k))-sin(2*x(1)*l)*((x(1)+rz1*k)*(x(2)+iz2*k)+... (x(2)+iz1*k)*(x(1)+rz2*k)) ) (x(1)-rz1*k)*(x(2)-iz2*k)+(x(2)-iz1*k)*(x(1)-rz2*k)-exp(-2*x(2)*l)*... ( cos(2*x(1)*l)*((x(1)+rz1*k)*(x(2)+iz2*k)+... (x(2)+iz1*k)*(x(1)+rz2*k))+sin(2*x(1)*l)*((x(1)+rz1*k)*(x(1)+... rz2*k)-(x(2)+iz1*k)*(x(2)+iz2*k)) )]; Published with MATLAB® R2013a 220 Appendix: B Implementation of theoretical/analytical modal approach clc; clear; tic format shortG; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Given parameter Lx=6.71;%length in x direction(m) Ly=5.3;%length in y direction(m) Lz=4.22;%length in z direction(m) V=Lx*Ly*Lz;% room volume S=2*(Lx*Ly+Ly*Lz+Lz*Lx);%total surface area L=4*(Lx+Ly+Lz);%total edge c=343; %soundspeed (m/s) rou=1.2; %air density f=100;% driving frequency around which eigenfrequencies are highly accurate k=2*pi*f/c;%driving wave number fr=infsup(89,112);% range of frequency band x_zeta1=389;%specific impedance of wall x1 x_zeta2=389;%specific impedance of wall x2 y_zeta1=389;%specific impedance of wall y1 y_zeta2=389;%specific impedance of wall y2 z_zeta1=389;%0.8-12.61i;%specific impedance of wall z1 z_zeta2=389;%specific impedance of wall z2 maxm=5;%maxm limit of real part of eigenvalues time=10;% allowed decay time global i Wnum; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues in x direction kx i=0; X=[infsup(0,maxm);infsup(1e-10,10*max([(1/2/Lx) abs((1/x_zeta1)*k) ... abs((1/x_zeta2)*k)]))];% initial interval Krawczyk('nleqn',X,Lx,k,x_zeta1,x_zeta2); p=i;%no of eigenvalues in x direction kx=Wnum(1:p); [sorted,rkx]=sort(real(kx)); 221 kx=kx(rkx); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues in y direction ky i=0;Wnum(:)=0; X=[infsup(0,maxm);infsup(1e-10,10*max([(1/2/Ly) abs((1/y_zeta1)*k) ... abs((1/y_zeta2)*k)]))];% initial interval Krawczyk('nleqn',X,Ly,k,y_zeta1,y_zeta2); q=i;%no of eigenvalues in y direction ky=Wnum(1:q); [sorted,rky]=sort(real(ky)); ky=ky(rky); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues in z direction kz i=0;Wnum(:)=0; X=[infsup(0,maxm);infsup(1e-10,10*max([(1/2/Lz) abs((1/z_zeta1)*k) ... abs((1/z_zeta2)*k)]))];% initial interval Krawczyk('nleqn',X,Lz,k,z_zeta1,z_zeta2); r=i;%no of eigenvalues in z direction kz=Wnum(1:r); [sorted,rkz]=sort(real(kz)); kz=kz(rkz); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Sound Pressure field calculation %source position xs=0.5;ys=0.5;zs=0.5; Source=[xs ys zs]; %receiver position x=3.5;y=2.5;z=0.5; Rec=[x y z]; %source volume velocity Q=10; [SP_total,SP_obliq,SP_tangen,SP_axial,SP_fundamental]=sound_pressure... (kx,ky,kz,k,c,rou,Q,x_zeta1,y_zeta1,z_zeta1,Lx,Ly,Lz,x,y,z,xs,ys,zs) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Modal Analysis for a given driven frequency [... band_modal_index,band_modal_eigenvalues,band_modal_frequencies,band_modal_damp_const,band_modal_theta,... band_axial_index,band_axial_eigenvalues,band_axial_frequencies,band_axial_damp_const,band_axial_theta,... 222 band_tangen_index,band_tangen_eigenvalues,band_tangen_frequencies,band_tangen_damp_const,band_tangen_theta,... band_obliq_index,band_obliq_eigenvalues,band_obliq_frequencies,band_obliq_damp_const,band_obliq_theta... Nx_calc,Ny_calc,Nz_calc,N_ideal... ]= modal_model(kx,ky,kz,k,c,Lx,Ly,Lz,fr); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Reverberation Time calculation %collective modal decay for the frequency band [RT]=reverberation_time(band_modal_damp_const,... band_axial_damp_const,band_tangen_damp_const,band_obliq_damp_const,time,c); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Published with MATLAB® R2013a 223 Appendix: C Subroutines used during implementation of theoretical modal approach Subroutine for modal pressure field calculation function [SP_total,SP_obliq,SP_tangen,SP_axial,SP_fundamental]=... sound_pressure(kx,ky,kz,k,c,rou,Q,x_zeta1,y_zeta1,z_zeta1,Lx,Ly,Lz,x,y,z,... xs,ys,zs) %'pressure_field.m' performs computation of sound pressure at a specified % receiver position for a given source position and for any given % sets ofeigenvalues of the room in x,y and z directions. % % INPUT: kx ky kz eigenvalues in x, y & z directions respectively % Lx Ly Lz room dimensions % x_zeta1 y_zeta1 z_zeta1 characteristic inpedance of the walls % x1,y1 z1 respectively % k driving frequency % x y z receiver position % Q source volume velocity % xs ys zs source position % % OUTPUT: SP_obliq,SP_tangen,SP_axial sound pressure for different modes % SP_total effective sound pressure for all three modes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Expressions for eigenfunction and normalization constant %eigen function expression EF=@(k_r,k,r_zeta,r) (k_r+(1/r_zeta)*k)*exp(1i*k_r*r)+(k_r-(1/r_zeta)*k)... *exp(-1i*k_r*r); %normalization constant expression NC=@(k_r,k,r_zeta,L) (2*L*(k_r^2-power((1/r_zeta)*k,2))-1i*(1/2/k_r)*... ( power(k_r+(1/r_zeta)*k,2)*(exp(1i*2*k_r*L)-1)-power(k_r-(1/r_zeta)... *k,2)*(exp(-1i*2*k_r*L)-1) )); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Modal sound pressure calculation l=0;% l represents total no of oblique modes 224 m=0;% m represents total no of tangential modes n=0;% n represents total no of axial modes for ii=1:length(kx) for jj=1:length(ky) for kk=1:length(kz) % Fundamental mode calculation if ii==1 && jj==1 && kk==1 fundamental_eigenvalues(1)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); lamda_fundamental(1)=feval(NC,kx(ii),k,x_zeta1,Lx)*... feval(NC,ky(jj),k,y_zeta1,Ly)*... feval(NC,kz(kk),k,z_zeta1,Lz); R_position_fundamental(1)=feval(EF,kx(ii),k,x_zeta1,x)*... feval(EF,ky(jj),k,y_zeta1,y)*... feval(EF,kz(kk),k,z_zeta1,z); S_position_fundamental(1)=feval(EF,kx(ii),k,x_zeta1,xs)*... feval(EF,ky(jj),k,y_zeta1,ys)*... feval(EF,kz(kk),k,z_zeta1,zs); SP_fundamental(1)=(1i*k*c*rou*Q)*(1/... (fundamental_eigenvalues(1)^2-k^2))*... (1/lamda_fundamental(1))*R_position_fundamental(1)*... S_position_fundamental(1); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Axial mode calculation if ii==1 && jj==1 && kk>=2 n=n+1; axial_eigenvalues(n)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); lamda_axial(n)=feval(NC,kx(ii),k,x_zeta1,Lx)*... feval(NC,ky(jj),k,y_zeta1,Ly)*... feval(NC,kz(kk),k,z_zeta1,Lz); R_position_axial(n)=feval(EF,kx(ii),k,x_zeta1,x)*... feval(EF,ky(jj),k,y_zeta1,y)*... feval(EF,kz(kk),k,z_zeta1,z); S_position_axial(n)=feval(EF,kx(ii),k,x_zeta1,xs)*... feval(EF,ky(jj),k,y_zeta1,ys)*... feval(EF,kz(kk),k,z_zeta1,zs); SP_axial(n)=(1i*k*c*rou*Q)*(1/(axial_eigenvalues(n)^2-k^2))*... (1/lamda_axial(n))*R_position_axial(n)*S_position_axial(n); end if ii==1 && jj>=2 && kk==1 225 n=n+1; axial_eigenvalues(n)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); lamda_axial(n)=feval(NC,kx(ii),k,x_zeta1,Lx)*... feval(NC,ky(jj),k,y_zeta1,Ly)*... feval(NC,kz(kk),k,z_zeta1,Lz); R_position_axial(n)=feval(EF,kx(ii),k,x_zeta1,x)*... feval(EF,ky(jj),k,y_zeta1,y)*... feval(EF,kz(kk),k,z_zeta1,z); S_position_axial(n)=feval(EF,kx(ii),k,x_zeta1,xs)*... feval(EF,ky(jj),k,y_zeta1,ys)*... feval(EF,kz(kk),k,z_zeta1,zs); SP_axial(n)=(1i*k*c*rou*Q)*(1/(axial_eigenvalues(n)^2-k^2))*... (1/lamda_axial(n))*R_position_axial(n)*S_position_axial(n); end if ii>=2 && jj==1 && kk==1 n=n+1; axial_eigenvalues(n)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); lamda_axial(n)=feval(NC,kx(ii),k,x_zeta1,Lx)*... feval(NC,ky(jj),k,y_zeta1,Ly)*... feval(NC,kz(kk),k,z_zeta1,Lz); R_position_axial(n)=feval(EF,kx(ii),k,x_zeta1,x)*... feval(EF,ky(jj),k,y_zeta1,y)*... feval(EF,kz(kk),k,z_zeta1,z); S_position_axial(n)=feval(EF,kx(ii),k,x_zeta1,xs)*... feval(EF,ky(jj),k,y_zeta1,ys)*... feval(EF,kz(kk),k,z_zeta1,zs); SP_axial(n)=(1i*k*c*rou*Q)*(1/(axial_eigenvalues(n)^2-k^2))*... (1/lamda_axial(n))*R_position_axial(n)*S_position_axial(n); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Tangential mode calculation if ii==1 && jj>=2 && kk>=2 m=m+1; tangen_eigenvalues(m)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); lamda_tangen(m)=feval(NC,kx(ii),k,x_zeta1,Lx)*... feval(NC,ky(jj),k,y_zeta1,Ly)*... feval(NC,kz(kk),k,z_zeta1,Lz); R_position_tangen(m)=feval(EF,kx(ii),k,x_zeta1,x)*... 226 feval(EF,ky(jj),k,y_zeta1,y)*... feval(EF,kz(kk),k,z_zeta1,z); S_position_tangen(m)=feval(EF,kx(ii),k,x_zeta1,xs)*... feval(EF,ky(jj),k,y_zeta1,ys)*... feval(EF,kz(kk),k,z_zeta1,zs); SP_tangen(m)=(1i*k*c*rou*Q)*(1/(tangen_eigenvalues(m)^2-k^2))*... (1/lamda_tangen(m))*R_position_tangen(m)*S_position_tangen(m); end if ii>=2 && jj==1 && kk>=2 m=m+1; tangen_eigenvalues(m)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); lamda_tangen(m)=feval(NC,kx(ii),k,x_zeta1,Lx)*... feval(NC,ky(jj),k,y_zeta1,Ly)*... feval(NC,kz(kk),k,z_zeta1,Lz); R_position_tangen(m)=feval(EF,kx(ii),k,x_zeta1,x)*... feval(EF,ky(jj),k,y_zeta1,y)*... feval(EF,kz(kk),k,z_zeta1,z); S_position_tangen(m)=feval(EF,kx(ii),k,x_zeta1,xs)*... feval(EF,ky(jj),k,y_zeta1,ys)*... feval(EF,kz(kk),k,z_zeta1,zs); SP_tangen(m)=(1i*k*c*rou*Q)*(1/(tangen_eigenvalues(m)^2-k^2))*... (1/lamda_tangen(m))*R_position_tangen(m)*S_position_tangen(m); end if ii>=2 && jj>=2 && kk==1 m=m+1; tangen_eigenvalues(m)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); lamda_tangen(m)=feval(NC,kx(ii),k,x_zeta1,Lx)*... feval(NC,ky(jj),k,y_zeta1,Ly)*... feval(NC,kz(kk),k,z_zeta1,Lz); R_position_tangen(m)=feval(EF,kx(ii),k,x_zeta1,x)*... feval(EF,ky(jj),k,y_zeta1,y)*... feval(EF,kz(kk),k,z_zeta1,z); S_position_tangen(m)=feval(EF,kx(ii),k,x_zeta1,xs)*... feval(EF,ky(jj),k,y_zeta1,ys)*... feval(EF,kz(kk),k,z_zeta1,zs); SP_tangen(m)=(1i*k*c*rou*Q)*(1/(tangen_eigenvalues(m)^2-k^2))*... (1/lamda_tangen(m))*R_position_tangen(m)*S_position_tangen(m); 227 end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % oblique mode calculation if ii>=2 && jj>=2 && kk>=2 l=l+1; obliq_eigenvalues(l)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); lamda_obliq(l)=feval(NC,kx(ii),k,x_zeta1,Lx)*... feval(NC,ky(jj),k,y_zeta1,Ly)*... feval(NC,kz(kk),k,z_zeta1,Lz); R_position_obliq(l)=feval(EF,kx(ii),k,x_zeta1,x)*... feval(EF,ky(jj),k,y_zeta1,y)*... feval(EF,kz(kk),k,z_zeta1,z); S_position_obliq(l)=feval(EF,kx(ii),k,x_zeta1,xs)*... feval(EF,ky(jj),k,y_zeta1,ys)*... feval(EF,kz(kk),k,z_zeta1,zs); SP_obliq(l)=(1i*k*c*rou*Q)*(1/(obliq_eigenvalues(l)^2-k^2))*... (1/lamda_obliq(l))*R_position_obliq(l)*S_position_obliq(l); end end end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Total Sound pressure SP_obliq=sum(SP_obliq(1:l));%oblique sound pressure at defined position SP_tangen=sum(SP_tangen(1:m));%tangential sound pressure at defined position SP_axial=sum(SP_axial(1:n));%axial sound pressure at defined position SP_total=SP_obliq+SP_tangen+SP_axial+SP_fundamental; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Subroutine for modal composition analysis function [... band_modal_index,band_modal_eigenvalues,band_modal_frequencies,... band_modal_damp_const,band_modal_theta,... band_axial_index,band_axial_eigenvalues,band_axial_frequencies,... band_axial_damp_const,band_axial_theta,... band_tangen_index,band_tangen_eigenvalues,band_tangen_frequencies,... band_tangen_damp_const,band_tangen_theta,... band_obliq_index,band_obliq_eigenvalues,band_obliq_frequencies,... 228 band_obliq_damp_const,band_obliq_theta... Nx_calc,Ny_calc,Nz_calc,N_ideal... ]= modal_model(kx,ky,kz,k,c,Lx,Ly,Lz,fr) % band_axial_index,band_axial_eigenvalues,band_axial_frequencies, % band_axial_damp_const,band_axial_theta,... % band_tangen_index,band_tangen_eigenvalues,band_tangen_frequencies, % band_tangen_damp_const,band_tangen_theta,... % band_obliq_index,band_obliq_eigenvalues,band_obliq_frequencies, % band_obliq_damp_const,band_obliq_theta... %'modal_analysis.m' performs complete modal analysis of a room for any given % sets ofeigenvalues of the room in x,y and z directions. % % INPUT: kx eigenvalues in x direction % ky eigenvalues in y direction % kz eigenvalues in z direction % c sound speed % fr band of frequency under consideration % % OUTPUT: mode number,eigenvalues, wave number, frequency and damping % constant - for oblique,tangential and axial modes for both the % complete and band under consideration. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Modal calculation l=0;% l represents total no of oblique modes m=0;% m represents total no of tangential modes n=0;% n represents total no of axial modes for ii=1:length(kx) for jj=1:length(ky) for kk=1:length(kz) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Fundamental mode calculation if ii==1 && jj==1 && kk==1 fundamental_eigenvalues(1)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); findex{1}=sprintf('%d%d%d',ii-1,jj-1,kk-1); ftheta(1:3,1)=(180/pi).*real([asin(kx(ii)/... fundamental_eigenvalues(1));... asin(ky(jj)/fundamental_eigenvalues(1));asin(kz(kk)/... 229 fundamental_eigenvalues(1))]); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Axial mode calculation if ii==1 && jj==1 && kk>=2 n=n+1; axial_eigenvalues(n)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); aindex{n}=sprintf('%d%d%d',ii-1,jj-1,kk-1); atheta(1:3,n)=(180/pi).*real([asin(kx(ii)/... axial_eigenvalues(n));... asin(ky(jj)/axial_eigenvalues(n));asin(kz(kk)/... axial_eigenvalues(n))]); end if ii==1 && jj>=2 && kk==1 n=n+1; axial_eigenvalues(n)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); aindex{n}=sprintf('%d%d%d',ii-1,jj-1,kk-1); atheta(1:3,n)=(180/pi).*real([asin(kx(ii)/... axial_eigenvalues(n));... asin(ky(jj)/axial_eigenvalues(n));asin(kz(kk)/... axial_eigenvalues(n))]); end if ii>=2 && jj==1 && kk==1 n=n+1; axial_eigenvalues(n)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); aindex{n}=sprintf('%d%d%d',ii-1,jj-1,kk-1); atheta(1:3,n)=(180/pi).*real([asin(kx(ii)/... axial_eigenvalues(n));... asin(ky(jj)/axial_eigenvalues(n));asin(kz(kk)/... axial_eigenvalues(n))]); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Tangential mode calculation if ii==1 && jj>=2 && kk>=2 m=m+1; tangen_eigenvalues(m)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); tindex{m}=sprintf('%d%d%d',ii-1,jj-1,kk-1); ttheta(1:3,m)=(180/pi).*real([asin(kx(ii)/... tangen_eigenvalues(m));... asin(ky(jj)/tangen_eigenvalues(m));asin(kz(kk)/... 230 tangen_eigenvalues(m))]); end if ii>=2 && jj==1 && kk>=2 m=m+1; tangen_eigenvalues(m)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); tindex{m}=sprintf('%d%d%d',ii-1,jj-1,kk-1); ttheta(1:3,m)=(180/pi).*real([asin(kx(ii)/... tangen_eigenvalues(m));... asin(ky(jj)/tangen_eigenvalues(m));asin(kz(kk)/... tangen_eigenvalues(m))]); end if ii>=2 && jj>=2 && kk==1 m=m+1; tangen_eigenvalues(m)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); tindex{m}=sprintf('%d%d%d',ii-1,jj-1,kk-1); ttheta(1:3,m)=(180/pi).*real([asin(kx(ii)/... tangen_eigenvalues(m));... asin(ky(jj)/tangen_eigenvalues(m));asin(kz(kk)/... tangen_eigenvalues(m))]); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % oblique mode calculation if ii>=2 && jj>=2 && kk>=2 l=l+1; obliq_eigenvalues(l)=sqrt(kx(ii)^2+ky(jj)^2+kz(kk)^2); oindex{l}=sprintf('%d%d%d',ii-1,jj-1,kk-1); otheta(1:3,l)=(180/pi).*real([asin(kx(ii)/... obliq_eigenvalues(l));... asin(ky(jj)/obliq_eigenvalues(l));asin(kz(kk)/... obliq_eigenvalues(l))]); end end end end oindex(1:l)=oindex;tindex(1:m)=tindex;aindex(1:n)=aindex; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Combining modal frequencies modal_eigenvalues(1)=fundamental_eigenvalues; 231 index(1)=findex; theta(1:3,1)=ftheta; modal_eigenvalues(2:n+1)=axial_eigenvalues; index(2:n+1)=aindex; theta(1:3,2:n+1)=atheta; modal_eigenvalues(n+2:n+1+m)=tangen_eigenvalues; index(n+2:n+1+m)=tindex; theta(1:3,n+2:n+1+m)=ttheta; modal_eigenvalues(n+1+m+1:n+1+m+l)=obliq_eigenvalues; index(n+1+m+1:n+1+m+l)=oindex; theta(1:3,n+1+m+1:n+1+m+l)=otheta; [sorted,rmodal_eigenvalues]=sort(real(modal_eigenvalues)); modal_eigenvalues=modal_eigenvalues(rmodal_eigenvalues); modal_frequencies=real(modal_eigenvalues)*c/(2*pi); index=index(rmodal_eigenvalues); theta=theta'; theta=theta(rmodal_eigenvalues,1:3); theta=theta'; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % counting all modes in the given frequency band j=0; for ll=1:length(modal_frequencies) if in(modal_frequencies(ll),fr) j=j+1; band_modal_frequencies(j)=modal_frequencies(ll);%frequency band_modal_eigenvalues(j)=modal_eigenvalues(ll);%eigenvalue band_modal_damp_const(j)=c*imag(modal_eigenvalues(ll));%damping constant band_modal_index(j)=index(ll); band_modal_theta(1:3,j)=theta(1:3,ll); end end [sorted,amodal_eigenvalues]=sort(real(axial_eigenvalues)); axial_eigenvalues=axial_eigenvalues(amodal_eigenvalues); axial_frequencies=real(axial_eigenvalues)*c/(2*pi); 232 aindex=aindex(amodal_eigenvalues); atheta=atheta'; atheta=atheta(amodal_eigenvalues,1:3); atheta=atheta'; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % counting axial modes in the given frequency band j=0; for ll=1:length(axial_frequencies) if in(axial_frequencies(ll),fr) j=j+1; band_axial_frequencies(j)=axial_frequencies(ll);%frequency band_axial_eigenvalues(j)=axial_eigenvalues(ll);%eigenvalue band_axial_damp_const(j)=c*imag(axial_eigenvalues(ll));%damping constant band_axial_index(j)=aindex(ll); band_axial_theta(1:3,j)=atheta(1:3,ll); end end [sorted,tmodal_eigenvalues]=sort(real(tangen_eigenvalues)); tangen_eigenvalues=tangen_eigenvalues(tmodal_eigenvalues); tangen_frequencies=real(tangen_eigenvalues)*c/(2*pi); tindex=tindex(tmodal_eigenvalues); ttheta=ttheta'; ttheta=ttheta(tmodal_eigenvalues,1:3); ttheta=ttheta'; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % counting tangential modes in the given frequency band j=0; for ll=1:length(tangen_frequencies) if in(tangen_frequencies(ll),fr) j=j+1; band_tangen_frequencies(j)=tangen_frequencies(ll);%frequency band_tangen_eigenvalues(j)=tangen_eigenvalues(ll);%eigenvalue band_tangen_damp_const(j)=c*imag(tangen_eigenvalues(ll));%damping constant band_tangen_index(j)=tindex(ll); band_tangen_theta(1:3,j)=ttheta(1:3,ll); 233 end end [sorted,omodal_eigenvalues]=sort(real(obliq_eigenvalues)); obliq_eigenvalues=obliq_eigenvalues(omodal_eigenvalues); obliq_frequencies=real(obliq_eigenvalues)*c/(2*pi); oindex=oindex(omodal_eigenvalues); otheta=otheta'; otheta=otheta(omodal_eigenvalues,1:3); otheta=otheta'; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % counting oblique modes in the given frequency band j=0; for ll=1:length(obliq_frequencies) if in(obliq_frequencies(ll),fr) j=j+1; band_obliq_frequencies(j)=obliq_frequencies(ll);%frequency band_obliq_eigenvalues(j)=obliq_eigenvalues(ll);%eigenvalue band_obliq_damp_const(j)=c*imag(obliq_eigenvalues(ll));%damping constant band_obliq_index(j)=oindex(ll); band_obliq_theta(1:3,j)=otheta(1:3,ll); end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Subroutine for modal decay and RT function [RT]=reverberation_time(band_modal_damp_const,... band_axial_damp_const,band_tangen_damp_const,band_obliq_damp_const,time,c) %'reverberation_time.m' performs computation of RT for a given band of modal % decay constants % % INPUT: band_total_mode_number, % band_mode_damp_const, % time, 234 % c sound speed % % OUTPUT: RT reverberation time from modal decay curve % TT average RT for the band normal modes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % collective modal decay t=0:0.1:time; for j=1:length(t) for jj=1:length(band_modal_damp_const) %RT for individual modes T(jj)=6*log(10)/(2*band_modal_damp_const(jj)); numerator(jj)=T(jj)*10^(-6*t(j)/T(jj)); %numerator(jj)=(1/band_mode_damp_const(jj))*... % exp(-2*band_mode_damp_const(jj)*t(j)); %denominator(jj)=1/band_mode_damp_const(jj); end numerator(j)=sum(numerator(1:length(band_modal_damp_const))); denominator(j)=sum(T(1:length(band_modal_damp_const))); %denominator(j)=sum(denominator(1:band_total_mode_number)); decay(j)=10*log10(numerator(j)/denominator(j)); if decay(j)<=-60 rt=j; RT=t(j); break; else RT=0; end end if RT==0 fprintf('Expontial decay does not reach upto 60 dB\n'); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % axial mode decay ta=0:0.1:time; for ja=1:length(ta) for jj=1:length(band_axial_damp_const) 235 %RT for individual modes T(jj)=6*log(10)/(2*band_axial_damp_const(jj)); numerator(jj)=T(jj)*10^(-6*ta(ja)/T(jj)); %numerator(jj)=(1/band_mode_damp_const(jj))*... % exp(-2*band_mode_damp_const(jj)*t(j)); %denominator(jj)=1/band_mode_damp_const(jj); end numerator(ja)=sum(numerator(1:length(band_axial_damp_const))); denominator(ja)=sum(T(1:length(band_axial_damp_const))); %denominator(j)=sum(denominator(1:band_total_mode_number)); Adecay(ja)=10*log10(numerator(ja)/denominator(ja)); if decay(ja)<=-60 art=ja; aRT=ta(ja); break; else aRT=0; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % tangential mode decay tt=0:0.1:time; for jt=1:length(tt) for jj=1:length(band_tangen_damp_const) %RT for individual modes T(jj)=6*log(10)/(2*band_tangen_damp_const(jj)); numerator(jj)=T(jj)*10^(-6*tt(jt)/T(jj)); %numerator(jj)=(1/band_mode_damp_const(jj))*... % exp(-2*band_mode_damp_const(jj)*t(j)); %denominator(jj)=1/band_mode_damp_const(jj); end numerator(jt)=sum(numerator(1:length(band_tangen_damp_const))); denominator(jt)=sum(T(1:length(band_tangen_damp_const))); %denominator(j)=sum(denominator(1:band_total_mode_number)); Tdecay(jt)=10*log10(numerator(jt)/denominator(jt)); 236 if decay(jt)<=-60 trt=jt; tRT=tt(jt); break; else tRT=0; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % oblique mode decay to=0:0.1:time; for jo=1:length(to) for jj=1:length(band_obliq_damp_const) %RT for individual modes T(jj)=6*log(10)/(2*band_obliq_damp_const(jj)); numerator(jj)=T(jj)*10^(-6*to(jo)/T(jj)); %numerator(jj)=(1/band_mode_damp_const(jj))*... % exp(-2*band_mode_damp_const(jj)*t(j)); %denominator(jj)=1/band_mode_damp_const(jj); end numerator(jo)=sum(numerator(1:length(band_obliq_damp_const))); denominator(jo)=sum(T(1:length(band_obliq_damp_const))); %denominator(j)=sum(denominator(1:band_total_mode_number)); Odecay(jo)=10*log10(numerator(jo)/denominator(jo)); if decay(jo)<=-60 ort=jo; oRT=to(jo); break; else oRT=0; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % plotting decay curves %TT=sum(T)/length(T); 237 plot(t(1:j),decay(1:j),'r',ta(1:ja),Adecay(1:ja),'g',tt(1:jt),... Tdecay(1:jt),'b',to(1:jo),Odecay(1:jo),'k','linewidth',2); ylim([-60 0]); legend('Collective Decay','Axial Decay','Tangential Decay','Oblique Decay'); grid on;xlabel 'Time (s)';ylabel 'Level (dB)'; title 'Collective modal decay curve'; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Published with MATLAB® R2013a 238 Appendix: D Post-processing analyses for numerical modal approach clc; clear all; clc; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Reading frequency data file [freq, pressure]=xlsread('sound_pressure.xlsx'); band_f=freq(:,:); fs=2*max(band_f); df=band_f(2)-band_f(1);%% must check this value%%%%%%%%%%%%%%%%% pressure=str2double(pressure); p1=pressure(:,1); p2=pressure(:,2); p3=pressure(:,3); p4=pressure(:,4); p5=pressure(:,5); spl=10*log10(0.5*pressure.*conj(pressure)./(2e-5)^2); SPL_1=spl(:,1); SPL_2=spl(:,2); SPL_3=spl(:,3); SPL_4=spl(:,4); SPL_5=spl(:,5); [~, eigen]=xlsread('eigen_frequency.xlsx'); eigen=str2double(eigen); eigen_f=eigen(:,:); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Analyses condition V=81.53; % room volume S=178.8; % room surface area specimen=6.502; % specimen area alpha=0.02; % average surface absorption 239 q1=10*(cosd(0)+1i*sind(0));q2=10*(cosd(30)+1i*sind(30)); % flow source volume velocity % octave_center =[50,63,80,100,125,160,200,250,315]; % CENTER FREQUENCIES % octave_limit =[44,56,70,89,112,141,178,224,282,355];% LOWER AND UPPER FREQUENCIES octave_center =[63,125,250]; % CENTER FREQUENCIES octave_limit =[44,88,177,355];% LOWER AND UPPER FREQUENCIES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 % SPL manipulation and standard deviation [spl_1_freq,p1_freq,f1]=SPL(SPL_1,p1,band_f,octave_center,octave_limit,1); [spl_2_freq,p2_freq,f2]=SPL(SPL_2,p2,band_f,octave_center,octave_limit,2); [spl_3_freq,p3_freq,f3]=SPL(SPL_3,p3,band_f,octave_center,octave_limit,3); [spl_4_freq,p4_freq,f4]=SPL(SPL_4,p4,band_f,octave_center,octave_limit,4); [spl_5_freq,p5_freq,f5]=SPL(SPL_5,p5,band_f,octave_center,octave_limit,5); for ii=1:length(octave_center) mean_spl_1_freq(ii)=10*log10(sum(10.^(spl_1_freq(ii,:)./10))/... length(spl_1_freq(ii,:))); mean_spl_2_freq(ii)=10*log10(sum(10.^(spl_2_freq(ii,:)./10))/... length(spl_2_freq(ii,:))); mean_spl_3_freq(ii)=10*log10(sum(10.^(spl_3_freq(ii,:)./10))/... length(spl_3_freq(ii,:))); mean_spl_4_freq(ii)=10*log10(sum(10.^(spl_4_freq(ii,:)./10))/... length(spl_4_freq(ii,:))); mean_spl_5_freq(ii)=10*log10(sum(10.^(spl_5_freq(ii,:)./10))/... length(spl_5_freq(ii,:))); end % mean_spl_1_freq contains the average SPL for different location for % bands for ii=1:length(octave_center) mean_spl_freq(ii)=10*log10( ( 10^(mean_spl_1_freq(ii)/10)+... 10^(mean_spl_2_freq(ii)/10)+... 10^(mean_spl_3_freq(ii)/10)+10^(mean_spl_4_freq(ii)/10)+... 10^(mean_spl_5_freq(ii)/10) )/5 ); spl_stnd_deviation(ii)=sqrt( ( (mean_spl_freq(ii)-mean_spl_1_freq(ii)).^2+... (mean_spl_freq(ii)-mean_spl_2_freq(ii)).^2+(mean_spl_freq(ii)-... mean_spl_3_freq(ii)).^2+... (mean_spl_freq(ii)-mean_spl_4_freq(ii)).^2+(mean_spl_freq(ii)-... mean_spl_5_freq(ii)).^2 )/4 ); 240 mean_p_freq(ii)=( mean(p1_freq(ii,:))+mean(p2_freq(ii,:))+... mean(p3_freq(ii,:))+... mean(p4_freq(ii,:))+mean(p5_freq(ii,:)) )/5; end % mean_spl_freq contains the overall average SPL for different % bands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Impulse response and decay, RT and standard deviation [p1_time,t1]=impulse(band_f,df,p1,octave_center,octave_limit,1); [p2_time,t2]=impulse(band_f,df,p2,octave_center,octave_limit,2); [p3_time,t3]=impulse(band_f,df,p3,octave_center,octave_limit,3); [p4_time,t4]=impulse(band_f,df,p4,octave_center,octave_limit,4); [p5_time,t5]=impulse(band_f,df,p5,octave_center,octave_limit,5); %p1_time contains the impulse signal for different banda%%%%%%%%%% for ii=1:length(octave_center) [y1{ii}(1,:),x1{ii}(1,:),T160(ii),R1(ii)]=level_decay(f1(ii,:),... p1_freq(ii,:),t1,p1_time(ii,:),fs,octave_center,ii,1); [y2{ii}(1,:),x2{ii}(1,:),T260(ii),R2(ii)]=level_decay(f2(ii,:),... p2_freq(ii,:),t2,p2_time(ii,:),fs,octave_center,ii,2); [y3{ii}(1,:),x3{ii}(1,:),T360(ii),R3(ii)]=level_decay(f3(ii,:),... p3_freq(ii,:),t3,p3_time(ii,:),fs,octave_center,ii,3); [y4{ii}(1,:),x4{ii}(1,:),T460(ii),R4(ii)]=level_decay(f4(ii,:),... p4_freq(ii,:),t4,p4_time(ii,:),fs,octave_center,ii,4); [y5{ii}(1,:),x5{ii}(1,:),T560(ii),R5(ii)]=level_decay(f5(ii,:),... p5_freq(ii,:),t5,p5_time(ii,:),fs,octave_center,ii,5); end %y1 is the IIDC data and x is the corresponding time data for different %bands for ii=1:length(octave_center) mean_T60(ii)=5/( 1/T160(ii)+1/T260(ii)+1/T360(ii)+1/T460(ii)+1/T560(ii) ); mean_R(ii)=(R1(ii)+R2(ii)+R3(ii)+R4(ii)+R5(ii))/5; mean_decay_rate(ii)=60/mean_T60(ii); T60_stnd_deviation(ii)=sqrt( ( (mean_T60(ii)-T160(ii)).^2+... (mean_T60(ii)-T260(ii)).^2+(mean_T60(ii)-T360(ii)).^2+... (mean_T60(ii)-T460(ii)).^2+(mean_T60(ii)-T560(ii)).^2 )/4 ); end % mean_T60 contains the overall average T60 for different % bands 241 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Modal analysis and decay [modes,delta,no_of_modes,modal_density,modal_spacing,modal_width,... modal_overlap,modal_overlap_factor]=eigen_value(eigen_f,... octave_center,octave_limit); % modes, delta contains the modal freq and damp constants for different % bands for ii=1:length(octave_center) [modal_y{ii}(1,:),modal_x{ii}(1,:),modal_T60(ii),modal_R(ii)]=... modal_decay(delta(ii,1:no_of_modes(ii)),octave_center,ii); end %modal_y is the modal decay data and modal_x is the corresponding time %data for different bands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Predicted Absorption and sound power level calculation A=0.161*V./mean_T60; % diffuse field formula for RT random_alpha(1:length(octave_center))=alpha; sabine_alpha=A./S; eyring_alpha=1-exp(-A./S); % Total power level of two sources using SPL and RT formulae Lw_total=mean_spl_freq-10*log10(mean_T60)+10*log10(V)+10*log10(1+S*343./... (8.*V.*octave_center))-13.9; for ii=1:length(octave_center) Lw_2(ii)=10*log10( (1./(2*2*pi*octave_center(ii)*1.2)).*... abs( real(mean_p_freq(ii)).*(imag(1i*2*pi*octave_center(ii)*1.2*q1)... +imag(1i*2*pi*octave_center(ii)*1.2*q2))-... imag(mean_p_freq(ii)).*(real(1i*2*pi*octave_center(ii)*1.2*q1)+... real(1i*2*pi*octave_center(ii)*1.2*q2)) ) )+120-3; end % power level of individual sources Lw=10*log10(10.^(Lw_total./10)./2); % regenerating T60 using predicted Lw and Lp T60_1 = 10.^( (mean_spl_freq - Lw_total + 10*log10(V) + 10*log10(1+S*343./... (8.*V.*octave_center))-13.9 )./10 ); T60_2 = 10.^( (mean_spl_freq - Lw_2 + 10*log10(V) + 10*log10(1+S*343./... (8.*V.*octave_center))-13.9 )./10 ); 242 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Expected Absorption and sound power level calculation % Expected reverberation time. T60_expected_sabine(1:length(octave_center)) = 0.161*V/(S*alpha); T60_expected_eyring(1:length(octave_center)) = 0.161*V/(-S*log(1-alpha)); A_expected_sabine = 0.161*V./T60_expected_sabine; A_expected_eyring = 0.161*V./T60_expected_eyring; Lp_expected_sabine=Lw_total+10*log10(2/(4*pi*1.5^2)+4*(1-alpha)./... A_expected_sabine); Lp_expected_eyring=Lw_total+10*log10(2/(4*pi*1.5^2)+4*(1-alpha)./... A_expected_eyring); for ii=1:length(octave_center) Lw_expected_band(ii) = 10*log10(sum(1.2*343*((2*pi.*f1(ii,:))./343).^2*... ((abs(q1))^2+(abs(q2))^2)./(8*pi)))+120;% Lw_expected(ii)=(inv(2*pi*octave_center(ii)/343) *... sqrt(2*pi*343/(V*(octave_limit(ii+1)-octave_limit(ii))))); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Level of accuracy T60_sabine_error=100*(abs(mean_T60-T60_expected_sabine)./... T60_expected_sabine); T60_eyring_error=100*(abs(mean_T60-T60_expected_eyring)./... T60_expected_eyring); Lp_sabine_error=100*(abs(mean_spl_freq-Lp_expected_sabine)./... Lp_expected_sabine); Lp_eyring_error=100*(abs(mean_spl_freq-Lp_expected_eyring)./... Lp_expected_eyring); A_sabine_error=100*(abs(A-A_expected_sabine)./A_expected_sabine); A_eyring_error=100*(abs(A-A_expected_eyring)./A_expected_eyring); d1=1;d2=-10./(log(10).*mean_T60); Lw_sabine_error=100*(sqrt( (d1.*Lp_sabine_error./100).^2+... (d2.*T60_sabine_error./100).^2 )); Lw_eyring_error=100*(sqrt( (d1.*Lp_eyring_error./100).^2+... (d2.*T60_eyring_error./100).^2 )); 243 alpha_sabine_error=100*(abs(random_alpha-sabine_alpha)./random_alpha); alpha_eyring_error=100*(abs(random_alpha-eyring_alpha)./random_alpha); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Final output variables % frequency response at 5 different positions for different bands p1_freq;p2_freq;p3_freq;p4_freq;p5_freq;% SIZE(no. of bands, no of frequencies) % corresponding frequencies of frequency response f1;f2;f3;f4;f5; % SIZE(no. of bands, no of frequencies) % impulse response at 5 different positions for different bands p1_time;p2_time;p3_time;p4_time;p5_time;% SIZE(no. of bands, time vector) t1;t2;t3;t4;t5; % corresponding time vector of impulse response % integrated impulse decay at 5 different positions for different bands y1;y2;y3;y4;y5;% SIZE{no. of bands} (1,time vector) % corresponding time vector of integrated impulse decay x1;x2;x3;x4;x5; % SIZE{no. of bands} (1,time vector) % modal decay for different bands modal_y;% SIZE{no. of bands} (1,time vector) % corresponding time vector of integrated impulse decay modal_x; % SIZE{no. of bands} (1,time vector) % reverberation time at 5 different positions for different bands T160;T260;T360;T460;T560;% SIZE(no of bands) % regression coefficient R1;R2;R3;R4;R5;mean_R; % SPL at 5 different positions for different bands mean_spl_1_freq;mean_spl_2_freq;mean_spl_3_freq;mean_spl_4_freq; mean_spl_5_freq; % modal reverberation time and regression coefficient for different bands modal_T60;modal_R;% SIZE(no of bands) % mean SPL and standard deviation mean_spl_freq;spl_stnd_deviation;% SIZE(no of bands) 244 Lp_expected_sabine;Lp_expected_eyring; Lp_sabine_error;Lp_eyring_error; % mean T60 and standard deviation mean_T60;T60_stnd_deviation;mean_decay_rate;% SIZE(no of bands) T60_expected_sabine;T60_expected_eyring; T60_sabine_error;T60_eyring_error; % modal analysis no_of_modes;modal_density;modal_spacing;modal_width; modal_overlap;modal_overlap_factor;% SIZE(no of bands) % sound power level Lw;Lw_total; Lw_sabine_error;Lw_eyring_error; % absorption A;A_expected_sabine;A_expected_eyring; A_sabine_error;A_eyring_error; % absorption coefficient random_alpha;sabine_alpha;eyring_alpha; alpha_sabine_error;alpha_eyring_error; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %writing in a file %writing modal composition statistics header_1={'Third-octave bands (Hz)'; 'Number of Modes'; 'Modal Density'; ... 'Modal Spacing'; ... 'Modal width'; 'Modal Overlap'; 'Modal Overlap Factor';}; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',header_1,'sheet 4','A2'); modal_composition=[(octave_center') (no_of_modes') (modal_density') ... (modal_spacing') ... (modal_width') (modal_overlap') (modal_overlap_factor')]; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',modal_composition',... 'sheet 4','C2'); %writing reverberation time statistics header_2={'Third-octave bands (Hz)'; 'Modal RT (s)';'RT (s) @ Rec#1'; ... 'RT (s) @ Rec#2';... 245 'RT (s) @ Rec#3';'RT (s) @ Rec#4'; 'RT (s) @ Rec#5';... 'Predicted Average RT (s)';... 'RT Standard deviation (s)';'Sabine Expeced RT (s)';... 'Sabine Error Level (%)';... 'Eyring Expeced RT (s)';'Eyring Error Level (%)';'Decay Rate (dB/s)';}; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',header_2,'sheet 4','A12'); reverberation_time=[(octave_center') (modal_T60') (T160') (T260') (T360') ... (T460') ... (T560') (mean_T60') (T60_stnd_deviation') (T60_expected_sabine') ... (T60_sabine_error') ... (T60_expected_eyring') (T60_eyring_error') (mean_decay_rate')]; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',reverberation_time',... 'sheet 4','C12'); %writing SPL statistics header_3={'Third-octave bands (Hz)'; 'SPL (dB) @ Rec#1'; 'SPL (dB) @ Rec#2';... 'SPL (dB) @ Rec#3';'SPL (dB) @ Rec#4'; 'SPL (dB) @ Rec#5';... 'Predicted Average SPL (dB)';... 'SPL Standard deviation (dB)';'Sabine Expeced SPL (dB)';... 'Sabine Error Level (%)';... 'Eyring Expeced SPL (dB)';'Eyring Error Level (%)';}; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',header_3,'sheet 4','A35'); sound_pressure_level=[(octave_center') (mean_spl_1_freq') (mean_spl_2_freq') ... (mean_spl_3_freq') (mean_spl_4_freq') (mean_spl_5_freq') (mean_spl_freq') ... (spl_stnd_deviation') (Lp_expected_sabine') (Lp_sabine_error') ... (Lp_expected_eyring') ... (Lp_eyring_error')]; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',sound_pressure_level',... 'sheet 4','C35'); %writing correlation coefficient statistics header_4={'Third-octave bands (Hz)'; 'R^2 @ Rec#1'; 'R^2 @ Rec#2';... 'R^2 @ Rec#3';'R^2 @ Rec#4'; 'R^2 @ Rec#5';'Average R^2';'R^2 Modal';}; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',header_4,'sheet 4','A50'); 246 correlation_coefficient=[(octave_center') (R1') (R2') (R3') (R4') (R5') ... (mean_R') (modal_R')]; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',correlation_coefficient',... 'sheet 4','C50'); %writing source power level statistics header_5={'Third-octave bands (Hz)'; 'Total Predicted Power Level (dB)'; ... 'Sabine Error Level (%)';... 'Eyring Error Level (%)';'Predicted Level for each source (dB)';}; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',header_5,'sheet 4','A60'); power_level=[(octave_center') (Lw_total') (Lw_sabine_error') ... (Lw_eyring_error') (Lw')]; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',power_level',... 'sheet 4','C60'); %writing absorption measurement statistics header_6={'Third-octave bands (Hz)'; 'Total Predicted Absorption (m^2)';... 'Expected Sabine Absorption (m^2)';... 'Sabine Error Level (%)';'Expected Eyring Absorption (m^2)';... 'Eyring Error Level (%)';... 'Random absorp coeff';'Sabine absorp coeff';'Sabine error';... 'Eyring absorp coeff';'Eyring error';}; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',header_6,'sheet 4','A70'); absorption_measurement=[(octave_center') (A') (A_expected_sabine') ... (A_sabine_error') ... (A_expected_eyring') (A_eyring_error') (random_alpha') (sabine_alpha') ... (alpha_sabine_error') ... (eyring_alpha') (alpha_eyring_error')]; xlswrite('Result_ASTM_E90_VOL_82_empty room.xlsx',absorption_measurement',... 'sheet 4','C70'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Published with MATLAB® R2013a 247 Appendix: E Subroutines used during post processing for numerical modal approach Subroutine for FR and SPL manipulation function [band_spl,band_p,f]=SPL(spl_p,p,band_f,octave_center,... octave_limit,receiver_location) for jj=1:length(octave_center) j=0; for i=1:length(band_f) if band_f(i)<=octave_limit(jj+1) && band_f(i)>=octave_limit(jj) j=j+1; f(jj,j)=band_f(i); band_spl(jj,j)=spl_p(i); band_p(jj,j)=p(i); end end % Plot the frequency response figure; plot(f(jj,:), real(band_p(jj,:)),'b-','LineWidth',1); set(gca,'FontSize',16); title(['FR for the ' num2str(octave_center(jj)) '-Hz band at receiver #' ... num2str(receiver_location) ''], 'FontSize', 16); axis tight;xlabel('Frequency (Hz)', 'FontSize', 16), ylabel('Pressure (Pa)',... 'FontSize', 16); end Subroutine for IFFT and IR manipulation function [p_time,tt]=impulse(band_f,df,p,octave_center,octave_limit,... receiver_location) f_nyquist=max(band_f); f=0:df:max(band_f); 248 n=length(f)-1; N=2*n; fs=N*df;%2*f_nyquist; dt = (1 / fs); tt = (0:N-1).*dt; % inverse fourier transform FR = zeros(1,N); FR(min(band_f)/df+1:N/2+1) = p; FR(N/2+2:N) = fliplr(conj(FR(2:N/2))); IR=(ifft(((FR)))); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %checking whether ifft is right by doing fft on IR subplot(311); plot(band_f,real(p)); subplot(312); plot(tt,real(IR)); subplot(313); pp=fft(IR); plot(band_f,real(pp(min(band_f)/df+1:max(band_f)/df+1))); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % calculation for third octave bands p_time=zeros(length(octave_center),N); for ii=1:length(octave_center) % third order butter worth filter design N_order = 3; % Filter order Qr = octave_center(ii)/(octave_limit(ii+1)-octave_limit(ii)); Qd = (pi/2/N_order)/(sin(pi/2/N_order))*Qr; alpha = (1 + sqrt(1+4*Qd^2))/2/Qd; W1 = octave_center(ii)/(f_nyquist)/alpha; % Lower bound for filter W2 = (octave_center(ii)/(f_nyquist)*alpha);% Upper bound for filter [Bc,Ac] = butter(N_order,[W1,W2]); % Filter coefficients tdomain = filtfilt(Bc,Ac,IR); % [Bc,Ac]=butter(N_order,[octave_limit(ii)/f_nyquist,octave_limit(ii+1)/... % f_nyquist],'bandpass');% 3rd order Butterworth filter % tdomain=filtfilt(Bc,Ac,IR);%Bandpass filtering of impulse p_time(ii,:)=tdomain; 249 % Plot the impulse response figure; plot(tt, real(p_time(ii,:)),'b-','LineWidth',1); set(gca,'FontSize',16); title(['IR for the ' num2str(octave_center(ii)) '-Hz band at receiver #' ... num2str(receiver_location) ''], 'FontSize', 16); axis tight;xlabel('Time (s)', 'FontSize', 16), ylabel('Pressure (Pa)', ... 'FontSize', 16); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % checking whether filtering is ok by comparing filtered signal to original figure; subplot(311); fr = zeros(1,N/2+1);fr(min(band_f)/df+1:N/2+1) = p; plot(f,real(fr));axis tight; subplot(312); pp=fft(p_time(5,:)); plot(f(2:end),real(pp(1:length(pp)/2))); subplot(313); plot(tt,real(p_time(5,:))); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Subroutine for EDC and RT manipulation function [Y,X,T60,R]=level_decay(f,transfer,t,impulse,S_Rate,octave_center,... band,receiver_location) p_time=impulse; impulse=abs(hilbert((impulse))); if length(impulse)/S_Rate < 1.5 %if impulse is not at least 1.5 sec T = length(impulse); %then use the whole waveform warning(['input vector less than 1.5 s in duration']); else T = round(S_Rate * 1.8); %otherwise, use first 1.5 sec end % decay calculation using Schroeder backward integration formula iidc(length(impulse):-1:1) = cumsum(impulse(length(impulse):-1:1).^2); 250 y = iidc(1:T); y = 10 * log10(abs(y)/max(abs(y))); %generate y xtime = linspace(1,(T/(S_Rate/1000)),T); %generate x %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Regression analysis and RT calculation Num_samples=length(xtime); % Calculate the linear regression %%%%%%%%%%%CHECK THIS BEFORE LINE FITTING%%%%%%%%%%%%%%%%%%%%%%%%%%%% Regression_Span = [round(Num_samples*0.02):round(Num_samples*0.5)]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 xx=xtime(Regression_Span);yy=y(Regression_Span); n=length(Regression_Span); % slope of the best fitted line %a(1)= ( n.*sum(xx.*yy)-sum(xx).*sum(yy) )/( n.*sum(xx.^2)-(sum(xx)).^2 ); % y- intercept of the best fitted line %a(2)=( sum(yy).*sum(xx.^2)-sum(xx).*sum(xx.*yy) )/( n.*sum(xx.^2)-(sum(xx)).^2 ); % alternate way of calculating slope and intercept using MATLAB function a = polyfit(xx,yy,1); %# Fit polynomial coefficients for line %yfit = a(2)+xtime.*a(1); %# Compute the best-fit line yfit=polyval(a,xx); %Compute the residual values and correlation coefficient yresid=yy-yfit;SSresid = sum(yresid.^2);SStotal = (length(yy)-1) * var(yy); r = (1 - SSresid/SStotal); R=1000*(1-abs(r)); % Calculate the T60 for each microphone %rt = -60/a(1); rt = round(abs(60/a(1))-a(2)); %rt = round((-(60)-a(2))/a(1)); % final data to be passed T60=rt/1000; % Reverberation time X=xtime; % Decay plot x-data Y=y; % Decay plot y-data %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Plot the decay figure; plot(xtime./1000, y(1:Num_samples),'b-','LineWidth',2); hold on; % Hold to overlay plot(xtime./1000,yfit, ... % Data fit 'k', 'LineWidth',1.5); % Set line width to 2 hold off; title('Integrated Impulse Decay Curve'); 251 ylim([-35 0]);xlim([0 max(xtime./1000)*1.1]); xlabel('Time (s)'), ylabel('Level (dB)'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Fit the tangent to decay curve clear xtime; xtime = linspace(1,rt,rt*(S_Rate/1000)); if length(y)<length(xtime) difference = length(xtime)-length(y); pad(1:difference) = nan; y = [y,pad]; else y = y(1:length(xtime)); end Num_samples=length(xtime); % Calculate the linear regression %%%%%%%%%%%CHECK THIS BEFORE LINE FITTING%%%%%%%%%%%%%%%%%%%%%%%%%%%% Regression_Span = [round(Num_samples*0.02):round(Num_samples*0.5)]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 xx=xtime(Regression_Span);yy=y(Regression_Span); n=length(Regression_Span); figure; subplot(2,2,[1,2]); plot(xtime./1000, y,'b-','LineWidth',1.5); set(gca,'FontSize',16); title(['Predicted EDC\RT, FR, IR: ' num2str(octave_center(band)) ... '-Hz band at receiver #' num2str(receiver_location) ''], 'FontSize', 16); ylim([-70 0]); xlim([0 (rt/1000)*1.1]); xlabel('Time (s)', 'FontSize', 16), ylabel('Level (dB)', 'FontSize', 16); xlimits = xlim; line([xlimits(1),xlimits(2)],[-60,-60],'Color',[.4,.4,.4],'LineWidth',1); text(.6 * xlimits(2),-8,['T_6_0 = ',num2str(T60),' s'],'FontSize',10,... 'color','k', 'FontSize', 16); line([xx(1)/1000,rt/1000],[yy(1),-60],'Color','r','LineWidth',1); subplot(224); plot(t, real(p_time),'b-','LineWidth',1); set(gca,'FontSize',16); axis tight;xlabel('Time (s)', 'FontSize', 16); 252 subplot(223); plot(f(find(f ~= 0)), real(transfer(find(abs(transfer) ~= 0))),... 'b-','LineWidth',1); set(gca,'FontSize',16); axis tight;xlabel('Frequency (Hz)', 'FontSize', 16), ylabel('Pressure (Pa)',... 'FontSize', 16); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Subroutine for modal composition analyses function [modes,delta,no_of_modes,modal_density,modal_spacing,modal_width,... modal_overlap,modal_overlap_factor]=... eigen_value(eigen_f,octave_center,octave_limit) [sorted,r]=sort(real(eigen_f)); eigen_f=eigen_f(r); eigen_f=complex(real(eigen_f),2*pi*imag(eigen_f)); freq=real(eigen_f); damp=imag(eigen_f); % counting modes in a given frequency band for ii=1:length(octave_center) j=0; for ll=1:length(eigen_f) if freq(ll)<=octave_limit(ii+1) && freq(ll)>=octave_limit(ii) j=j+1; modes(ii,j)=freq(ll);%frequency delta(ii,j)=damp(ll);%damping constant width(ii,j)=delta(ii,j)/pi; end end no_of_modes(ii)=j; modal_density(ii)=no_of_modes(ii)/(octave_limit(ii+1)-octave_limit(ii)); modal_width(ii)=sum(width(ii,1:no_of_modes(ii)))/no_of_modes(ii); modal_overlap(ii)=modal_width(ii).*modal_density(ii); end 253 % calculating modal spacing for ii=1:length(octave_center) for jj=2:no_of_modes(ii) spacing(ii,jj-1)=modes(ii,jj)-modes(ii,jj-1); end modal_spacing(ii)=sum(spacing(ii,:))/length(spacing(ii,:)); modal_overlap_factor(ii)=modal_width(ii)/modal_spacing(ii); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Subroutine for modal decay and RT manipulation function [Y,X,T60,R]=modal_decay(damp,octave_center,band) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % collective modal decay t=0:0.01:20; for j=1:length(t) for jj=1:length(damp) T(jj)=6*log(10)/(2*damp(jj));%RT for individual modes numerator(jj)=T(jj)*10^(-6*t(j)/T(jj)); end numerator(j)=sum(numerator(1:length(damp))); denominator(j)=sum(T(1:length(damp))); decay(j)=10*log10(numerator(j)/denominator(j)); if decay(j)<=-60 rt=j; RT=t(rt); break; else RT=0; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % RT calculation % final data to be passed T60=RT; % Reverberation time 254 X=t(1:rt); % Decay plot x-data Y=decay(1:rt); % Decay plot y-data %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Regression analysis and RT calculation Num_samples=length(X); % Calculate the linear regression %%%%%%%%%%%CHECK THIS BEFORE LINE FITTING%%%%%%%%%%%%%%%%%%%%%%%%%%%% Regression_Span = [round(Num_samples*0.05):round(Num_samples*0.8)]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 xx=X(Regression_Span);yy=Y(Regression_Span); n=length(Regression_Span); % slope and intercept of the best fitted line a = polyfit(xx,yy,1); %# Fit polynomial coefficients for line %yfit = a(2)+xtime.*a(1); %# Compute the best-fit line yfit=polyval(a,xx); %Compute the residual values and correlation coefficient yresid=yy-yfit;SSresid = sum(yresid.^2);SStotal = (length(yy)-1) * var(yy); r = (1 - SSresid/SStotal); R=1000*(1-abs(r)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Plot the decay figure; plot(X, Y,'b-','LineWidth',2); set(gca,'FontSize',16); title(['Modal RT for the ' num2str(octave_center(band)) '-Hz band '], ... 'FontSize', 16); ylim([-70 0]);xlim([0 max(X)*1.1]); xlimits = xlim; xlabel('Time (s)', 'FontSize', 16), ylabel('Level (dB)', 'FontSize', 16); line([xlimits(1),xlimits(2)],[-60,-60],'Color',[.4,.4,.4],'LineWidth',1); text(.6 * xlimits(2),-8,['T_6_0 = ',num2str(T60),' s'],'FontSize',10,... 'color','k', 'FontSize', 16); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Published with MATLAB® R2013a 255 Appendix: F F.1 Third-octave-band prediction for the ISO 354 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: The predicted modal compositions as shown in Table F.1 reveal that a value for the minimum number of modes of around 20 (exact number is 19) occurs from the 100-Hz third-octave band. However, the statistically-based cut-off frequency limit is 103 Hz, which makes the 100-Hz third-octave band disqualified as the first low-frequency prediction band and, at the same time, suggests that prediction can be done accurately from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 125-Hz third-octave are 1.24, 0.8, 0.37 and 0.46, respectively. The Schroeder low-frequency limit, considering a band-averaged value of 5.6 s for the RT, as shown in Table F.3, is 390 Hz. Table F.1: Predicted third-octave-band modal compositions for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 5 5 11 19 36 61 116 219 421 Modal density, 𝑛(𝑓) 0.42 0.36 0.58 0.83 1.24 1.65 2.52 3.78 5.77 Modal spacing, 𝛿𝑓 2.46 2.13 1.61 1.25 0.80 0.59 0.40 0.27 0.17 Modal width, 𝛿𝑤 0.35 0.34 0.34 0.36 0.37 0.38 0.39 0.41 0.42 Modal overlap, 𝑀 0.15 0.12 0.20 0.30 0.46 0.63 0.99 1.56 2.43 Modal overlap factor, 𝑀𝑓 0.14 0.16 0.21 0.29 0.46 0.65 0.99 1.56 2.43 Spatial uniformity of SPL: From Table F.2, the predicted standard deviations of SPL, a measure of spatial variation, reveal that in lower frequency bands – especially in the 50- and 63-Hz third-octave bands – this variation is greater than 3 dB. On the other hand, a deviation of around 1.5 dB occurs from the 80-Hz third-octave band. These deviations reduce further as the frequency increases. 256 Table F.2: Predicted third-octave-band SPL statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 152.6 148.0 153.7 153.2 152.6 157.9 163.4 166.5 170.2 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 144.2 154.1 152.2 152.6 151.6 157.3 161.9 165.5 169.4 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 153.7 148.0 153.1 153.7 153.3 156.7 164.2 165.1 169.7 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 147.4 153.6 149.8 154.8 155.8 159.0 163.0 167.0 170.3 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 151.0 147.4 153.6 151.7 151.7 158.2 162.0 166.2 170.1 Predicted average SPL, ?̅?𝑝 (dB) 151.0 151.2 152.7 153.3 153.3 157.9 163.0 166.1 170.0 Standard deviation, 𝜎𝐿𝑝 (dB) 4.1 3.5 1.6 1.2 1.7 0.9 1.0 0.8 0.4 Expected Sabine SPL, 𝐿𝑝 (dB) 154.4 153.8 154.8 155.3 154.5 159.1 164.1 167.1 170.9 Sabine error (%) 2.2 1.7 1.3 1.3 0.8 0.8 0.7 0.6 0.6 Expected Eyring SPL, 𝐿𝑝 (dB) 154.4 153.7 154.7 155.3 154.4 159.1 164.1 167.0 170.9 Eyring error (%) 2.21 1.63 1.31 1.27 0.73 0.75 0.69 0.56 0.55 Accuracy of the predicted SPLs: The accuracy of the predicted steady-state SPLs is checked against the Sabine and Eyring versions of the diffuse-field formulae as presented in Table F.2. The errors associated with the predicted SPLs are always less than 1% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases. Spatial uniformity of RT: From Table F.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.4 s from the 80-Hz third-octave band. Accuracy of the predicted RTs: The accuracy of the predicted RTs is checked against the expected values of the RTs calculated using the Sabine and Eyring versions of the diffuse-field formulae as presented in Table F.3. The predicted RTs in third-octave bands range from 6.18 to 5.08 s, while the expected Sabine and Eyring RTs are 4.72 s and 4.67 s, respectively. In both the cases, the errors are greater in the low-frequency bands. The errors are less than or around 20% from the 160-Hz third-octave band in both cases. It is also obvious from Table F.3 that the errors associated with the Eyring version are greater than with the Sabine version. 257 Table F.3: Predicted third-octave-band RT statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.08 7.35 7.01 6.53 6.44 6.16 5.95 5.72 5.56 Predicted RT @ Rec#1: 𝑇60 (s) 5.44 5.73 6.12 5.53 6.08 5.83 5.39 5.35 4.94 Predicted RT @ Rec#2: 𝑇60 (s) 6.48 6.84 6.01 5.62 5.75 5.64 5.37 5.36 5.08 Predicted RT @ Rec#3: 𝑇60 (s) 5.37 5.70 5.86 5.79 5.99 6.06 4.93 4.98 4.91 Predicted RT @ Rec#4: 𝑇60 (s) 5.66 6.64 6.34 5.35 6.50 4.92 5.50 5.42 5.34 Predicted RT @ Rec#5: 𝑇60 (s) 5.63 6.01 6.05 5.58 6.68 5.68 5.49 5.33 4.86 Predicted average RT: ?̅?60 (s) 5.69 6.15 6.07 5.57 6.18 5.60 5.33 5.28 5.02 Standard deviation, 𝜎𝑇60 (s) 0.45 0.53 0.18 0.16 0.38 0.43 0.23 0.17 0.19 Expected Sabine RT: 𝑇60 (s) 4.72 4.72 4.72 4.72 4.72 4.72 4.72 4.72 4.72 Sabine error (%) 20.5 30.2 28.6 17.9 30.9 18.6 12.8 11.9 6.34 Expected Eyring RT, 𝑇60 (s) 4.67 4.67 4.67 4.67 4.67 4.67 4.67 4.67 4.67 Eyring error (%) 21.7 31.6 29.9 19.1 32.2 19.8 14.0 13.0 7.42 Decay rate (dB/s) 10.5 9.8 9.9 10.8 9.7 10.7 11.3 11.4 11.9 Linearity of temporal decay curves: From Table F.4, the predicted curvatures of the energy-decay curves vary within 7 to 11 from the 100-Hz third-octave band. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix G.1. Table F.4: Predicted third-octave-band decay-curve curvature statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 5.0 4.1 7.9 6.9 5.8 8.3 6.4 3.0 11.5 𝜅 @ Rec#2 5.7 5.4 4.0 5.9 12.7 14.0 21.5 6.8 6.6 𝜅 @ Rec#3 6.7 4.9 12.8 6.9 18.4 9.6 9.0 12.0 5.1 𝜅 @ Rec#4 2.8 5.4 55.1 3.6 7.2 6.0 4.5 3.7 8.0 𝜅 @ Rec#5 9.5 11.9 7.9 11.9 6.8 16.3 5.9 13.0 4.7 Average, 𝜅 5.95 6.34 17.6 7.1 10.2 10.9 9.45 7.72 7.16 Modal, 𝜅 0.54 0.34 0.18 0.10 0.17 0.14 0.10 0.19 0.15 258 Table F.5: Predicted third-octave-band source-power level statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 155.1 154.5 155.5 156.0 155.2 159.8 164.8 167.8 171.6 Sabine error (%) 15.8 21.4 20.5 14.1 21.7 14.4 10.5 9.8 5.5 Eyring error (%) 16.7 22.3 21.4 15.0 22.7 15.4 11.4 10.7 6.4 Predicted, ?̅?𝑤 (dB) (1 source) 152.1 151.5 152.5 153.0 152.2 156.8 161.8 164.8 168.6 Accuracy of the predicted source-power levels: The predicted source-power levels and the corresponding errors in third-octave bands are presented in Table F.5. The errors associated with the predicted source-power levels are less than 15% from the 160-Hz third-octave band. The errors are greater than 20% in the rest of the bands. Accuracy of the predicted room absorptions: It is seen from Table F.6 that the errors are less than or around 15% from the 160-Hz third-octave band, and greater than 20% in the rest of the bands. The same is true for the room absorption-coefficient prediction case, as the absorption-coefficient predictions depend on the accuracy of the absorption predictions. Table F.6: Predicted third-octave-band room absorption statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 4.17 3.86 3.91 4.26 3.84 4.24 4.45 4.49 4.73 Sabine expected absorption, 𝐴 (m2) 5.03 5.03 5.03 5.03 5.03 5.03 5.03 5.03 5.03 Sabine error (%) 17.0 23.2 22.3 15.2 23.6 15.7 11.4 10.6 5.96 Eyring expected absorption, 𝐴 (m2) 5.08 5.08 5.08 5.08 5.08 5.08 5.08 5.08 5.08 Eyring error (%) 17.8 24.0 23.0 16.1 24.4 16.5 12.3 11.5 6.91 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 17.0 23.2 22.3 15.2 23.6 15.7 11.4 10.6 5.96 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 17.7 23.8 22.9 15.9 24.2 16.4 12.2 11.4 6.84 259 F.2 Third-octave-band prediction for the ISO 354 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table F.7. The cut-off limit will remain be the same as before. Table F.7: Predicted third-octave-band modal compositions for ISO 354 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 5 6 11 21 35 63 120 221 421 Modal density, 𝑛(𝑓) 0.42 0.43 0.58 0.91 1.21 1.70 2.61 3.81 5.77 Modal spacing, 𝛿𝑓 2.46 1.62 1.67 1.06 0.82 0.58 0.39 0.26 0.17 Modal width, 𝛿𝑤 1.14 6.14 1.17 5.38 2.06 3.28 2.72 2.50 2.55 Modal overlap, 𝑀 0.48 2.63 0.68 4.91 2.48 5.58 7.10 9.51 14.7 Modal overlap factor, 𝑀𝑓 0.46 3.80 0.70 5.05 2.50 5.69 7.06 9.58 14.8 Table F.8: Predicted third-octave-band SPL statistics for ISO 354 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 147.2 145.7 149.3 148.0 150.8 155.5 157.5 162.7 166.1 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 142.1 146.0 144.8 148.5 150.5 154.1 158.8 163.5 165.2 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 149.1 143.9 148.3 149.6 150.0 152.9 157.1 161.3 165.1 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 143.2 146.3 139.5 149.8 147.3 155.1 159.0 161.7 166.3 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 146.5 142.8 148.7 145.8 148.6 152.5 157.0 162.3 164.3 Predicted average SPL, ?̅?𝑝 (dB) 146.3 145.2 147.3 148.5 149.6 154.2 158.0 162.4 165.5 Standard deviation, 𝜎𝐿𝑝 (dB) 3.02 1.56 4.29 1.62 1.47 1.34 0.95 0.85 0.80 260 Table F.9: Predicted third-octave-band RT statistics for ISO 354 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 4.40 4.82 3.97 3.12 3.87 3.04 2.50 2.45 2.19 Predicted RT @ Rec#1: 𝑇60 (s) 3.25 3.17 3.25 3.23 3.87 2.66 2.02 2.15 2.45 Predicted RT @ Rec#2: 𝑇60 (s) 3.86 2.37 2.54 2.49 2.98 2.48 2.31 2.24 2.68 Predicted RT @ Rec#3: 𝑇60 (s) 3.44 2.51 3.20 2.97 3.24 2.74 1.89 2.40 2.32 Predicted RT @ Rec#4: 𝑇60 (s) 3.82 2.92 4.69 3.12 5.59 3.15 2.77 2.98 3.64 Predicted RT @ Rec#5: 𝑇60 (s) 3.40 2.56 3.28 2.85 3.17 2.93 2.06 2.35 2.42 Predicted average RT: ?̅?60 (s) 3.54 2.68 3.26 2.91 3.58 2.77 2.17 2.40 2.63 Standard deviation, 𝜎𝑇60 (s) 0.27 0.33 0.80 0.29 1.09 0.26 0.35 0.33 0.55 Decay rate (dB/s) 17.0 22.4 18.4 20.6 16.8 21.6 27.6 25.0 22.8 Spatial uniformity of SPL: The standard deviations of the SPL for this ‘with absorber’ case, as a measure of spatial variation, are presented in Table F.8. It is noticeable that the deviations are greater in the low-frequency bands, and are below the ISO-prescribed limit of 1.5 dB from the 125-Hz third-octave band. Overall, the deviations are greater than the values obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT for this ‘with absorber’ case, as a measure of spatial variation, are presented in Table F.9. It is obvious that the deviations are greater than in the previous ‘without absorber’ case in almost all third-octave bands. Table F.10: Predicted third-octave-band decay-curve curvature statistics for ISO 354 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 25.3 58.2 7.6 15.7 12.7 16.1 10.8 18.0 16.7 𝜅 @ Rec#2 44.9 113.2 25.2 20.9 34.7 12.8 33.9 19.2 8.3 𝜅 @ Rec#3 23.2 11.5 7.0 8.3 56.4 17.2 12.2 11.6 6.3 𝜅 @ Rec#4 41.0 190.4 26.6 34.1 71.9 24.8 44.4 62.6 77.5 𝜅 @ Rec#5 31.0 39.4 6.4 12.1 29.5 23.2 11.1 33.3 10.6 Average, 𝜅 33.1 82.5 14.6 18.2 41.0 18.8 22.5 28.9 23.9 Modal, 𝜅 0.07 0.12 0.80 2.50 1.11 1.89 1.84 3.19 1.87 261 Linearity of decay curves: The curvatures of the energy-decay curves as presented in Table F.10 reveal that the curvatures are greater than those obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave-bands are presented in Appendix G.2. Table F.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ISO 354 rectangular reverberation room with the longest z-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.29 0.58 0.39 0.45 0.32 0.50 0.75 0.62 0.49 Sabine error (%) 61.6 24.1 49.1 41.1 57.8 34.7 1.9 18.0 35.0 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.25 0.44 0.32 0.36 0.27 0.39 0.53 0.46 0.39 Eyring error (%) 66.7 42.3 57.8 52.5 63.9 48.5 30.8 39.0 48.7 Accuracy of the sample absorption coefficients: The accuracy of the predicted absorption coefficients of an absorber sample are presented in Table F.11. It is noticeable that the predicted values deviate from the theoretical values to a great extent. However, comparing the predicted absorption coefficients shown in Table 3.12 and Table F.11, it is easy to say that in Case #2, there is an overall improvement in the prediction accuracy than in Case #1 prediction. 262 F.3 Octave-band prediction for the ISO 354 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: The modal compositions data shown in Table F.12 implies that prediction can be done accurately from the 63-Hz octave band, as the required minimum number of modes occurs from that band. However, the suggested statistically-based cut-off frequency limit is 65 Hz, which suggests that prediction has to be started from the 125-Hz band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz octave band are 1.28, 0.77, 0.37 and 0.48, respectively. On the other hand, the pure-tone-based Schroeder limit corresponding to a modal overlap factor of 3 and a band-averaged RT of 5.9 s, as shown in Table F.14, is 400 Hz Table F.12: Predicted octave-band modal compositions for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 21 114 758 Modal density, 𝑛(𝑓) 0.48 1.28 4.26 Modal spacing, 𝛿𝑓 2.08 0.77 0.23 Modal width, 𝛿𝑤 0.34 0.37 0.41 Modal overlap, 𝑀 0.16 0.48 1.76 Modal overlap factor, 𝑀𝑓 0.17 0.48 1.77 Spatial uniformity of SPL: The predicted SPL standard deviations are within 0.4 to 0.85 dB in all octave bands as shown in Table F.13. Accuracy of the predicted SPLs: As shown in Table F.13, the predicted SPLs show errors below 1% from the 125-Hz octave band, while the errors are around 1.5% in the 63-Hz band for both Sabine and Eyring cases. 263 Table F.13: Predicted octave-band SPL statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 152.6 156.1 168.5 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 152.6 155.3 167.6 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 152.9 155.7 167.9 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 151.8 157.5 168.6 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 152.1 155.9 168.2 Predicted average SPL, ?̅?𝑝 (dB) 152.4 156.2 168.2 Standard deviation, 𝜎𝐿𝑝 (dB) 0.43 0.85 0.42 Expected Sabine SPL, 𝐿𝑝 (dB) 154.6 157.6 169.3 Sabine error (%) 1.40 0.87 0.66 Expected Eyring SPL, 𝐿𝑝 (dB) 154.6 157.5 169.2 Eyring error (%) 1.37 0.85 0.63 Table F.14: Predicted octave-band RT statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.12 6.34 5.69 Predicted RT @ Rec#1: 𝑇60 (s) 6.57 6.11 5.18 Predicted RT @ Rec#2: 𝑇60 (s) 7.10 5.81 5.14 Predicted RT @ Rec#3: 𝑇60 (s) 5.64 5.93 4.87 Predicted RT @ Rec#4: 𝑇60 (s) 8.74 5.84 5.29 Predicted RT @ Rec#5: 𝑇60 (s) 6.21 5.82 5.09 Predicted average RT: ?̅?60 (s) 6.70 5.90 5.11 Standard deviation, 𝜎𝑇60 (s) 1.19 0.13 0.16 Expected Sabine RT: 𝑇60 (s) 4.72 4.72 4.72 Sabine error (%) 42.0 24.9 8.20 Expected Eyring RT, 𝑇60 (s) 4.67 4.67 4.67 Eyring error (%) 43.4 26.2 9.30 Decay rate (dB/s) 8.95 10.2 11.7 264 Table F.15: Predicted octave-band curvature statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 1.28 3.43 5.32 𝜅 @ Rec#2 1.55 5.91 8.58 𝜅 @ Rec#3 4.70 6.54 3.16 𝜅 @ Rec#4 4.01 3.49 2.86 𝜅 @ Rec#5 2.98 7.39 3.35 Average, 𝜅 2.90 5.35 4.66 Modal, 𝜅 0.31 0.16 0.18 Spatial uniformity of RT: The predicted RT standard deviations are less than 0.2 s from the 125-Hz band, while 1.19 s in the 63-Hz band. These are shown in Table F.14. Accuracy of the predicted RTs: As shown in Table F.14, the predicted RTs show errors below or around 25% from the 125-Hz octave band. Linearity of decay curves: The curvatures are within 2 to 6 in all octave bands as shown in Table F.15. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix G.3. Accuracy of the predicted source-power levels: The predicted source-power levels show errors below 20% from the 125-Hz band, while the errors are greater than 20% in the rest of the bands as shown in Table F.16. Table F.16: Predicted octave-band source-power level statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 155.3 158.3 170.0 Sabine error (%) 27.2 18.4 7.0 Eyring error (%) 28.2 19.3 7.9 Predicted, ?̅?𝑤 (dB) (1 source) 152.3 155.2 167.0 265 Accuracy of the predicted room absorptions: The predicted room absorptions, as well as the absorption coefficients, show errors below 20% from the 125-Hz band, while the errors are greater than 20% in the rest of the bands. These are shown in Table F.17. Table F.17: Predicted octave-band room absorption statistics for ISO 354 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 3.54 4.02 4.65 Sabine expected absorption, 𝐴 (m2) 5.03 5.03 5.03 Sabine error (%) 29.6 19.9 7.58 Eyring expected absorption, 𝐴 (m2) 5.08 5.08 5.08 Eyring error (%) 30.3 20.7 8.51 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Sabine error (%) 29.6 19.9 7.58 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 30.1 20.6 8.43 266 F.4 Octave-band prediction for the ISO 354 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit as there no change in the modal compositions except the modal width and, hence, the modal overlap as shown in Table F.18. Table F.18: Predicted octave-band modal compositions for ISO 354 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 22 118 763 Modal density, 𝑛(𝑓) 0.50 1.33 4.29 Modal spacing, 𝛿𝑓 2.02 0.75 0.23 Modal width, 𝛿𝑤 2.52 3.30 2.56 Modal overlap, 𝑀 1.26 4.37 11.0 Modal overlap factor, 𝑀𝑓 1.25 4.41 11.0 Table F.19: Predicted octave-band SPL statistics for ISO 354 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 148.2 153.4 164.3 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 145.4 152.4 164.2 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 148.2 151.9 163.2 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 144.5 152.8 164.3 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 147.4 150.5 163.0 Predicted average SPL, ?̅?𝑝 (dB) 147.0 152.3 163.8 Standard deviation, 𝜎𝐿𝑝 (dB) 1.72 1.09 0.60 267 Table F.20: Predicted octave-band RT statistics for ISO 354 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 4.45 3.57 2.36 Predicted RT @ Rec#1: 𝑇60 (s) 3.24 3.20 2.38 Predicted RT @ Rec#2: 𝑇60 (s) 4.89 2.83 2.43 Predicted RT @ Rec#3: 𝑇60 (s) 3.24 3.05 2.27 Predicted RT @ Rec#4: 𝑇60 (s) 13.0 4.12 3.01 Predicted RT @ Rec#5: 𝑇60 (s) 3.58 3.15 2.34 Predicted average RT: ?̅?60 (s) 4.25 3.22 2.46 Stnd. deviation, 𝜎𝑇60 (s) 4.48 0.50 0.30 Decay rate (dB/s) 14.1 18.7 24.4 Spatial uniformity of SPL: The predicted SPL deviations are within 0.6 to 1.1 dB from the 125-Hz band, while 1.72 dB in the 63-Hz band, as shown in Table F.19. However, overall deviations are greater than in the ‘without absorber’ case. Spatial uniformity of RT: The predicted RT deviations are within 0.3 to 0.5 s from the 125-Hz band, while 4.48 s in the 63-Hz band, as shown in Table F.20. However, overall deviations are greater than in the ‘without absorber’ case. Table F.21: Predicted octave-band decay-curve curvature statistics for ISO 354 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 11.1 13.7 17.0 𝜅 @ Rec#2 152.0 21.1 5.7 𝜅 @ Rec#3 12.7 9.3 3.1 𝜅 @ Rec#4 244.3 54.1 51.3 𝜅 @ Rec#5 15.7 6.9 10.2 Average, 𝜅 87.2 21.0 17.5 Modal, 𝜅 0.60 2.61 2.55 268 Linearity of decay curves: The curvatures are within 15 to 21 from the 125-Hz band, as shown in Table F.21. However, overall curvatures are greater than in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix G.4. Table F.22: Predicted octave-band sample absorber absorption coefficient statistics for ISO 354 rectangular reverberation room with the longest z-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.24 0.39 0.58 Sabine error (%) 69.0 49.3 24.2 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.21 0.32 0.44 Eyring error (%) 72.4 57.9 42.4 Accuracy of the predicted sample absorption coefficients: As shown in Table F.22, the e predicted Sabin absorption coefficients show errors around 24% in the 250-Hz band, while the errors are greater than 50% in the rest of the bands. On the other hand, the Eyring predictions yield errors around 40% in the 250-Hz band, while the errors are than 55% in the rest of the bands. 269 Appendix: G G.1 Third-octave-band prediction for the ISO 354 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber G.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 7.08 s0 2 4 6 8-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 7.35 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 7.01 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 6.53 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.44 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 6.16 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 5.95 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.72 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 5.56 s270 G.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs in third-octave bands, at receiver #5 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 5.63 s0 1-20020Time (s)45 50 55-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.013 s0 1-40-20020Time (s)60 65 70-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 6.052 s0 1-50050Time (s)70 80-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 5.58 s0 1-40-2002040Time (s)90 100 110-2000020004000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 6.678 s0 1-50050Time (s)120 130 140-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 5.678 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 5.491 s0 1-1000100Time (s)180 200 220-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.33 s0 1-2000200Time (s)240 260 280-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 4.86 s0 1-5000500Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)271 G.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 5.66 s0 1-40-2002040Time (s)45 50 55020004000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.638 s0 1-40-2002040Time (s)60 65 70-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 6.34 s0 1-50050Time (s)70 80-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 5.351 s0 1-50050Time (s)90 100 1100500010000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 6.502 s0 1-50050Time (s)120 130 140-8000-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 4.923 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 5.498 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.419 s0 1-2000200400Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 5.343 s0 1-400-2000200Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)272 G.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 5.368 s0 1-40-2002040Time (s)45 50 55-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.697 s0 1-20020Time (s)60 65 70-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.862 s0 1-50050Time (s)70 80-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 5.785 s0 1-50050Time (s)90 100 110-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.986 s0 1-50050Time (s)120 130 140-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 6.058 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 4.93 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.982 s0 1-2000200Time (s)240 260 280-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 4.911 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)273 G.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 6.484 s0 1-40-2002040Time (s)45 50 55-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.841 s0 1-50050Time (s)60 65 70-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 6.009 s0 1-40-2002040Time (s)70 80-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.621 s0 1-50050Time (s)90 100 110-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.752 s0 1-40-2002040Time (s)120 130 140-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.644 s0 1-50050Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 5.371 s0 1-1000100Time (s)180 200 220-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.355 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 5.081 s0 1-400-2000200400Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)274 G.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 5.437 s0 1-20020Time (s)45 50 5502000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.732 s0 1-20020Time (s)60 65 70-10000100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 6.123 s0 1-50050Time (s)70 80-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 5.525 s0 1-50050Time (s)90 100 110-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 6.081 s0 1-50050Time (s)120 130 140-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 5.83 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 5.393 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.345 s0 1-2000200Time (s)240 260 280-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 4.94 s0 1-5000500Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)275 G.2 Third-octave-band prediction for the ISO 354 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber G.2.1 Theoretical modal decay curves and the corresponding RTs, in third-octave bands 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 4.4 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.82 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 3.97 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 3.12 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.87 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 3.04 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.5 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.45 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.19 s276 G.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 3.402 s0 1-20020Time (s)45 50 55-3000-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 2.556 s0 1-20020Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 3.283 s0 1-40-2002040Time (s)70 80-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 2.852 s0 1-40-20020Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 3.173 s0 1-50050Time (s)120 130 140-1000010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 2.929 s0 1-1000100Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.058 s0 1-1000100Time (s)180 200 220-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.354 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 2.42 s0 1-2000200Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)277 G.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 3.82 s0 1-40-2002040Time (s)45 50 55010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 2.922 s0 1-20020Time (s)60 65 70-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 4.686 s0 1-40-2002040Time (s)70 80-50005001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 3.122 s0 1-50050Time (s)90 100 110020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.591 s0 1-40-2002040Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 3.151 s0 1-1000100Time (s)150 160 170-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.769 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.982 s0 1-2000200400Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 3.638 s0 1-400-2000200400Time (s)300 350-50000500010000Frequency (Hz)Pressure (Pa)278 G.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 3.437 s0 1-40-2002040Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 2.512 s0 1-10010Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 3.201 s0 1-20020Time (s)70 80020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 2.971 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 3.242 s0 1-50050Time (s)120 130 140-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 2.736 s0 1-1000100Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 1.892 s0 1-1000100Time (s)180 200 220-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.402 s0 1-2000200Time (s)240 260 280-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 2.318 s0 1-2000200Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)279 G.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 3.858 s0 1-40-2002040Time (s)45 50 55-1000-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 2.369 s0 1-40-2002040Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 2.535 s0 1-20020Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 2.494 s0 1-40-2002040Time (s)90 100 110-10000100020003000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.976 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.483 s0 1-1000100Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.314 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.242 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 2.679 s0 1-400-2000200Time (s)300 350-10000-500005000Frequency (Hz)Pressure (Pa)280 G.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 3.246 s0 1-20020Time (s)45 50 55020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 3.173 s0 1-2002040Time (s)60 65 70010002000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 3.252 s0 1-40-2002040Time (s)70 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 3.226 s0 1-40-2002040Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.865 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.662 s0 1-1000100Time (s)150 160 170-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.018 s0 1-1000100Time (s)180 200 220-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.152 s0 1-2000200Time (s)240 260 280-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 2.452 s0 1-400-2000200400Time (s)300 3500500010000Frequency (Hz)Pressure (Pa)281 G.3 Octave-band prediction for the ISO 354 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber G.3.1 Theoretical modal decay curves and the corresponding RTs, in octave bands G.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #5 G.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #4 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 7.12 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.34 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.69 s0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.207 s0 1-50050Time (s)60 80-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.815 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.087 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 8.737 s0 1-50050Time (s)60 80-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.841 s0 1-1000100Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.286 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)282 G.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #3 G.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #2 G.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.644 s0 1-100-50050Time (s)60 80-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.925 s0 1-1000100200Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.867 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 7.098 s0 1-50050Time (s)60 80-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.809 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.143 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 6.574 s0 1-50050Time (s)60 80-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 6.109 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.181 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)283 G.4 Octave-band prediction for the ISO 354 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber G.4.1 Theoretical modal decay curves and the corresponding RTs, in octave bands G.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 G.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.45 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.57 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.36 s0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 3.584 s0 1-50050Time (s)60 80-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 3.148 s0 1-1000100Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.342 s0 1-5000500Time (s)200 300-500005000Frequency (Hz)Pressure (Pa)0 5 10-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 13.035 s0 1-50050Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 4.121 s0 1-1000100Time (s)100 150-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 3.005 s0 1-5000500Time (s)200 300-50000500010000Frequency (Hz)Pressure (Pa)284 G.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 G.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs in octave bands, at receiver #2 G.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 3.239 s0 1-50050Time (s)60 80-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 3.047 s0 1-1000100Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.269 s0 1-5000500Time (s)200 300-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 4.889 s0 1-50050Time (s)60 80-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.828 s0 1-1000100Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.429 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 3.238 s0 1-50050Time (s)60 80-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.202 s0 1-1000100Time (s)100 150-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.376 s0 1-5000500Time (s)200 300-50000500010000Frequency (Hz)Pressure (Pa)285 Appendix: H H.1 Third-octave-band prediction for the ISO 354 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table H.1, the corresponding statistically-based cut-off frequency limit is 103 Hz, which suggests that prediction can be done from the 125-Hz band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 125-Hz third-octave band are 1.17, 0.82, 0.37 and 0.45, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 5.6 s for the RT, as shown in Table H.3, is 390 Hz. Table H.1: Predicted third-octave-band modal compositions for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 6 11 19 34 63 118 221 418 Modal density, 𝑛(𝑓) 0.25 0.43 0.58 0.83 1.17 1.70 2.57 3.81 5.73 Modal spacing, 𝛿𝑓 3.41 1.91 1.74 1.23 0.82 0.59 0.39 0.26 0.17 Modal width, 𝛿𝑤 0.35 0.35 0.34 0.36 0.37 0.38 0.39 0.41 0.42 Modal overlap, 𝑀 0.09 0.15 0.20 0.30 0.43 0.64 1.01 1.55 2.38 Modal overlap factor, 𝑀𝑓 0.10 0.18 0.20 0.29 0.45 0.65 1.01 1.57 2.39 Spatial uniformity of SPL: From Table H.2, the predicted standard deviations of SPL are less than 0.9 dB from the 160-Hz band, while the deviations are greater than 1.5 dB in the rest of the bands. Accuracy of the predicted SPLs: As presented in Table H.2, the errors associated with the predicted SPLs are always less than or around 1% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases. In the rest of the bands, the errors are greater than 1.2%. 286 Table H.2: Predicted third-octave-band SPL statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 145.3 149.8 145.8 151.1 151.3 160.1 164.5 166.7 170.7 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 140.1 146.3 152.5 154.7 154.2 158.8 163.2 165.7 171.0 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 148.9 138.2 150.2 152.4 152.0 160.1 162.9 165.2 169.9 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 147.2 142.6 146.7 152.2 155.9 158.1 162.6 165.7 169.9 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 152.2 138.5 151.0 154.8 157.1 158.9 163.3 167.1 169.7 Predicted average SPL, ?̅?𝑝 (dB) 148.3 145.3 149.9 153.3 154.7 159.3 163.3 166.1 170.3 Standard deviation, 𝜎𝐿𝑝 (dB) 4.8 5.6 3.0 1.6 2.5 0.9 0.7 0.8 0.6 Expected Sabine SPL, 𝐿𝑝 (dB) 151.8 147.8 151.9 155.2 156.3 160.8 164.8 167.2 171.1 Sabine error (%) 2.3 1.7 1.3 1.2 1.1 0.9 0.9 0.6 0.5 Expected Eyring SPL, 𝐿𝑝 (dB) 151.8 147.8 151.9 155.1 156.3 160.8 164.8 167.2 171.1 Eyring error (%) 2.26 1.65 1.29 1.19 1.03 0.92 0.88 0.63 0.46 Table H.3: Predicted third-octave-band RT statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.06 6.48 6.56 6.39 6.13 6.04 5.77 5.57 5.40 Predicted RT @ Rec#1: 𝑇60 (s) 5.84 5.85 6.09 6.00 5.33 5.44 5.02 5.18 4.74 Predicted RT @ Rec#2: 𝑇60 (s) 5.79 5.78 5.56 5.51 5.70 5.59 5.32 5.33 4.89 Predicted RT @ Rec#3: 𝑇60 (s) 5.47 7.44 5.64 5.80 5.64 4.75 5.00 5.40 5.25 Predicted RT @ Rec#4: 𝑇60 (s) 5.63 6.19 7.22 5.76 5.56 5.51 4.54 5.46 5.58 Predicted RT @ Rec#5: 𝑇60 (s) 5.68 6.32 6.82 5.79 5.67 5.23 5.15 4.65 6.01 Predicted average RT: ?̅?60 (s) 5.68 6.26 6.20 5.77 5.58 5.29 4.99 5.19 5.25 Standard deviation, 𝜎𝑇60 (s) 0.15 0.67 0.73 0.18 0.15 0.34 0.29 0.33 0.52 Expected Sabine RT: 𝑇60 (s) 4.78 4.78 4.78 4.78 4.78 4.78 4.78 4.78 4.78 Sabine error (%) 18.9 31.1 29.8 20.7 16.7 10.6 4.5 8.5 10.0 Expected Eyring RT, 𝑇60 (s) 4.73 4.73 4.73 4.73 4.73 4.73 4.73 4.73 4.73 Eyring error (%) 20.1 32.4 31.1 22.0 17.9 11.8 5.6 9.6 11.1 Decay rate (dB/s) 10.6 9.6 9.7 10.4 10.8 11.4 12.0 11.6 11.4 287 Table H.4: Predicted third-octave-band decay curve curvature statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 9.0 6.4 25.5 7.4 13.5 15.7 3.5 7.6 3.1 𝜅 @ Rec#2 12.4 9.2 20.9 10.9 9.2 4.3 12.5 5.6 3.8 𝜅 @ Rec#3 7.5 6.9 12.6 5.8 16.0 20.3 13.2 6.9 4.2 𝜅 @ Rec#4 2.8 6.6 19.5 11.9 3.4 19.3 4.8 10.6 4.3 𝜅 @ Rec#5 5.6 9.4 58.9 5.7 6.5 12.3 4.0 7.1 17.2 Average, 𝜅 7.47 7.70 27.5 8.34 9.72 14.4 7.64 7.58 6.51 Modal, 𝜅 0.22 0.02 0.02 0.04 0.02 0.03 0.03 0.03 0.01 Spatial uniformity of RT: From Table H.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.52 s from the 100-Hz third-octave band. Accuracy of the predicted RTs: As presented in Table H.3, the predicted RTs show errors less than or around 15% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases; while the errors are greater than 20% in the rest of the bands. Linearity of temporal decay curves: From Table H.4, the predicted curvatures of the energy-decay curves vary within 6 to 15 from the 100-Hz band. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix I.1. Accuracy of the predicted source-power levels: As presented in Table H.5, the predicted source-power level errors are less than or around 15% from the 100-Hz third-octave band. Table H.5: Predicted third-octave-band source-power level statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 152.5 148.5 152.6 155.9 157.0 161.5 165.5 167.9 171.8 Sabine error (%) 14.6 21.6 20.9 15.7 13.1 8.8 4.0 7.2 8.3 Eyring error (%) 15.5 22.5 21.8 16.6 14.0 9.7 4.9 8.1 9.2 Predicted, ?̅?𝑤 (dB) (1 source) 149.5 145.5 149.6 152.8 154.0 158.5 162.5 164.9 168.8 288 Table H.6: Predicted third-octave-band room absorption statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 4.20 3.81 3.85 4.14 4.28 4.51 4.78 4.60 4.54 Sabine expected absorption, 𝐴 (m2) 4.99 4.99 4.99 4.99 4.99 4.99 4.99 4.99 4.99 Sabine error (%) 15.9 23.7 22.9 17.2 14.3 9.62 4.30 7.85 9.07 Eyring expected absorption, 𝐴 (m2) 5.04 5.04 5.04 5.04 5.04 5.04 5.04 5.04 5.04 Eyring error (%) 16.7 24.5 23.7 18.0 15.2 10.5 5.26 8.78 9.98 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 15.9 23.7 22.9 17.2 14.3 9.62 4.30 7.85 9.07 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 16.6 24.3 23.5 17.9 15.1 10.4 5.21 8.70 9.89 Accuracy of the predicted room absorptions: As presented in Table H.6, the errors are less than or around 15% from the 100-Hz third-octave band, and greater than 20% in the rest of the bands. The same is true for the room absorption coefficient prediction case, as the absorption coefficient predictions depend on the accuracy of the absorption predictions. 289 H.2 Third-octave-band prediction for the ISO 354 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table H.7. The cut-off frequency limit will remain be the same as before. Table H.7: Predicted third-octave-band modal compositions for ISO 354 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 6 13 19 36 63 121 224 418 Modal density, 𝑛(𝑓) 0.25 0.43 0.68 0.83 1.24 1.70 2.63 3.86 5.73 Modal spacing, 𝛿𝑓 3.99 1.85 1.42 1.25 0.79 0.59 0.38 0.26 0.17 Modal width, 𝛿𝑤 0.91 1.20 5.56 2.81 3.83 2.90 3.03 2.83 2.58 Modal overlap, 𝑀 0.23 0.52 3.80 2.32 4.75 4.94 7.96 10.9 14.8 Modal overlap factor, 𝑀𝑓 0.23 0.65 3.92 2.25 4.86 4.92 7.94 11.0 14.9 Spatial uniformity of SPL: The standard deviations of the SPL are presented in Table H.8. It is noticeable that the deviations are greater than the ISO-prescribed limit of 1.5 dB in all third-octave bands with the exception in the 250- and 315-Hz third-octave band. Overall, the deviations are greater than those obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT are presented in Table H.9. It is obvious that the deviations are greater than the previous ‘without absorber’ case in almost all third-octave bands. 290 Table H.8: Predicted third-octave-band SPL statistics for ISO 354 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 144.0 142.5 141.0 147.1 148.9 155.7 161.7 162.4 164.2 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 139.6 140.6 145.2 146.7 149.9 155.8 160.6 160.2 165.5 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 145.0 136.4 143.6 148.2 145.5 154.4 156.8 160.7 166.3 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 144.2 138.1 141.9 151.0 147.1 151.8 158.4 161.1 164.7 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 148.6 141.4 149.2 152.4 149.7 151.8 159.1 162.0 165.4 Predicted average SPL, ?̅?𝑝 (dB) 145.2 140.3 145.2 149.7 148.5 154.2 159.6 161.4 165.3 Standard deviation, 𝜎𝐿𝑝 (dB) 3.39 2.57 3.46 2.59 1.91 2.03 1.93 0.95 0.80 Linearity of decay curves: The curvatures of the decay curves as presented in Table H.10 reveal that the curvatures are greater than the values obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix I.2. Table H.9: Predicted third-octave-band RT statistics for ISO 354 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 3.32 3.67 4.30 3.57 3.01 2.78 2.73 2.29 2.06 Predicted RT @ Rec#1: 𝑇60 (s) 3.20 1.76 3.50 2.88 2.57 2.41 2.21 2.10 2.71 Predicted RT @ Rec#2: 𝑇60 (s) 3.69 2.22 3.60 2.61 2.52 2.21 2.37 2.04 1.97 Predicted RT @ Rec#3: 𝑇60 (s) 3.47 6.10 4.12 3.52 3.49 2.71 2.77 2.27 2.71 Predicted RT @ Rec#4: 𝑇60 (s) 3.28 2.92 4.09 3.37 2.93 2.70 2.27 2.52 2.63 Predicted RT @ Rec#5: 𝑇60 (s) 3.46 2.04 3.52 3.54 2.95 3.28 2.44 2.05 3.53 Predicted average RT: ?̅?60 (s) 3.41 2.48 3.75 3.14 2.85 2.62 2.40 2.18 2.62 Standard deviation, 𝜎𝑇60 (s) 0.19 1.87 0.31 0.42 0.39 0.41 0.22 0.20 0.56 Decay rate (dB/s) 17.6 24.2 16.0 19.1 21.0 22.9 25.0 27.5 22.9 291 Table H.10: Predicted third-octave-band decay curve curvature statistics for ISO 354 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 10.7 5.3 61.6 9.2 14.2 6.7 16.8 8.9 21.8 𝜅 @ Rec#2 19.7 23.8 34.6 11.0 66.5 46.3 17.7 8.0 10.5 𝜅 @ Rec#3 2.7 198.5 18.9 15.8 23.0 10.6 7.3 16.0 64.6 𝜅 @ Rec#4 5.9 66.1 7.1 19.1 41.8 18.7 7.7 17.5 62.5 𝜅 @ Rec#5 1.9 38.9 35.5 12.0 60.1 11.9 8.6 3.4 57.9 Average, 𝜅 8.2 66.5 31.6 13.4 41.1 18.8 11.6 10.8 43.5 Modal, 𝜅 0.5 0.4 2.2 0.8 0.9 2.1 3.1 4.1 2.8 Accuracy of the sample absorption coefficients: The predicted absorption coefficients of an absorber sample are presented in Table H.11. It is noticeable that the predicted values deviate from the theoretical values to a great extent. However, comparing the predicted absorption coefficients shown in Table 3.12, F.11 and H.11, it is easy to say that in Case #3, there is an overall improvement in the prediction accuracy than in Case #1 and Case #2 predictions. Table H.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ISO 354 oblique reverberation room with the longest z-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.29 0.61 0.26 0.36 0.43 0.48 0.54 0.67 0.48 Sabine error (%) 61.5 19.54 65.2 52.0 43.5 36.3 28.3 12.3 36.8 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.25 0.46 0.23 0.31 0.35 0.38 0.42 0.49 0.38 Eyring error (%) 66.6 39.81 69.4 59.8 54.1 49.5 44.7 36.0 49.8 292 H.3 Octave-band prediction for the ISO 354 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table H.12, the statistically-based cut-off frequency limit is 65 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz band are 1.28, 0.77, 0.37 and 0.48, respectively. On the other hand, the pure-tone-based Schroeder limit corresponding to a modal overlap factor of 3 and a band-averaged RT of 6.1 s, as shown in Table H.14, is 406 Hz. Table H.12: Predicted octave-band modal compositions for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 20 114 759 Modal density, 𝑛(𝑓) 0.45 1.28 4.26 Modal spacing, 𝛿𝑓 2.12 0.77 0.23 Modal width, 𝛿𝑤 0.35 0.37 0.41 Modal overlap, 𝑀 0.16 0.48 1.75 Modal overlap factor, 𝑀𝑓 0.16 0.48 1.75 Spatial uniformity of SPL: The predicted SPL standard deviations are less than or around 1.5 dB in all octave-bands as shown in Table H.13. Accuracy of the predicted SPLs: As shown in Table H.13, the predicted SPLs show errors below 1% from the 125-Hz octave band, while the error is around 1.3% in the 63-Hz band for both Sabine and Eyring cases. 293 Table H.13: Predicted octave-band SPL statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 148.3 157.2 169.0 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 149.0 157.3 168.8 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 146.6 157.4 167.9 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 146.7 156.9 168.0 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 150.1 158.1 168.3 Predicted average SPL, ?̅?𝑝 (dB) 148.4 157.4 168.4 Standard deviation, 𝜎𝐿𝑝 (dB) 1.51 0.44 0.47 Expected Sabine SPL, 𝐿𝑝 (dB) 150.3 159.1 169.5 Sabine error (%) 1.28 1.04 0.63 Expected Eyring SPL, 𝐿𝑝 (dB) 150.2 159.0 169.5 Eyring error (%) 1.26 1.01 0.61 Table H.14: Predicted octave-band RT statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 6.64 6.15 5.53 Predicted RT @ Rec#1: 𝑇60 (s) 6.08 5.52 4.95 Predicted RT @ Rec#2: 𝑇60 (s) 6.21 5.73 5.16 Predicted RT @ Rec#3: 𝑇60 (s) 8.12 5.37 5.23 Predicted RT @ Rec#4: 𝑇60 (s) 9.09 5.70 5.39 Predicted RT @ Rec#5: 𝑇60 (s) 6.94 5.59 5.32 Predicted average RT: ?̅?60 (s) 7.11 5.58 5.20 Standard deviation, 𝜎𝑇60 (s) 1.31 0.15 0.17 Expected Sabine RT: 𝑇60 (s) 4.78 4.78 4.78 Sabine error (%) 48.9 16.7 8.9 Expected Eyring RT, 𝑇60 (s) 4.73 4.73 4.73 Eyring error (%) 50.4 17.9 10.0 Decay rate (dB/s) 8.43 10.76 11.53 294 Table H.15: Predicted octave-band decay curve curvature statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 3.95 8.84 0.88 𝜅 @ Rec#2 4.11 3.15 1.42 𝜅 @ Rec#3 5.93 10.59 2.09 𝜅 @ Rec#4 7.88 4.87 1.24 𝜅 @ Rec#5 4.08 4.86 1.90 Average, 𝜅 5.19 6.46 1.51 Modal, 𝜅 0.05 0.04 0.03 Spatial uniformity of RT: The predicted RT standard deviations are within 0.15 to 0.17 s from the 125-Hz band, while the value is 1.31 s in the 63-Hz band. These are shown in Table H.14. Accuracy of the predicted RTs: As shown in Table H.14, the predicted RTs show errors below or around 15% from the 125-Hz band, while the errors are greater than 15% in the rest of the bands. Degree of linearity of decay curves: The curvatures are within 1 to 7 in all octave bands as shown in Table H.15. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix I.3. Table H.16: Predicted octave-band source-power level statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 151.0 159.7 170.2 Sabine error (%) 29.9 13.1 7.5 Eyring error (%) 30.8 14.0 8.4 Predicted, ?̅?𝑤 (dB) (1 source) 148.0 156.7 167.2 295 Accuracy of the predicted source-power levels: The predicted source-power levels show errors below 15% from the 125-Hz band as shown in Table H.16. Table H.17: Predicted octave-band room absorption statistics for ISO 354 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 3.35 4.28 4.58 Sabine expected absorption, 𝐴 (m2) 4.99 4.99 4.99 Sabine error (%) 32.8 14.4 8.2 Eyring expected absorption, 𝐴 (m2) 5.04 5.04 5.04 Eyring error (%) 33.5 15.2 9.1 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Sabine error (%) 32.8 14.4 8.2 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 33.3 15.1 9.0 Accuracy of the predicted room absorptions: The predicted room absorptions, as well as the absorption coefficients, show errors below 15% from the 125-Hz band, while the errors are greater than 15% in the rest of the bands. These are shown in Table H.17. 296 H.4 Octave-band prediction for the ISO 354 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit as there no change in the modal compositions except the modal-width and, hence, the modal overlap as shown in Table H.18. Table H.18: Predicted octave-band modal compositions for ISO 354 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 22 116 765 Modal density, 𝑛(𝑓) 0.50 1.30 4.30 Modal spacing, 𝛿𝑓 1.92 0.76 0.23 Modal width, 𝛿𝑤 3.74 3.20 2.72 Modal overlap, 𝑀 1.87 4.17 11.7 Modal overlap factor, 𝑀𝑓 1.94 4.22 11.7 Table H.19: Predicted octave-band SPL statistics for ISO 354 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 143.4 153.0 163.8 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 142.6 153.2 163.7 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 142.5 151.8 163.9 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 142.9 151.2 163.1 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 147.7 152.7 163.9 Predicted average SPL, ?̅?𝑝 (dB) 144.3 152.5 163.7 Standard deviation, 𝜎𝐿𝑝 (dB) 2.25 0.87 0.34 297 Table H.20: Predicted octave-band RT statistics for ISO 354 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 4.11 3.19 2.40 Predicted RT @ Rec#1: 𝑇60 (s) 2.67 2.48 2.23 Predicted RT @ Rec#2: 𝑇60 (s) 2.68 2.36 2.02 Predicted RT @ Rec#3: 𝑇60 (s) 8.14 3.17 2.64 Predicted RT @ Rec#4: 𝑇60 (s) 5.56 3.21 2.11 Predicted RT @ Rec#5: 𝑇60 (s) 3.79 3.33 2.76 Predicted average RT: ?̅?60 (s) 3.80 2.85 2.32 Standard deviation, 𝜎𝑇60 (s) 2.47 0.46 0.33 Decay rate (dB/s) 15.8 21.1 25.9 Spatial uniformity of SPL: The predicted SPL deviations are less than 1.5 dB in all octave bands as shown in Table H.19. However, the overall deviations are greater than in the ‘without absorber’ case. Spatial uniformity of RT: The predicted RT deviations are within 0.3 to 0.5 s from the 125-Hz band, while 2.47 s in the 63-Hz band as shown in Table H.20. However, the overall deviations are greater than in the ‘without absorber’ case. Table H.21: Predicted octave-band decay curve curvature statistics for ISO 354 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 14.4 3.4 3.0 𝜅 @ Rec#2 32.8 26.0 3.4 𝜅 @ Rec#3 62.6 24.5 52.4 𝜅 @ Rec#4 50.5 9.6 13.5 𝜅 @ Rec#5 19.5 8.4 19.7 Average, 𝜅 36.0 14.4 18.4 Modal, 𝜅 2.3 2.2 5.2 298 Linearity of decay curves: The curvatures are within 14 to 36 from the 125-Hz band as shown in Table H.21. However, the overall curvatures are greater than in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix I.4. Table H.22: Predicted octave-band sample absorber absorption coefficient statistics for ISO 354 oblique reverberation room with the longest z-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.31 0.43 0.60 Sabine error (%) 59.7 43.4 20.8 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.26 0.35 0.45 Eyring error (%) 65.3 54.0 40.5 Accuracy of the predicted sample absorption coefficients: As shown in Table H.22, the predicted Sabine absorption coefficients show errors around 20% in the 250-Hz band, while the errors are greater than 40% in the rest of the bands. On the other hand, the Eyring predictions yield errors around 40% in the 250-Hz band, while the errors are greater than 50% in the rest of the bands. 299 Appendix: I I.1 Third-octave-band prediction for the ISO 354 reverberation room – Case #3: oblique-shaped with the longest z-dimension and without absorber I.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 7.06 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.48 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 6.56 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 6.39 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.13 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 6.04 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 5.77 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.57 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 5.4 s300 I.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, IN third-octave bands, at receiver #5 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 5.684 s0 1-40-2002040Time (s)45 50 55-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.317 s0 1-10010Time (s)60 65 70-1000-5000500Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 6.816 s0 1-20020Time (s)70 80-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 5.788 s0 1-50050Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.671 s0 1-50050Time (s)120 130 1400500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 5.228 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 5.152 s0 1-2000200Time (s)180 200 220-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.648 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 6.013 s0 1-5000500Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)301 I.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 5.625 s0 1-20020Time (s)45 50 551000200030004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.187 s0 1-10010Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 7.221 s0 1-50050Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 5.763 s0 1-50050Time (s)90 100 110-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.563 s0 1-50050Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 5.513 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 4.538 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.456 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 5.576 s0 1-5000500Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)302 I.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 5.467 s0 1-20020Time (s)45 50 55-6000-4000-20000Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 7.441 s0 1-40-2002040Time (s)60 65 70-50005001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.643 s0 1-20020Time (s)70 800200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 5.801 s0 1-40-2002040Time (s)90 100 110-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.636 s0 1-40-2002040Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 4.748 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 5.003 s0 1-1000100Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 5.396 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 5.253 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)303 I.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 5.789 s0 1-20020Time (s)45 50 55-1500-1000-5000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.775 s0 1-20020Time (s)60 65 70-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 5.563 s0 1-50050Time (s)70 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.507 s0 1-50050Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.703 s0 1-50050Time (s)120 130 140-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.593 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 5.323 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.332 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 4.89 s0 1-400-2000200400Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)304 I.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 5.843 s0 1-10010Time (s)45 50 55-10000100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.848 s0 1-20020Time (s)60 65 70-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 6.088 s0 1-20020Time (s)70 80-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 6.003 s0 1-40-2002040Time (s)90 100 110-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.328 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 5.441 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 5.018 s0 1-400-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.181 s0 1-400-2000200Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 4.738 s0 1-400-2000200400Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)305 I.2 Third-octave-band prediction for the ISO 354 reverberation room – Case #3: oblique-shaped with the longest z-dimension and with absorber I.2.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 1 2 3-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 3.32 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 3.67 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 4.3 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 3.57 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.01 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 2.78 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.73 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.29 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.06 s306 I.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 3.461 s0 1-20020Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 2.039 s0 1-10010Time (s)60 65 70-1000-5000500Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 3.522 s0 1-20020Time (s)70 80020004000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 3.544 s0 1-50050Time (s)90 100 110-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.949 s0 1-50050Time (s)120 130 140-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 3.282 s0 1-50050Time (s)150 160 170-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.437 s0 1-200-1000100Time (s)180 200 220-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.053 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 3.529 s0 1-400-2000200400Time (s)300 350-10000-500005000Frequency (Hz)Pressure (Pa)307 I.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 3.279 s0 1-20020Time (s)45 50 550100020003000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 2.917 s0 1-10010Time (s)60 65 70-2000200400600Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 4.091 s0 1-40-2002040Time (s)70 80-50005001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 3.369 s0 1-50050Time (s)90 100 110-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.933 s0 1-50050Time (s)120 130 140-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 2.703 s0 1-50050Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.269 s0 1-1000100Time (s)180 200 220-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.522 s0 1-2000200400Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 2.627 s0 1-400-2000200400Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)308 I.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 3.472 s0 1-20020Time (s)45 50 55-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.097 s0 1-20020Time (s)60 65 70-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 4.118 s0 1-20020Time (s)70 80-50005001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 3.524 s0 1-20020Time (s)90 100 110-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 3.487 s0 1-20020Time (s)120 130 140-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 2.714 s0 1-1000100Time (s)150 160 170-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 2.768 s0 1-1000100Time (s)180 200 220-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.267 s0 1-200-1000100Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 2.708 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)309 I.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 3.691 s0 1-20020Time (s)45 50 55-1000-5000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 2.219 s0 1-10010Time (s)60 65 70-1000-5000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 3.598 s0 1-20020Time (s)70 80-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 2.606 s0 1-20020Time (s)90 100 110-1000010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.52 s0 1-50050Time (s)120 130 140-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.21 s0 1-1000100Time (s)150 160 170-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.366 s0 1-2000200Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.04 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 1.973 s0 1-400-2000200400Time (s)300 350-10000-500005000Frequency (Hz)Pressure (Pa)310 I.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 3.201 s0 1-20020Time (s)45 50 55-1000010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 1.76 s0 1-10010Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 3.502 s0 1-20020Time (s)70 80-1000-5000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 2.878 s0 1-20020Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 2.573 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.408 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.21 s0 1-2000200Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.1 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 2.708 s0 1-2000200Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)311 I.3 Octave-band prediction for the ISO 354 reverberation room – Case #3: oblique-shaped with the longest z-dimension and without absorber I.3.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands I.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in bands, at receiver #5 I.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.64 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.15 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.53 s0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.938 s0 1-50050Time (s)60 80-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.59 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.32 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 9.091 s0 1-50050Time (s)60 80-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.7 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.386 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)312 I.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 I.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 I.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 8.122 s0 1-50050Time (s)60 80-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.368 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 5.229 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.211 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.729 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.158 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 6.079 s0 1-50050Time (s)60 80-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.521 s0 1-1000100Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.952 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)313 I.4 Octave-band prediction for the ISO 354 reverberation room – Case #3: oblique-shaped with the longest z-dimension and with absorber I.4.1 Theoretical modal decay curves and the corresponding RTs, in octave bands I.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 I.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.11 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.19 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.4 s0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 3.791 s0 1-40-2002040Time (s)60 80-500005000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 3.331 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.761 s0 1-5000500Time (s)200 300-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.556 s0 1-40-2002040Time (s)60 800100020003000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.208 s0 1-50050100Time (s)100 150-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.108 s0 1-5000500Time (s)200 300-500005000Frequency (Hz)Pressure (Pa)314 I.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 I.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 I.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 8.138 s0 1-40-2002040Time (s)60 80-3000-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 3.173 s0 1-100-50050Time (s)100 150-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.637 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 2.683 s0 1-50050Time (s)60 80-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.359 s0 1-1000100Time (s)100 150-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.022 s0 1-5000500Time (s)200 300-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 2.668 s0 1-40-2002040Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 2.479 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.227 s0 1-5000500Time (s)200 300-500005000Frequency (Hz)Pressure (Pa)315 Appendix: J J.1 Third-octave-band prediction for the ISO 354 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table J.1, the corresponding statistically-based cut-off frequency limit is 103 Hz, which suggests that prediction can be done from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 125-Hz third-octave are 1.14, 0.87, 0.36 and 0.41, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 5.8 s for the RT, as shown in Table J.3, is 395 Hz. Table J.1: Predicted third-octave-band modal compositions for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 5 11 18 33 64 118 224 420 Modal density, 𝑛(𝑓) 0.25 0.36 0.58 0.78 1.14 1.73 2.57 3.86 5.75 Modal spacing, 𝛿𝑓 3.20 2.74 1.69 1.28 0.87 0.57 0.39 0.26 0.17 Modal width, 𝛿𝑤 0.32 0.31 0.32 0.34 0.36 0.37 0.39 0.40 0.41 Modal overlap, 𝑀 0.08 0.11 0.19 0.27 0.41 0.65 1.01 1.55 2.36 Modal overlap factor, 𝑀𝑓 0.10 0.11 0.19 0.26 0.41 0.65 1.00 1.56 2.37 Spatial uniformity of SPL: From Table J.2, the predicted standard deviations of SPL are less than 1.5 dB from the 160-Hz band, while the deviations are greater than 1.5 dB in the rest of the bands. Accuracy of the predicted SPLs: As presented in Table J.2, the errors associated with the predicted SPLs are always less than or around 1% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases. In the rest of the bands, the errors are greater than 1.2%. 316 Table J.2: Predicted third-octave-band SPL statistics for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 139.5 141.9 148.2 153.5 153.5 157.3 162.8 165.4 167.6 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 140.7 141.0 149.0 156.1 157.2 158.8 162.7 168.2 169.3 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 146.8 137.4 145.8 150.7 153.7 158.6 164.0 167.2 168.0 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 142.9 145.9 148.8 157.6 159.6 160.0 163.6 169.0 168.5 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 146.9 143.8 148.5 152.8 156.1 157.2 164.1 168.4 168.0 Predicted average SPL, ?̅?𝑝 (dB) 144.4 142.9 148.2 154.8 156.6 158.5 163.5 167.8 168.3 Standard deviation, 𝜎𝐿𝑝 (dB) 3.6 3.3 1.3 2.9 2.7 1.1 0.7 1.4 0.7 Expected Sabine SPL, 𝐿𝑝 (dB) 147.6 145.2 150.0 156.8 158.5 160.0 165.0 169.0 169.3 Sabine error (%) 2.2 1.6 1.2 1.3 1.2 1.0 0.9 0.7 0.6 Expected Eyring SPL, 𝐿𝑝 (dB) 147.5 145.1 150.0 156.8 158.4 160.0 165.0 169.0 169.3 Eyring error (%) 2.14 1.55 1.20 1.25 1.13 0.94 0.91 0.69 0.58 Table J.3: Predicted third-octave-band RT statistics for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.14 7.15 7.05 6.61 6.34 6.00 5.68 5.55 5.41 Predicted RT @ Rec#1: 𝑇60 (s) 6.21 6.68 6.41 6.50 5.65 5.36 5.77 5.76 5.58 Predicted RT @ Rec#2: 𝑇60 (s) 6.18 7.10 6.95 5.03 5.71 5.68 4.67 4.81 4.91 Predicted RT @ Rec#3: 𝑇60 (s) 5.84 5.95 5.75 5.06 4.91 4.82 4.79 4.59 4.91 Predicted RT @ Rec#4: 𝑇60 (s) 6.26 6.61 7.17 5.54 5.66 5.72 5.03 5.28 5.06 Predicted RT @ Rec#5: 𝑇60 (s) 5.97 6.71 6.40 6.62 5.19 5.16 4.82 5.35 5.08 Predicted average RT: ?̅?60 (s) 6.09 6.59 6.50 5.67 5.40 5.33 4.99 5.12 5.09 Standard deviation, 𝜎𝑇60 (s) 0.18 0.42 0.55 0.77 0.36 0.38 0.44 0.47 0.28 Expected Sabine RT: 𝑇60 (s) 4.87 4.87 4.87 4.87 4.87 4.87 4.87 4.87 4.87 Sabine error (%) 24.9 35.2 33.4 16.4 10.92 9.35 2.37 5.14 4.54 Expected Eyring RT, 𝑇60 (s) 4.82 4.82 4.82 4.82 4.82 4.82 4.82 4.82 4.82 Eyring error (%) 26.2 36.6 34.7 17.6 12.0 10.5 3.41 6.20 5.60 Decay rate (dB/s) 9.86 9.10 9.23 10.6 11.1 11.3 12.0 11.7 11.8 317 Table J.4: Predicted third-octave-band decay-curve curvature statistics for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 3.0 12.4 22.5 7.2 3.9 9.7 13.5 6.0 2.6 𝜅 @ Rec#2 7.7 3.6 52.4 42.1 8.0 5.9 26.1 11.8 4.8 𝜅 @ Rec#3 7.6 19.4 21.1 12.8 11.8 9.2 8.6 5.4 1.6 𝜅 @ Rec#4 5.6 15.9 26.3 7.1 4.0 21.4 7.5 9.0 3.7 𝜅 @ Rec#5 10.7 8.4 22.1 7.3 30.2 8.0 33.5 11.4 8.1 Average, 𝜅 6.92 12.0 28.9 15.3 11.6 10.8 17.8 8.70 4.15 Modal, 𝜅 0.04 0.01 0.02 0.01 0.02 0.01 0.00 0.00 0.00 Spatial uniformity of RT: From Table J.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.5 s from the 125-Hz third-octave band. Accuracy of the predicted RTs: As presented in Table J.3, the predicted RTs show errors less than or around 10% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases, while the errors are greater than 15% in the rest of the bands. Linearity of temporal decay curves: From Table J.4, the predicted curvatures of the energy-decay curves vary within 4 to 18 from the 100-Hz band. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix K.1. Accuracy of the predicted source-power levels: As presented in Table J.5, the source-power level errors are less than or around 10% from the 100-Hz third-octave band. Table J.5: Predicted third-octave-band source-power level statistics for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 148.2 145.8 150.6 157.4 159.1 160.7 165.7 169.6 170.0 Sabine error (%) 17.9 23.3 22.3 12.6 8.9 7.7 2.3 4.4 3.9 Eyring error (%) 18.8 24.2 23.2 13.5 9.7 8.6 3.1 5.3 4.8 Predicted, ?̅?𝑤 (dB) (1 source) 145.2 142.8 147.6 154.4 156.1 157.7 162.6 166.6 167.0 318 Table J.6: Predicted third-octave-band room absorption statistics for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 3.94 3.64 3.69 4.23 4.44 4.50 4.81 4.68 4.71 Sabine expected absorption, 𝐴 (m2) 4.92 4.92 4.92 4.92 4.92 4.92 4.92 4.92 4.92 Sabine error (%) 20.0 26.1 25.0 14.1 9.85 8.55 2.32 4.89 4.34 Eyring expected absorption, 𝐴 (m2) 4.97 4.97 4.97 4.97 4.97 4.97 4.97 4.97 4.97 Eyring error (%) 20.8 26.8 25.8 14.9 10.8 9.47 3.30 5.84 5.30 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 20.0 26.1 25.0 14.1 9.85 8.55 2.32 4.89 4.34 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 20.6 26.6 25.6 14.8 10.7 9.38 3.26 5.79 5.25 Accuracy of the predicted room absorptions: As presented in Table J.6, the errors are less than or around 15% from the 100-Hz third-octave band, and greater than 20% in the rest of the bands. The same is true for the room absorption coefficient prediction case, as the absorption coefficient predictions depend on the accuracy of the absorption predictions. 319 J.2 Third-octave-band prediction for the ISO 354 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table J.7. The cut-off frequency limit will remain be the same as before. Table J.7: Predicted third-octave-band modal compositions for ISO 354 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 6 12 20 34 64 123 225 426 Modal density, 𝑛(𝑓) 0.25 0.43 0.63 0.87 1.17 1.73 2.67 3.88 5.84 Modal spacing, 𝛿𝑓 2.68 2.14 1.45 1.17 0.81 0.57 0.37 0.26 0.17 Modal width, 𝛿𝑤 1.01 6.58 6.79 3.04 4.47 3.20 3.95 3.01 3.21 Modal overlap, 𝑀 0.25 2.82 4.29 2.64 5.24 5.53 10.6 11.7 18.7 Modal overlap factor, 𝑀𝑓 0.38 3.07 4.68 2.60 5.50 5.61 10.6 11.6 18.7 Table J.8: Predicted third-octave-band SPL statistics for ISO 354 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 136.3 136.2 143.7 152.0 149.7 153.2 159.2 161.1 164.4 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 138.4 137.8 144.9 150.7 150.5 153.1 157.3 164.0 165.9 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 141.6 133.3 139.2 145.3 149.4 150.2 158.1 160.0 162.6 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 141.4 142.8 143.8 150.4 153.1 153.8 159.5 163.2 165.5 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 140.3 136.2 141.0 150.9 150.1 152.4 161.6 163.2 162.1 Predicted average SPL, ?̅?𝑝 (dB) 140.0 138.5 143.0 150.3 150.8 152.7 159.4 162.5 164.3 Standard deviation, 𝜎𝐿𝑝 (dB) 2.26 3.76 2.41 2.66 1.51 1.42 1.65 1.72 1.71 320 Table J.9: Predicted third-octave-band RT statistics for ISO 354 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 3.43 2.82 3.17 2.87 3.06 3.50 2.58 2.39 2.18 Predicted RT @ Rec#1: 𝑇60 (s) 3.44 4.56 3.72 2.91 3.07 2.57 2.73 2.64 3.16 Predicted RT @ Rec#2: 𝑇60 (s) 3.40 4.30 2.54 2.28 2.45 2.48 2.48 2.24 1.92 Predicted RT @ Rec#3: 𝑇60 (s) 3.02 2.75 2.72 2.38 2.67 2.45 2.36 1.93 2.42 Predicted RT @ Rec#4: 𝑇60 (s) 3.76 3.39 2.71 2.67 2.83 2.55 2.08 1.95 2.60 Predicted RT @ Rec#5: 𝑇60 (s) 2.77 2.76 2.44 2.47 2.49 2.77 2.59 2.16 2.24 Predicted average RT: ?̅?60 (s) 3.24 3.40 2.76 2.52 2.68 2.56 2.43 2.16 2.40 Stnd. deviation, 𝜎𝑇60 (s) 0.39 0.87 0.52 0.25 0.26 0.13 0.25 0.29 0.47 Decay rate (dB/s) 18.5 17.7 21.7 23.8 22.4 23.4 24.7 27.8 25.0 Spatial uniformity of SPL: The standard deviations of the SPL are presented in Table J.8. It is noticeable that the deviations are greater than those obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT are presented in Table J.9. It is obvious that the deviations are greater than the ‘without absorber’ case in almost all third-octave bands. Table J.10: Predicted third-octave-band decay-curve curvature statistics for ISO 354 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 30.3 177.9 6.9 26.1 22.5 29.9 43.4 18.8 97.3 𝜅 @ Rec#2 14.3 38.5 6.1 10.7 11.2 16.2 23.3 8.2 2.5 𝜅 @ Rec#3 9.2 10.7 16.0 18.3 22.5 6.3 10.1 17.5 11.3 𝜅 @ Rec#4 5.4 20.8 9.5 22.2 11.3 22.3 10.8 5.4 63.4 𝜅 @ Rec#5 56.5 33.1 16.8 47.6 17.2 28.6 16.4 16.5 31.3 Average, 𝜅 23.2 56.2 11.1 25.0 16.9 20.7 20.8 13.3 41.1 Modal, 𝜅 0.42 0.74 0.82 1.72 2.29 5.25 2.57 3.07 3.97 321 Table J.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ISO 354 oblique reverberation room with the longest x-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.32 0.32 0.46 0.49 0.42 0.45 0.47 0.60 0.49 Sabine error (%) 57.8 58.3 39.1 35.5 45.1 40.6 38.0 21.4 35.7 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.27 0.27 0.37 0.39 0.34 0.36 0.38 0.45 0.39 Eyring error (%) 63.9 64.2 51.2 49.0 55.1 52.2 50.6 40.8 49.1 Linearity of decay curves: The curvatures of the energy-decay curves as presented in Table J.10 reveal that the curvatures are greater than the values obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix K.2. Accuracy of the sample absorption coefficients: The predicted absorption coefficients of an absorber sample are presented in Table J.11. It is noticeable that the predicted values deviate from the theoretical values to a great extent. 322 J.3 Octave-band prediction for the ISO 354 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table J.12, the statistically-based cut-off frequency limit is 65 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz band are 1.28, 0.78, 0.36 and 0.47, respectively. On the other hand, the pure-tone-based Schroeder limit, corresponding to a modal overlap factor of 3 and a band-averaged RT of 5.8 s, as shown in Table J.14, is 398 Hz. Table J.12: Predicted octave-band modal compositions for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 19 114 763 Modal density, 𝑛(𝑓) 0.43 1.28 4.29 Modal spacing, 𝛿𝑓 2.17 0.78 0.23 Modal width, 𝛿𝑤 0.32 0.36 0.41 Modal overlap, 𝑀 0.14 0.46 1.74 Modal overlap factor, 𝑀𝑓 0.15 0.47 1.74 Spatial uniformity of SPL: The predicted SPL standard deviations are less than or around 1.5 dB in all octave bands as shown in Table J.13. Accuracy of the predicted SPLs: As shown in Table J.13, the predicted SPLs show errors below or around 1% from the 125-Hz octave-band. Spatial uniformity of RT: The predicted RT standard deviations are less than 0.35 s from the 125-Hz band as shown in Table J.14. 323 Table J.13: Predicted octave-band SPL statistics for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 145.6 156.0 166.6 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 146.2 158.4 168.4 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 145.4 156.6 167.7 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 147.4 160.1 168.5 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 147.6 156.7 168.0 Predicted average SPL, ?̅?𝑝 (dB) 146.5 157.8 167.9 Standard deviation, 𝜎𝐿𝑝 (dB) 0.99 1.69 0.80 Expected Sabine SPL, 𝐿𝑝 (dB) 148.6 159.6 169.1 Sabine error (%) 1.39 1.11 0.70 Expected Eyring SPL, 𝐿𝑝 (dB) 148.6 159.5 169.1 Eyring error (%) 1.36 1.09 0.68 Table J.14: Predicted octave-band RT statistics for ISO 354 oblique reverberation room with the longest x-dimension tion and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.10 6.26 5.50 Predicted RT @ Rec#1: 𝑇60 (s) 8.35 5.71 5.62 Predicted RT @ Rec#2: 𝑇60 (s) 6.74 5.56 4.93 Predicted RT @ Rec#3: 𝑇60 (s) 6.57 5.06 4.80 Predicted RT @ Rec#4: 𝑇60 (s) 6.96 5.65 5.17 Predicted RT @ Rec#5: 𝑇60 (s) 6.44 5.48 5.24 Predicted average RT: ?̅?60 (s) 6.95 5.48 5.14 Standard deviation, 𝜎𝑇60 (s) 0.78 0.26 0.32 Expected Sabine RT: 𝑇60 (s) 4.87 4.87 4.87 Sabine error (%) 42.7 12.5 5.47 Expected Eyring RT, 𝑇60 (s) 4.82 4.82 4.82 Eyring error (%) 44.1 13.7 6.54 Decay rate (dB/s) 8.63 10.9 11.7 324 Table J.15: Predicted octave-band decay-curve curvature statistics for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 10.0 3.5 1.4 𝜅 @ Rec#2 27.0 5.9 5.5 𝜅 @ Rec#3 3.3 3.4 0.9 𝜅 @ Rec#4 11.0 8.2 4.0 𝜅 @ Rec#5 7.4 10.9 7.5 Average, 𝜅 11.8 6.4 3.9 Modal, 𝜅 0.02 0.03 0.01 Accuracy of the predicted RTs: As shown in Table J.14, the predicted RTs show errors below or around 15% from the 125-Hz band, while the errors are greater than 15% in the rest of the bands. Degree of linearity of decay curves: The curvature values are within 3 to 12 in all the octave-bands as shown in Table J.15. The modal-decay curves and the energy-decay curves predicted in octave bands are given in Appendix K.3. Accuracy of the predicted source-power levels: The predicted source-power levels show errors below or around 10% from the 125-Hz band as shown in Table J.16. Table J.16: Predicted octave-band source-power level statistics for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 149.2 160.2 169.7 Sabine error (%) 26.7 10.0 4.7 Eyring error (%) 27.6 10.9 5.6 Predicted, ?̅?𝑤 (dB) (1 source) 146.2 157.2 166.7 325 Table J.17: Predicted octave-band room absorption statistics for ISO 354 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 3.45 4.37 4.66 Sabine expected absorption, 𝐴 (m2) 4.92 4.92 4.92 Sabine error (%) 29.9 11.1 5.2 Eyring expected absorption, 𝐴 (m2) 4.97 4.97 4.97 Eyring error (%) 30.6 12.0 6.1 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Sabine error (%) 29.9 11.1 5.2 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 30.4 11.9 6.1 Accuracy of the predicted room absorptions: The predicted room absorptions, as well as the absorption coefficients, show errors below or around 10% from the 125-Hz band for both Sabine and Eyring prediction cases as shown in Table J.17. 326 J.4 Octave-band prediction for the ISO 354 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit as there no change in the modal compositions except the modal-width and, hence, the modal-overlap as shown in Table J.18. Table J.18: Predicted octave-band modal compositions for ISO 354 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 21 117 775 Modal density, 𝑛(𝑓) 0.48 1.31 4.35 Modal spacing, 𝛿𝑓 1.87 0.75 0.23 Modal width, 𝛿𝑤 5.90 3.55 3.27 Modal overlap, 𝑀 2.82 4.66 14.2 Modal overlap factor, 𝑀𝑓 3.16 4.74 14.2 Table J.19: Predicted octave-band SPL statistics for ISO 354 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 141.1 152.6 163.0 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 142.5 152.4 164.5 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 139.8 149.6 161.5 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 143.7 153.5 164.3 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 140.5 152.1 163.2 Predicted average SPL, ?̅?𝑝 (dB) 141.7 152.2 163.4 Standard deviation, 𝜎𝐿𝑝 (dB) 1.57 1.47 1.21 327 Table J.20: Predicted octave-band RT statistics for ISO 354 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 3.19 3.34 2.37 Predicted RT @ Rec#1: 𝑇60 (s) 8.55 3.09 2.82 Predicted RT @ Rec#2: 𝑇60 (s) 2.79 2.43 2.11 Predicted RT @ Rec#3: 𝑇60 (s) 3.17 2.58 2.14 Predicted RT @ Rec#4: 𝑇60 (s) 3.01 2.79 2.16 Predicted RT @ Rec#5: 𝑇60 (s) 2.87 2.63 2.33 Predicted average RT: ?̅?60 (s) 3.40 2.68 2.28 Standard deviation, 𝜎𝑇60 (s) 2.62 0.25 0.30 Decay rate (dB/s) 17.7 22.4 26.3 Spatial uniformity of SPL: The predicted SPL deviations are greater than in the ‘without absorber’ case as shown in Table J.19. Spatial uniformity of RT: The predicted RT deviations are greater than in the ‘without absorber’ case as shown in Table J.20. Linearity of temporal decay curves: The curvatures are greater than in the ‘without absorber’ case as shown in Table J.21. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix K.4. Table J.21: Predicted octave-band decay-curve curvature statistics for ISO 354 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 76.2 25.8 49.6 𝜅 @ Rec#2 4.9 11.6 6.5 𝜅 @ Rec#3 4.6 8.6 9.5 𝜅 @ Rec#4 5.4 13.6 8.7 𝜅 @ Rec#5 24.1 31.8 18.1 Average, 𝜅 23.0 18.3 18.5 Modal, 𝜅 1.0 4.6 3.9 328 Table J.22: Predicted octave-band sample absorber absorption coefficient statistics for ISO 354 oblique reverberation room with the longest x-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.33 0.42 0.54 Sabine error (%) 56.0 44.3 28.8 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.28 0.35 0.42 Eyring error (%) 62.6 54.6 45.0 Accuracy of the predicted sample absorption coefficients: As shown in Table J.22, the predicted Sabine absorption coefficients show a band-averaged error of 40%. On the other hand, while the Eyring predictions yield a band-averaged error around 55%. 329 Appendix: K K.1 Third-octave-band prediction for the ISO 354 reverberation room – Case #4: oblique-shaped with the longest x-dimension and without absorber K.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 7.14 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 7.15 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 7.05 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 6.61 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.34 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 6 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 5.68 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.55 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 5.41 s330 K.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 5.966 s0 1-20020Time (s)45 50 55-10000100020003000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.713 s0 1-20020Time (s)60 65 70-10000100020003000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 6.404 s0 1-20020Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 6.619 s0 1-50050Time (s)90 100 110-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.189 s0 1-50050Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 5.164 s0 1-50050Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 4.824 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.345 s0 1-2000200Time (s)240 260 280-10123x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 5.075 s0 1-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)331 K.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 6.257 s0 1-20-10010Time (s)45 50 55-100001000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.611 s0 1-2002040Time (s)60 65 70-200002000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 7.174 s0 1-50050Time (s)70 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 5.542 s0 1-50050Time (s)90 100 110-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.658 s0 1-1000100Time (s)120 130 140-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 5.724 s0 1-50050Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 5.033 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.276 s0 1-5000500Time (s)240 260 280-10123x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 5.059 s0 1-2000200Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)332 K.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 5.844 s0 1-20020Time (s)45 50 55-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.949 s0 1-10010Time (s)60 65 70-1000-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.752 s0 1-20020Time (s)70 80020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 5.064 s0 1-20020Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 4.912 s0 1-50050Time (s)120 130 140-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 4.816 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 4.792 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.589 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 4.909 s0 1-2000200Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)333 K.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 6.182 s0 1-1001020Time (s)45 50 55-1500-1000-5000500Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 7.104 s0 1-20020Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 6.947 s0 1-20020Time (s)70 80020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.031 s0 1-50050Time (s)90 100 110-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.708 s0 1-1000100Time (s)120 130 140-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.683 s0 1-100-50050Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 4.667 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.809 s0 1-400-2000200400Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 4.906 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)334 K.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 6.209 s0 1-10-505Time (s)45 50 5505001000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 6.682 s0 1-1001020Time (s)60 65 70-3000-2000-10000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 6.412 s0 1-20020Time (s)70 80-4000-200002000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 6.497 s0 1-50050Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.654 s0 1-50050Time (s)120 130 140-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 5.363 s0 1-50050Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 5.765 s0 1-1000100Time (s)180 200 220-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.763 s0 1-2000200Time (s)240 260 280-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 5.576 s0 1-2000200Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)335 K.2 Third-octave-band prediction for the ISO 354 reverberation room – Case #4: oblique-shaped with the longest x-dimension and with absorber K.2.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 1 2 3-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 3.43 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 2.82 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 3.17 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 2.87 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.06 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 3.5 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.58 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.39 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.18 s336 K.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 2.765 s0 1-10010Time (s)45 50 55-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 2.762 s0 1-10010Time (s)60 65 70-2000200400600Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 2.438 s0 1-10010Time (s)70 8005001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 2.465 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.486 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 2.772 s0 1-50050Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.59 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.157 s0 1-2000200Time (s)240 260 280-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 2.243 s0 1-2000200Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)337 K.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 3.755 s0 1-20020Time (s)45 50 55-1000-5000500Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 3.391 s0 1-40-2002040Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 2.713 s0 1-40-2002040Time (s)70 80-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 2.667 s0 1-50050Time (s)90 100 110-10000100020003000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.829 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 2.55 s0 1-50050Time (s)150 160 170-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.077 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 1.95 s0 1-2000200Time (s)240 260 280-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 2.599 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)338 K.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 3.024 s0 1-10010Time (s)45 50 55-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 2.75 s0 1-505Time (s)60 65 70-400-2000200400Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 2.718 s0 1-10010Time (s)70 80-50005001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 2.38 s0 1-20020Time (s)90 100 110-1500-1000-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.674 s0 1-50050Time (s)120 130 1400100020003000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 2.45 s0 1-50050Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 2.357 s0 1-1000100Time (s)180 200 220-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 1.931 s0 1-200-1000100Time (s)240 260 280-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 2.416 s0 1-2000200Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)339 K.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 3.397 s0 1-10010Time (s)45 50 55-1000-5000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 4.302 s0 1-20020Time (s)60 65 70-800-600-400-200Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 2.539 s0 1-20020Time (s)70 80010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 2.28 s0 1-50050Time (s)90 100 110-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.45 s0 1-50050Time (s)120 130 140-10000100020003000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.48 s0 1-50050Time (s)150 160 170-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.48 s0 1-1000100Time (s)180 200 220-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.241 s0 1-2000200Time (s)240 260 280-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 1.924 s0 1-400-2000200400Time (s)300 350-50000500010000Frequency (Hz)Pressure (Pa)340 K.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 3.443 s0 1-10010Time (s)45 50 55-2000200400600Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 4.562 s0 1-10010Time (s)60 65 70-400-2000200Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 3.718 s0 1-20020Time (s)70 80-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 2.912 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.067 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.57 s0 1-1000100Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.727 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.644 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 3.162 s0 1-2000200Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)341 K.3 Octave-band prediction for the ISO 354 reverberation room – Case #4: oblique-shaped with the longest x-dimension and without absorber K.3.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands K.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 K.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 7.1 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.26 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.5 s0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.443 s0 1-40-2002040Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.483 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.242 s0 1-5000500Time (s)200 300-10123x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.96 s0 1-50050Time (s)60 80-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.649 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.172 s0 1-5000500Time (s)200 300-10123x 104Frequency (Hz)Pressure (Pa)342 K.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 K.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 K.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.57 s0 1-2002040Time (s)60 80-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.063 s0 1-1000100Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.804 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.743 s0 1-20020Time (s)60 80-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.556 s0 1-2000200Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.931 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 8.351 s0 1-40-20020Time (s)60 80-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.711 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.623 s0 1-400-2000200400Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)343 K.4 Octave-band prediction for the ISO 354 reverberation room – Case #4: oblique-shaped with the longest x-dimension and with absorber K.4.1 Theoretical modal decay curves and the corresponding RTs, in octave bands K.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 K.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 3.19 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.34 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.37 s0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 2.867 s0 1-20020Time (s)60 80-50005001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.627 s0 1-1000100Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.326 s0 1-400-2000200400Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 3.014 s0 1-50050Time (s)60 80-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.787 s0 1-1000100Time (s)100 150-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.163 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)344 K.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 K.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 K.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 3.173 s0 1-20020Time (s)60 80-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.575 s0 1-1000100Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.138 s0 1-2000200Time (s)200 300-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 2.79 s0 1-40-20020Time (s)60 80-1000010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.428 s0 1-1000100Time (s)100 150-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.108 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 8.554 s0 1-2002040Time (s)60 80-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.086 s0 1-1000100Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.823 s0 1-5000500Time (s)200 300-500005000Frequency (Hz)Pressure (Pa)345 Appendix: L L.1 Third-octave-band prediction for the ASTM C423 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table L.1, the corresponding statistically-based cut-off frequency limit is 108 Hz, which suggests that prediction can be done from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 125-Hz third-octave are 0.97, 1.02, 0.36 and 0.35, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 6.1 s for the RT, as shown in Table L.3, is 441 Hz. Table L.1: Predicted third-octave-band modal compositions for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 5 8 17 28 53 99 188 353 Modal density, 𝑛(𝑓) 0.25 0.36 0.42 0.74 0.97 1.43 2.15 3.24 4.84 Modal spacing, 𝛿𝑓 1.83 2.24 1.91 1.41 1.02 0.70 0.46 0.31 0.21 Modal width, 𝛿𝑤 0.32 0.32 0.33 0.35 0.36 0.39 0.40 0.41 0.42 Modal overlap, 𝑀 0.08 0.11 0.14 0.26 0.35 0.55 0.86 1.34 2.04 Modal overlap factor, 𝑀𝑓 0.17 0.14 0.17 0.25 0.35 0.56 0.87 1.33 2.03 Spatial uniformity of SPL: From Table L.2, the predicted standard deviations of SPL, a measure of spatial variation, reveal that deviations smaller than 1.5 dB occurs from the 160-Hz third-octave band, while deviations are greater than 2 dB in the rest of the bands. According to the ISO 354 standard, a maximum of 1.5 dB deviation is allowable, which is achieved in this case from the 160-Hz third-octave band. 346 Table L.2: Predicted third-octave-band SPL statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 136.2 144.4 147.1 151.0 158.8 159.4 164.4 167.6 169.4 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 134.3 143.9 147.3 151.8 155.6 157.6 162.7 169.5 170.1 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 152.9 152.6 145.6 146.5 157.5 157.0 163.7 167.5 171.1 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 150.7 143.8 148.8 149.9 153.7 158.6 162.3 166.3 169.9 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 143.0 145.0 146.6 151.1 159.9 157.8 161.6 167.8 169.1 Predicted average SPL, ?̅?𝑝 (dB) 148.3 147.6 147.2 150.4 157.6 158.2 163.1 167.9 170.0 Standard deviation, 𝜎𝐿𝑝 (dB) 10.0 4.2 1.2 2.1 2.5 0.9 1.1 1.2 0.8 Expected Sabine SPL, 𝐿𝑝 (dB) 150.7 150.3 149.2 151.8 159.3 159.8 164.3 169.1 170.9 Sabine error (%) 1.6 1.8 1.4 1.0 1.0 1.0 0.7 0.7 0.5 Expected Eyring SPL, 𝐿𝑝 (dB) 150.7 150.3 149.2 151.8 159.3 159.8 164.3 169.1 170.8 Eyring error (%) 1.56 1.79 1.34 0.93 1.02 0.98 0.73 0.70 0.50 Table L.3: Predicted third-octave-band RT statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.81 6.99 6.84 6.45 6.34 5.94 5.70 5.44 5.37 Predicted RT @ Rec#1: 𝑇60 (s) 13.41 5.63 5.97 6.56 5.09 5.44 5.55 5.24 5.84 Predicted RT @ Rec#2: 𝑇60 (s) 6.00 5.91 7.02 6.15 7.61 5.06 5.29 4.41 4.88 Predicted RT @ Rec#3: 𝑇60 (s) 6.56 5.98 5.78 7.24 4.70 5.67 5.25 5.16 5.06 Predicted RT @ Rec#4: 𝑇60 (s) 6.65 5.98 5.72 5.80 5.47 4.90 4.96 5.31 5.51 Predicted RT @ Rec#5: 𝑇60 (s) 6.61 5.65 5.98 5.90 5.27 4.66 5.10 4.82 4.48 Predicted average RT: ?̅?60 (s) 7.19 5.83 6.06 6.29 5.48 5.12 5.22 4.96 5.11 Standard deviation, 𝜎𝑇60 (s) 3.21 0.18 0.53 0.59 1.16 0.41 0.22 0.37 0.54 Expected Sabine RT: 𝑇60 (s) 4.78 4.78 4.78 4.78 4.78 4.78 4.78 4.78 4.78 Sabine error (%) 50.4 22.0 26.9 31.7 14.6 7.1 9.4 3.9 6.9 Expected Eyring RT, 𝑇60 (s) 4.73 4.73 4.73 4.73 4.73 4.73 4.73 4.73 4.73 Eyring error (%) 52.0 23.2 28.2 33.0 15.8 8.2 10.5 4.9 8.0 Decay rate (dB/s) 8.35 10.30 9.90 9.54 10.96 11.72 11.4 12.0 11.7 347 Accuracy of the predicted SPLs: The accuracy of the predicted steady-state SPLs is checked against the Sabine and Eyring versions of the diffuse-field formulae as presented in Table L.2. The errors associated with the predicted SPL are always less than 1% from the 100-Hz third-octave band for both Sabine and Eyring prediction cases. Spatial uniformity of RT: From Table L.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than, or around, 0.5 s from the 63-Hz third-octave band with the exception in the 125-Hz band.. Accuracy of the predicted RTs: As presented in Table L.3, the predicted RTs show errors less than, or around, 10% from the 160-Hz third-octave band for both Sabine and Eyring prediction cases, while the errors are greater than 15% in the rest of the bands. Linearity of temporal decay curves: From Table L.4, the predicted curvatures of the energy-decay curves vary within 4 to 23 in all third-octave bands with the maximum in the 125-Hz band. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix M.1. Accuracy of the predicted source-power levels: As presented in Table L.5, the source-power level errors are less than or around 10% from the 160-Hz third-octave band. Table L.4: Predicted third-octave-band decay curve curvature statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 47.2 7.9 28.2 23.9 29.7 13.8 6.1 4.4 4.8 𝜅 @ Rec#2 4.0 14.2 16.3 16.4 14.9 12.8 16.2 5.7 5.5 𝜅 @ Rec#3 6.3 4.8 9.4 16.1 12.6 11.6 5.8 4.5 2.7 𝜅 @ Rec#4 6.7 12.6 18.9 5.1 48.1 6.3 17.8 5.8 5.1 𝜅 @ Rec#5 2.1 5.9 6.0 3.7 9.9 6.3 19.3 12.3 4.9 Average, 𝜅 13.3 9.1 15.8 13.1 23.0 10.2 13.1 6.5 4.6 Modal, 𝜅 0.35 0.01 0.01 0.03 0.03 0.07 0.05 0.01 0.03 348 Table L.5: Predicted third-octave-band source-power level statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 150.7 150.3 149.2 151.8 159.3 159.8 164.3 169.1 170.9 Sabine error (%) 30.5 16.5 19.3 21.9 11.7 6.1 7.8 3.5 5.9 Eyring error (%) 31.4 17.4 20.2 22.8 12.6 7.1 8.7 4.4 6.8 Predicted, ?̅?𝑤 (dB) (1 source) 147.7 147.3 146.2 148.8 156.3 156.8 161.3 166.1 167.8 Accuracy of the predicted room absorptions: Table L.6 shows that the errors associated with the room-averaged absorption are less than 10% from the 160-Hz third-octave band, and greater than 10% in the rest of the bands. The same is true for the room absorption-coefficient prediction case, as the absorption-coefficient predictions depend on the accuracy of the absorption predictions. The room-averaged absorption-coefficient prediction statistics are also presented in Table L.6. Table L.6: Predicted third-octave-band room absorption statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 2.78 3.43 3.30 3.18 3.65 3.91 3.83 4.03 3.91 Sabine expected absorption, 𝐴 (m2) 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 4.19 Sabine error (%) 33.5 18.0 21.2 24.0 12.8 6.7 8.6 3.7 6.5 Eyring expected absorption, 𝐴 (m2) 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 4.23 Eyring error (%) 34.2 18.8 22.0 24.8 13.7 7.6 9.5 4.7 7.4 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 33.5 18.0 21.2 24.0 12.8 6.7 8.6 3.7 6.5 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 34.0 18.7 21.8 24.6 13.5 7.5 9.4 4.7 7.3 349 L.2 Third-octave-band prediction for the ASTM C423 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table L.7. The cut-off frequency limit will remain be the same as before. Table L.7: Predicted third-octave-band modal compositions for ASTM C423 rectangular reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 4 6 8 19 28 56 100 191 351 Modal density, 𝑛(𝑓) 0.33 0.43 0.42 0.83 0.97 1.51 2.17 3.29 4.81 Modal spacing, 𝛿𝑓 2.83 1.89 1.96 1.23 1.04 0.67 0.46 0.31 0.21 Modal width, 𝛿𝑤 1.71 5.99 1.08 5.88 2.36 3.01 2.65 2.8 2.5 Modal overlap, 𝑀 0.57 2.57 0.45 4.86 2.28 4.56 5.76 9.2 12.1 Modal overlap factor, 𝑀𝑓 0.60 3.17 0.55 4.79 2.27 4.51 5.80 9.1 12.1 Table L.8: Predicted third-octave-band SPL statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 136.1 141.9 142.6 146.2 151.7 155.2 161.2 163.8 166.4 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 134.6 141.1 148.4 146.1 148.7 154.9 157.5 163.9 164.1 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 142.8 143.0 144.9 142.8 144.9 154.2 158.0 162.0 167.0 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 140.4 143.6 143.4 146.5 145.1 152.3 158.4 162.6 166.2 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 139.2 142.9 147.2 148.3 156.0 153.2 158.2 163.5 166.5 Predicted average SPL, ?̅?𝑝 (dB) 139.5 142.6 145.9 146.3 151.3 154.1 158.8 163.2 166.1 Standard deviation, 𝜎𝐿𝑝 (dB) 3.45 1.01 2.52 2.03 5.21 1.19 1.48 0.84 1.12 350 Table L.9: Predicted third-octave-band RT statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 2.03 4.67 4.16 3.94 4.07 2.77 2.60 2.48 2.31 Predicted RT @ Rec#1: 𝑇60 (s) 12.7 3.5 4.0 4.0 3.0 2.7 2.1 2.4 3.0 Predicted RT @ Rec#2: 𝑇60 (s) 4.88 4.15 3.02 3.57 2.30 2.02 2.22 2.25 2.15 Predicted RT @ Rec#3: 𝑇60 (s) 1.84 3.42 2.82 3.99 2.54 2.47 2.27 2.09 2.21 Predicted RT @ Rec#4: 𝑇60 (s) 2.90 2.81 3.15 3.51 3.38 2.25 2.37 2.28 2.90 Predicted RT @ Rec#5: 𝑇60 (s) 5.41 3.77 1.99 3.44 2.56 2.97 2.42 2.44 1.93 Predicted average RT: ?̅?60 (s) 3.69 3.48 2.85 3.69 2.71 2.43 2.28 2.29 2.36 Standard deviation, 𝜎𝑇60 (s) 4.71 0.50 0.73 0.28 0.44 0.37 0.12 0.15 0.48 Decay rate (dB/s) 16.3 17.3 21.1 16.3 22.2 24.7 26.3 26.2 25.4 Spatial uniformity of SPL: The standard deviations of the SPL for this ‘with absorber’ case, as a measure of the spatial variation are presented in Table L.8. Overall, the deviations are greater than the values obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT for this ‘with absorber’ case, as a measure of the spatial variation are presented in Table L.9. The overall deviations are greater than in the previous ‘without absorber’ case in almost all third-octave bands. Table L.10: Predicted third-octave-band decay-curve curvature statistics for ASTM C423 rectangular reverberation room with the longest dimension in the x-direction and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 230.1 22.0 69.1 17.1 24.6 7.8 13.1 24.7 3.2 𝜅 @ Rec#2 142.9 16.5 5.6 5.0 24.1 5.4 26.0 7.2 15.7 𝜅 @ Rec#3 27.9 27.1 21.2 14.4 9.3 13.1 11.9 6.6 5.2 𝜅 @ Rec#4 106.2 12.4 7.6 6.7 60.3 11.5 9.3 7.0 5.5 𝜅 @ Rec#5 208.0 9.3 11.6 2.7 81.7 7.2 6.9 10.8 10.2 Average, 𝜅 143.0 17.5 23.0 9.2 40.0 9.0 13.5 11.3 8.0 Modal, 𝜅 0.49 0.51 2.42 0.72 5.71 3.59 1.88 2.57 2.28 351 Linearity of decay curves: The curvatures of the energy-decay curves as presented in Table L.10 reveal that the curvatures are greater than those obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix M.2. Accuracy of the sample absorption coefficients: The accuracy of the absorption coefficients of an absorber sample predicted using the Sabine and Eyring versions of the diffuse-field formulae is checked against the theoretical random absorption-coefficient values predicted using Eq. (3.17); these are presented in Table L.11. It is noticeable that predicted values deviate from the theoretical values to a great extent. The Sabine predictions show an error of around 25% from the 160-Hz third-octave band, while the errors are greater than 30% in the rest of the low-frequency bands. Table L.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ASTM C423 rectangular reverberation room with the longest x-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.35 0.31 0.49 0.30 0.49 0.57 0.65 0.62 0.60 Sabine error (%) 54.1 59.7 35.3 61.1 35.1 25.0 13.9 18.3 21.0 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.29 0.26 0.39 0.26 0.39 0.43 0.48 0.46 0.45 Eyring error (%) 61.3 65.3 48.9 66.3 48.8 42.8 36.8 39.1 40.6 352 L.3 Octave-band prediction for the ASTM C423 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table L.12, the statistically-based cut-off frequency limit is 69 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz band are 1.08, 0.92, 0.37 and 0.40, respectively. On the other hand, the pure-tone-based Schroeder limit, corresponding to a modal overlap factor of 3 and a band-averaged RT of 5.8 s, as shown in Table L.14, is 430 Hz. Spatial uniformity of SPL: The predicted SPL standard deviations are smaller than 1.5 dB from the 125-Hz octave band, as shown in Table L.13. Table L.12: Predicted octave-band modal compositions for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 16 96 642 Modal density, 𝑛(𝑓) 0.36 1.08 3.61 Modal spacing, 𝛿𝑓 2.53 0.92 0.28 Modal width, 𝛿𝑤 0.32 0.37 0.42 Modal overlap, 𝑀 0.12 0.40 1.50 Modal overlap factor, 𝑀𝑓 0.13 0.40 1.50 Accuracy of the predicted SPLs: The accuracy of the predicted steady-state SPLs is checked against the Sabine and Eyring versions of the diffuse-field as shown in Table L.13. The predicted results show errors below or around, 1% from the 125-Hz octave band, while the error is around 1.5% in the 63-Hz band for both the Sabine and Eyring cases. 353 Table L.13: Predicted octave-band SPL statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 144.9 158.3 168.6 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 145.1 156.1 169.4 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 152.3 156.4 169.3 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 149.2 156.3 168.1 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 145.8 158.2 168.1 Predicted average SPL, ?̅?𝑝 (dB) 148.5 157.2 168.7 Standard deviation, 𝜎𝐿𝑝 (dB) 3.40 1.09 0.65 Expected Sabine SPL, 𝐿𝑝 (dB) 150.6 158.9 169.9 Sabine error (%) 1.41 1.08 0.69 Expected Eyring SPL, 𝐿𝑝 (dB) 150.6 158.8 169.9 Eyring error (%) 1.38 1.05 0.67 Table L.14: Predicted octave-band RT statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.19 6.21 5.46 Predicted RT @ Rec#1: 𝑇60 (s) 6.79 5.47 5.41 Predicted RT @ Rec#2: 𝑇60 (s) 7.90 6.32 4.72 Predicted RT @ Rec#3: 𝑇60 (s) 6.27 5.21 5.05 Predicted RT @ Rec#4: 𝑇60 (s) 6.74 5.28 5.27 Predicted RT @ Rec#5: 𝑇60 (s) 6.14 5.01 4.68 Predicted average RT: ?̅?60 (s) 6.71 5.42 5.01 Standard deviation, 𝜎𝑇60 (s) 0.70 0.51 0.32 Expected Sabine RT: 𝑇60 (s) 4.78 4.78 4.78 Sabine error (%) 40.57 13.51 4.91 Expected Eyring RT, 𝑇60 (s) 4.73 4.73 4.73 Eyring error (%) 41.99 14.66 5.97 Decay rate (dB/s) 8.94 11.07 11.97 354 Table L.15: Predicted octave-band decay-curve curvature statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 4.07 10.75 1.94 𝜅 @ Rec#2 9.53 5.67 3.79 𝜅 @ Rec#3 5.43 13.28 0.98 𝜅 @ Rec#4 3.02 14.79 1.39 𝜅 @ Rec#5 1.19 7.11 3.33 Average, 𝜅 4.65 10.32 2.29 Modal, 𝜅 0.12 0.07 0.04 Spatial uniformity of RT: The predicted RT standard deviations are less than 0.5 s from the 125-Hz band; while 0.7 s in the 63-Hz octave band. These are shown in Table L.14. Accuracy of the predicted RTs: The accuracy of the predicted RTs is checked against the Sabine and Eyring versions of the diffuse-field formulae as shown in Table L.14. The predicted results show errors below 15% from the 125-Hz band, while the errors are greater than 20% in the rest of the bands. Linearity of decay curves: The curvatures are within 2 to 11 in all octave bands as shown in Table L.15. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix M.3. Table L.16: Predicted octave-band source-power level statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 150.59 158.85 169.88 Sabine error (%) 26.27 10.87 4.31 Eyring error (%) 27.19 11.79 5.22 Predicted, ?̅?𝑤 (dB) (1 source) 147.58 155.84 166.87 355 Table L.17: Predicted octave-band room absorption statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 2.98 3.69 3.99 Sabine expected absorption, 𝐴 (m2) 4.19 4.19 4.19 Sabine error (%) 28.86 11.90 4.68 Eyring expected absorption, 𝐴 (m2) 4.23 4.23 4.23 Eyring error (%) 29.57 12.78 5.63 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Sabine error (%) 28.86 11.90 4.68 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 29.36 12.67 5.58 Accuracy of the predicted source-power levels: The predicted source-power levels show errors below 12% from the 125-Hz band; the errors are greater than 15% in the rest of the bands, as shown in Table L.16. Accuracy of the predicted room absorptions: The predicted room absorptions, as well as the absorption coefficients, show errors below 13% from the 125-Hz band, while errors are greater than 16% in the rest of the bands. These are shown in Table L.17. 356 L.4 Octave-band prediction for the ASTM C423 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit. There is also no change in the modal compositions except the modal-width and, hence, the modal-overlap as shown in Table L.18. The same reasoning is applicable here what is discussed in the ‘third-octave-band prediction section with absorber’ case. Table L.18: Predicted octave-band modal compositions for ASTM C423 rectangular reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 17 101 645 Modal density, 𝑛(𝑓) 0.39 1.13 3.62 Modal spacing, 𝛿𝑓 2.68 0.88 0.28 Modal width, 𝛿𝑤 2.95 3.41 2.61 Modal overlap, 𝑀 1.14 3.87 9.47 Modal overlap factor, 𝑀𝑓 1.10 3.87 9.47 Spatial uniformity of SPL: The predicted SPL deviations are are higher than in the ‘without absorber’ case, as shown in Table L.19. Spatial uniformity of RT: The predicted RT deviations are greater than in the ‘without absorber’ case, as shown in Table L.20. Linearity of decay curves: The curvatures are greater than in the ‘without absorber’ case, as shown in Table L.21. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix M.4. 357 Table L.19: Predicted octave-band SPL statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 141.6 152.9 165.2 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 145.1 151.3 163.7 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 144.5 150.7 164.7 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 143.0 149.7 164.4 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 143.7 154.3 164.8 Predicted average SPL, ?̅?𝑝 (dB) 143.8 152.1 164.6 Standard deviation, 𝜎𝐿𝑝 (dB) 1.36 1.84 0.57 Accuracy of the sample absorption coefficients: The accuracy of the predicted absorption coefficients of an absorber sample is checked against the theoretical results as shown in Table L.22. The errors show a greater deviation from the theoretical values for both Sabine and Eyring prediction cases. Table L.20: Predicted octave-band RT statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 4.43 3.80 2.44 Predicted RT @ Rec#1: 𝑇60 (s) 7.83 3.21 2.40 Predicted RT @ Rec#2: 𝑇60 (s) 3.57 2.49 2.20 Predicted RT @ Rec#3: 𝑇60 (s) 3.82 2.89 2.18 Predicted RT @ Rec#4: 𝑇60 (s) 2.83 2.84 2.54 Predicted RT @ Rec#5: 𝑇60 (s) 3.32 2.74 2.20 Predicted average RT: ?̅?60 (s) 3.78 2.81 2.30 Stnd. deviation, 𝜎𝑇60 (s) 2.09 0.26 0.16 Decay rate (dB/s) 15.88 21.32 26.13 358 Table L.21: Predicted octave-band decay-curve curvature statistics for ASTM C423 rectangular reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 77.46 21.92 2.96 𝜅 @ Rec#2 8.76 8.18 4.16 𝜅 @ Rec#3 42.25 3.53 3.51 𝜅 @ Rec#4 8.32 11.20 1.82 𝜅 @ Rec#5 12.44 35.19 7.70 Average, 𝜅 29.84 16.00 4.03 Modal, 𝜅 1.29 5.07 2.54 Table L.22: Predicted octave-band sample absorber absorption coefficient statistics for ASTM C423 rectangular reverberation room with the longest x-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.31 0.45 0.62 Sabine error (%) 59.80 40.63 18.00 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.26 0.36 0.46 Eyring error (%) 65.36 52.22 38.97 359 Appendix: M M.1 Third-octave-band prediction for the ASTM C423 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and without absorber M.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 2 4 6 8-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 7.81 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.99 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 6.84 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 6.45 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.34 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 5.94 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 5.7 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.44 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 5.37 s360 M.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 6.606 s0 1-20020Time (s)45 50 55-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 5.65 s0 1-20020Time (s)60 65 70020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 5.977 s0 1-20020Time (s)70 80-10000100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 5.901 s0 1-40-2002040Time (s)90 100 110-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.269 s0 1-50050100Time (s)120 130 140-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 4.66 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 5.101 s0 1-1000100Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.818 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 4.476 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)361 M.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 6.646 s0 1-20020Time (s)45 50 55-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.982 s0 1-10010Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 5.724 s0 1-20020Time (s)70 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 5.804 s0 1-20020Time (s)90 100 110-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.474 s0 1-40-2002040Time (s)120 130 140-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 4.896 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 4.96 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.308 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 5.514 s0 1-400-2000200400Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)362 M.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 6.555 s0 1-20020Time (s)45 50 55-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.984 s0 1-40-2002040Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.784 s0 1-40-2002040Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 7.236 s0 1-40-2002040Time (s)90 100 110-10000100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 4.701 s0 1-50050Time (s)120 130 140-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 5.666 s0 1-50050Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 5.252 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 5.157 s0 1-2000200Time (s)240 260 280-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 5.058 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)363 M.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 5.997 s0 1-505Time (s)45 50 55-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.908 s0 1-10010Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 7.022 s0 1-20020Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 6.145 s0 1-40-2002040Time (s)90 100 110-4000-2000020004000Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 7.613 s0 1-50050Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.057 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 5.292 s0 1-1000100Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.411 s0 1-400-2000200400Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 4.875 s0 1-400-2000200400Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)364 M.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 5 10-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 13.409 s0 1-20020Time (s)45 50 550200400Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.626 s0 1-20-10010Time (s)60 65 70-10000100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 5.967 s0 1-20020Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 6.564 s0 1-40-20020Time (s)90 100 110-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.091 s0 1-50050Time (s)120 130 140-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 5.441 s0 1-1000100Time (s)150 160 170-8000-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 5.553 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.238 s0 1-2000200400Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 5.838 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)365 M.2 Third-octave-band prediction for the ASTM C423 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and with absorber M.2.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 2.03 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.67 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 4.16 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 3.94 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 4.07 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 2.77 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.6 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.48 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.31 s366 M.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 5.413 s0 1-20020Time (s)45 50 55-400-2000200400Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 3.765 s0 1-20020Time (s)60 65 70010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 1.99 s0 1-20020Time (s)70 80020004000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 3.441 s0 1-20020Time (s)90 100 110-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.564 s0 1-1000100Time (s)120 130 140-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 2.965 s0 1-50050Time (s)150 160 170020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.42 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.437 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 1.932 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)367 M.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 2.902 s0 1-10010Time (s)45 50 55-1000-5000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 2.813 s0 1-10010Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 3.147 s0 1-20-10010Time (s)70 80-100001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 3.511 s0 1-20020Time (s)90 100 110-10000100020003000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.381 s0 1-20020Time (s)120 130 140-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 2.248 s0 1-50050Time (s)150 160 170-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.372 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.283 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 2.9 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)368 M.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 1.843 s0 1-20020Time (s)45 50 55-1500-1000-5000500Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 3.423 s0 1-10010Time (s)60 65 70-1500-1000-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 2.822 s0 1-40-2002040Time (s)70 80-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 3.987 s0 1-20020Time (s)90 100 110010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.543 s0 1-20020Time (s)120 130 140-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 2.472 s0 1-50050Time (s)150 160 170-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 2.272 s0 1-1000100Time (s)180 200 220-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.085 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 2.211 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)369 M.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 4.884 s0 1-10010Time (s)45 50 55-2000200Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 4.151 s0 1-10010Time (s)60 65 70-1500-1000-5000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 3.023 s0 1-40-2002040Time (s)70 80020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.299 s0 1-20020Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.021 s0 1-50050Time (s)150 160 170-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.222 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 3.566 s0 1-20020Time (s)90 100 110-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.253 s0 1-400-2000200400Time (s)240 260 280-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 2.152 s0 1-2000200Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)370 M.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 2 4 6 8 10 12-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 12.657 s0 1-20-10010Time (s)45 50 550100200Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 3.52 s0 1-10010Time (s)60 65 70010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 3.989 s0 1-20020Time (s)70 80-1000-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 4.022 s0 1-20020Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.008 s0 1-50050Time (s)120 130 140-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.668 s0 1-50050Time (s)150 160 170-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.125 s0 1-1000100Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.431 s0 1-2000200400Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 2.984 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)371 M.3 Octave-band prediction for the ASTM C423 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and without absorber M.3.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands M.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 M.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 7.19 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.21 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.46 s0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.136 s0 1-40-2002040Time (s)60 80020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.013 s0 1-1000100Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.682 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.741 s0 1-50050Time (s)60 80-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.282 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.274 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)372 M.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 M.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 M.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.267 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.207 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 5.049 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 7.904 s0 1-40-2002040Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 6.315 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.724 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 6.792 s0 1-2002040Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.466 s0 1-1000100Time (s)100 150-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.411 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)373 M.4 Octave-band prediction for the ASTM C423 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and with absorber M.4.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands M.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 M.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.43 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.8 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.44 s0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 3.323 s0 1-40-2002040Time (s)60 80010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.744 s0 1-1000100Time (s)100 150-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.204 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 2.832 s0 1-20020Time (s)60 80-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.835 s0 1-50050Time (s)100 150-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.543 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)374 M.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 M.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 M.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 3.824 s0 1-40-2002040Time (s)60 80-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.888 s0 1-50050Time (s)100 150-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.182 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 3.571 s0 1-40-2002040Time (s)60 80020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.493 s0 1-100-50050Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.197 s0 1-5000500Time (s)200 300-50000500010000Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 7.827 s0 1-40-2002040Time (s)60 80-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.206 s0 1-50050100Time (s)100 150-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.4 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)375 Appendix: N N.1 Third-octave-band prediction for the ASTM C423 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table N.1, the corresponding statistically-based cut-off frequency limit is 108 Hz, which suggests that prediction can be done from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 125-Hz third-octave are 1.07, 0.92, 0.39 and 0.42, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 5.6 s for the RT, as shown in Table N.3, is 423 Hz. Table N.1: Predicted third-octave-band modal compositions for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 4 5 9 14 31 54 100 188 352 Modal density, 𝑛(𝑓) 0.33 0.36 0.47 0.61 1.07 1.46 2.17 3.24 4.82 Modal spacing, 𝛿𝑓 2.49 2.77 2.00 1.55 0.92 0.69 0.46 0.31 0.21 Modal width, 𝛿𝑤 0.36 0.33 0.35 0.37 0.39 0.39 0.41 0.42 0.43 Modal overlap, 𝑀 0.12 0.12 0.17 0.22 0.41 0.57 0.89 1.37 2.08 Modal overlap factor, 𝑀𝑓 0.14 0.12 0.18 0.24 0.42 0.57 0.90 1.38 2.07 Spatial uniformity of SPL: From Table N.2, the predicted standard deviations of SPL are less than 1.5 dB from the 125-Hz third-octave band. Accuracy of the predicted SPLs: The accuracy of the predicted steady-state SPLs is checked against the Sabine and Eyring versions of the diffuse-field formulae as presented in Table N.2. The errors associated with the predicted SPLs are always less than 1% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases. 376 Table N.2: Predicted third-octave-band SPL statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 140.5 152.9 150.7 151.6 156.3 158.3 163.2 167.8 170.2 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 142.7 156.2 151.3 157.0 156.7 159.6 163.3 167.0 169.8 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 139.5 153.3 150.3 154.7 156.7 159.2 161.8 168.3 170.6 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 142.2 153.8 151.8 152.6 155.9 159.0 163.0 167.3 171.2 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 141.7 156.4 153.3 157.3 155.6 157.7 163.7 167.0 170.0 Predicted average SPL, ?̅?𝑝 (dB) 141.5 154.8 151.6 155.2 156.3 158.8 163.1 167.5 170.4 Standard deviation, 𝜎𝐿𝑝 (dB) 1.3 1.7 1.2 2.6 0.5 0.8 0.7 0.6 0.6 Expected Sabine SPL, 𝐿𝑝 (dB) 144.8 157.4 153.6 157.1 157.7 160.1 164.1 168.5 171.1 Sabine error (%) 2.3 1.7 1.3 1.2 0.9 0.8 0.6 0.6 0.4 Expected Eyring SPL, 𝐿𝑝 (dB) 144.8 157.4 153.5 157.1 157.7 160.0 164.1 168.5 171.1 Eyring error (%) 2.31 1.65 1.26 1.18 0.91 0.77 0.62 0.59 0.41 Table N.3: Predicted third-octave-band RT statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.08 6.73 6.46 6.09 6.10 5.66 5.64 5.44 7.08 Predicted RT @ Rec#1: 𝑇60 (s) 5.95 5.82 5.87 5.31 5.55 5.50 4.83 4.67 5.95 Predicted RT @ Rec#2: 𝑇60 (s) 6.12 5.95 5.35 5.19 5.59 5.10 5.31 5.30 6.12 Predicted RT @ Rec#3: 𝑇60 (s) 5.56 5.96 5.54 5.53 4.95 5.36 5.24 6.31 5.56 Predicted RT @ Rec#4: 𝑇60 (s) 5.44 6.25 5.48 5.84 5.66 5.07 4.91 4.90 5.44 Predicted RT @ Rec#5: 𝑇60 (s) 6.35 6.33 5.66 6.27 5.28 5.58 4.93 4.84 6.35 Predicted average RT: ?̅?60 (s) 5.86 6.05 5.57 5.60 5.39 5.31 5.04 5.14 5.86 Standard deviation, 𝜎𝑇60 (s) 0.38 0.22 0.20 0.44 0.29 0.23 0.22 0.66 0.38 Expected Sabine RT: 𝑇60 (s) 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 4.6 Sabine error (%) 27.41 31.58 21.15 21.72 17.18 15.47 9.45 11.7 27.4 Expected Eyring RT, 𝑇60 (s) 4.56 4.56 4.56 4.56 4.56 4.56 4.56 4.56 4.56 Eyring error (%) 28.7 32.9 22.4 23.0 18.4 16.6 10.6 12.9 28.7 Decay rate (dB/s) 10.2 9.9 10.8 10.7 11.1 11.3 11.9 11.7 10.2 377 Spatial uniformity of RT: From Table N.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.5 s in all third-octave bands with the exception in the 250-Hz third-octave band. Accuracy of the predicted RTs: The accuracy of the predicted RTs is checked against the expected values of the RTs calculated using the Sabine and Eyring versions of the diffuse-field formulae as presented in Table N.3. The errors are less than or around 15% from the 160-Hz third-octave band in both Sabine and Eyring prediction cases. Linearity of temporal decay curves: From Table N.4, the predicted curvatures of the energy-decay curves vary within 5 to 17 in all third-octave bands. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix O.1. Table N.4: Predicted third-octave-band decay-curve curvature statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 4.0 3.2 2.3 9.9 7.8 17.9 7.1 11.9 4.0 𝜅 @ Rec#2 5.4 14.9 6.2 18.8 8.8 10.0 7.6 3.3 5.4 𝜅 @ Rec#3 3.2 16.4 6.2 7.7 11.6 5.7 14.0 10.9 3.2 𝜅 @ Rec#4 6.8 9.7 4.6 27.7 5.5 3.3 6.3 5.1 6.8 𝜅 @ Rec#5 7.5 40.6 6.6 17.9 13.3 9.2 9.1 3.6 7.5 Average, 𝜅 5.4 16.9 5.2 16.4 9.4 9.2 8.8 7.0 5.4 Modal, 𝜅 0.11 0.16 0.15 0.08 0.20 0.08 0.29 0.15 0.11 Table N.5: Predicted third-octave-band source-power level statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 157.5 153.7 157.2 157.8 160.2 164.2 168.6 171.2 157.5 Sabine error (%) 20.4 22.7 16.5 16.9 13.9 12.7 8.2 10.0 20.4 Eyring error (%) 21.3 23.6 17.5 17.8 14.8 13.6 9.1 10.9 21.3 Predicted, ?̅?𝑤 (dB) (1 source) 154.5 150.7 154.2 154.8 157.2 161.2 165.6 168.2 154.5 378 Accuracy of the predicted source-power levels: The predicted source-power levels and the corresponding errors in third-octave bands are presented in Table N.5. The errors associated with the predicted source-power levels are less than 13% from the 125-Hz third-octave band. Accuracy of the predicted room absorptions: It is seen from Table N.6 that the errors are less than or around 15% from the 125-Hz third-octave band, and greater than 20% in the rest of the bands. The same is true for the room absorption-coefficient prediction case, as the absorption-coefficient predictions depend on the accuracy of the absorption predictions. Table N.6: Predicted third-octave-band room absorption statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 3.40 3.29 3.57 3.56 3.69 3.75 3.95 3.87 3.40 Sabine expected absorption, 𝐴 (m2) 4.33 4.33 4.33 4.33 4.33 4.33 4.33 4.33 4.33 Sabine error (%) 21.5 24.0 17.5 17.8 14.7 13.4 8.6 10.5 21.5 Eyring expected absorption, 𝐴 (m2) 4.37 4.37 4.37 4.37 4.37 4.37 4.37 4.37 4.37 Eyring error (%) 22.3 24.8 18.3 18.7 15.5 14.3 9.6 11.4 22.3 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 21.5 24.0 17.5 17.8 14.7 13.4 8.6 10.5 21.5 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 22.1 24.6 18.1 18.5 15.4 14.1 9.5 11.3 22.1 379 N.2 Third-octave-band prediction for the ASTM C423 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table N.7. The cut-off limit will remain be the same as before. Table N.7: Predicted third-octave-band modal compositions for ASTM C423 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 4 4 10 15 32 56 101 191 351 Modal density, 𝑛(𝑓) 0.33 0.29 0.53 0.65 1.10 1.51 2.20 3.29 4.81 Modal spacing, 𝛿𝑓 2.59 3.18 1.95 1.60 0.92 0.66 0.45 0.30 0.21 Modal width, 𝛿𝑤 1.03 0.86 3.77 3.08 2.91 2.87 2.14 2.44 2.28 Modal overlap, 𝑀 0.34 0.25 1.99 2.01 3.21 4.35 4.70 8.0 11.0 Modal overlap factor, 𝑀𝑓 0.40 0.27 1.94 1.93 3.17 4.33 4.72 8.0 11.0 Table N.8: Predicted third-octave-band SPL statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 138.9 145.7 145.2 148.8 152.1 155.2 160.1 165.7 166.8 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 140.8 146.2 145.9 150.3 152.8 153.9 157.8 165.3 166.1 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 137.9 149.0 143.5 146.5 150.7 153.0 161.6 164.5 166.9 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 142.0 149.2 146.1 149.9 152.6 154.8 159.4 163.3 166.8 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 137.0 143.2 147.0 149.6 152.1 153.9 159.3 163.1 165.1 Predicted average SPL, ?̅?𝑝 (dB) 139.7 147.2 145.7 149.2 152.1 154.2 159.8 164.5 166.4 Standard deviation, 𝜎𝐿𝑝 (dB) 2.09 2.57 1.31 1.53 0.81 0.87 1.41 1.18 0.75 380 Table N.9: Predicted third-octave-band RT statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 4.66 5.62 4.55 3.17 4.24 3.73 2.92 3.28 2.64 Predicted RT @ Rec#1: 𝑇60 (s) 4.30 2.77 3.12 3.36 3.06 2.67 2.61 2.43 2.80 Predicted RT @ Rec#2: 𝑇60 (s) 2.82 2.82 3.19 3.55 3.75 3.16 2.52 2.43 3.10 Predicted RT @ Rec#3: 𝑇60 (s) 3.65 3.19 4.15 3.98 5.06 4.40 2.40 2.83 3.07 Predicted RT @ Rec#4: 𝑇60 (s) 4.59 2.90 3.80 3.65 2.96 4.32 2.69 2.56 3.29 Predicted RT @ Rec#5: 𝑇60 (s) 3.51 2.31 3.08 2.83 2.66 3.05 2.40 2.51 2.70 Predicted average RT: ?̅?60 (s) 3.67 2.77 3.42 3.43 3.32 3.38 2.52 2.54 2.98 Standard deviation, 𝜎𝑇60 (s) 0.71 0.32 0.48 0.43 0.98 0.81 0.13 0.16 0.24 Decay rate (dB/s) 16.4 21.7 17.5 17.5 18.1 17.7 23.8 23.6 20.2 Spatial uniformity of SPL: The standard deviations of the SPL for this ‘with absorber’ case, as a measure of spatial variation, are presented in Table N.8. Overall, the deviations are greater than the values obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT for this ‘with absorber’ case, as a measure of spatial variation, are presented in Table N.9. It is obvious that the deviations are greater than in the previous ‘without absorber’ case in almost all third-octave bands. Table N.10: Predicted third-octave-band decay-curve curvature statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 7.8 2.3 47.8 6.5 22.7 12.6 10.1 10.6 8.6 𝜅 @ Rec#2 22.8 2.6 15.4 18.3 6.0 26.2 11.7 5.6 11.5 𝜅 @ Rec#3 57.8 4.5 38.5 4.5 3.1 20.7 12.6 16.5 29.6 𝜅 @ Rec#4 10.6 1.1 16.2 4.7 21.8 15.5 8.2 19.2 36.0 𝜅 @ Rec#5 45.4 108.6 72.1 28.4 42.8 11.2 14.7 13.1 10.7 Average, 𝜅 28.9 23.8 38.0 12.5 19.3 17.2 11.5 13.0 19.3 Modal, 𝜅 0.04 0.37 0.83 1.68 0.97 2.05 2.05 9.28 3.85 381 Linearity of decay curves: The curvatures of the energy-decay curves as presented in Table N.10 reveal that the curvatures are greater than those obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave-bands are presented in Appendix O.2. Table N.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ASTM C423 rectangular reverberation room with the longest z-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.31 0.61 0.41 0.36 0.39 0.35 0.67 0.62 0.45 Sabine error (%) 59.9 19.9 46.6 53.0 48.5 53.8 12.4 18.2 40.7 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.26 0.46 0.33 0.30 0.32 0.30 0.49 0.46 0.36 Eyring error (%) 65.4 40.0 56.1 60.5 57.4 61.1 36.0 39.1 52.2 Accuracy of the sample absorption coefficients: The accuracy of the predicted absorption coefficients of an absorber sample are presented in Table N.11. It is noticeable that the predicted values deviate from the theoretical values to a great extent for both Sabine and Eyring prediction cases. 382 N.3 Octave-band prediction for the ISO 354 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: The modal compositions data shown in Table N.12 suggests that prediction can be done from the 125-Hz band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz octave band are 1.08, 0.9, 0.39 and 0.43, respectively. On the other hand, the pure-tone-based Schroeder limit corresponding to a modal overlap factor of 3 and a band-averaged RT of 5.8 s, as shown in Table N.14, is 430 Hz. Table N.12: Predicted octave-band modal compositions for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 18 96 643 Modal density, 𝑛(𝑓) 0.41 1.08 3.61 Modal spacing, 𝛿𝑓 2.31 0.90 0.28 Modal width, 𝛿𝑤 0.35 0.39 0.42 Modal overlap, 𝑀 0.14 0.42 1.54 Modal overlap factor, 𝑀𝑓 0.15 0.43 1.54 Spatial uniformity of SPL: The predicted SPL standard deviations are smaller than 1.5 dB in all octave bands as shown in Table N.13. Accuracy of the predicted SPLs: As shown in Table N.13, the predicted SPLs show errors below 1% from the 125-Hz octave band for both Sabine and Eyring cases. Spatial uniformity of RT: The predicted RT standard deviations are less than 0.3 s in al octave bands. These are shown in Table N.14. Accuracy of the predicted RTs: As shown in Table N.14, the predicted RTs show errors below 12% from the 250-Hz octave band. 383 Table N.13: Predicted octave-band SPL statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 151.13 157.09 168.81 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 153.36 158.85 168.32 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 151.28 158.07 169.09 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 152.22 157.33 169.28 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 154.06 157.69 168.55 Predicted average SPL, ?̅?𝑝 (dB) 152.56 157.85 168.83 Standard deviation, 𝜎𝐿𝑝 (dB) 1.29 0.69 0.39 Expected Sabine SPL, 𝐿𝑝 (dB) 154.78 159.26 169.84 Sabine error (%) 1.43 0.89 0.60 Expected Eyring SPL, 𝐿𝑝 (dB) 154.74 159.22 169.80 Eyring error (%) 1.41 0.86 0.57 Table H.14: Predicted octave-band RT statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 6.91 6.18 5.54 Predicted RT @ Rec#1: 𝑇60 (s) 6.34 5.61 4.80 Predicted RT @ Rec#2: 𝑇60 (s) 6.26 5.49 5.27 Predicted RT @ Rec#3: 𝑇60 (s) 6.73 5.55 5.45 Predicted RT @ Rec#4: 𝑇60 (s) 6.61 6.00 4.88 Predicted RT @ Rec#5: 𝑇60 (s) 6.43 5.79 4.98 Predicted average RT: ?̅?60 (s) 6.47 5.68 5.06 Standard deviation, 𝜎𝑇60 (s) 0.19 0.21 0.27 Expected Sabine RT: 𝑇60 (s) 4.60 4.60 4.60 Sabine error (%) 40.52 23.53 10.03 Expected Eyring RT, 𝑇60 (s) 4.56 4.56 4.56 Eyring error (%) 41.95 24.78 11.15 Decay rate (dB/s) 9.28 10.56 11.85 384 Table N.15: Predicted octave-band curvature statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 1.81 3.33 8.08 𝜅 @ Rec#2 4.00 5.22 2.03 𝜅 @ Rec#3 0.89 5.91 3.82 𝜅 @ Rec#4 1.00 5.53 2.08 𝜅 @ Rec#5 9.34 4.71 4.30 Average, 𝜅 3.41 4.94 4.06 Modal, 𝜅 0.20 0.16 0.18 Linearity of decay curves: The curvatures are within 3 to 5 in all octave bands as shown in Table N.15. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix O.3. Accuracy of the predicted source-power levels: The predicted source-power levels show errors below 10% from the 250-Hz band, while the errors are greater than 15% in the rest of the bands as shown in Table N.16. Table N.16: Predicted octave-band source-power level statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 154.89 159.37 169.94 Sabine error (%) 27.25 18.00 8.63 Eyring error (%) 28.21 18.95 9.58 Predicted, ?̅?𝑤 (dB) (1 source) 151.88 156.36 166.93 Accuracy of the predicted room absorptions: The predicted room absorptions, as well as the absorption coefficients, show errors below 10% from the 250-Hz band, while the errors are greater than 20% in the rest of the bands. These are shown in Table N.17. 385 Table N.17: Predicted octave-band room absorption statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 3.08 3.50 3.93 Sabine expected absorption, 𝐴 (m2) 4.33 4.33 4.33 Sabine error (%) 28.84 19.05 9.12 Eyring expected absorption, 𝐴 (m2) 4.37 4.37 4.37 Eyring error (%) 29.55 19.86 10.03 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Sabine error (%) 28.84 19.05 9.12 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 29.34 19.70 9.94 386 N.4 Octave-band prediction for the ASTM C423 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit as there no change in the modal compositions except the modal width and, hence, the modal overlap as shown in Table N.18. Table N.18: Predicted octave-band modal compositions for ASTM C423 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 18 100 646 Modal density, 𝑛(𝑓) 0.41 1.12 3.63 Modal spacing, 𝛿𝑓 2.34 0.89 0.28 Modal width, 𝛿𝑤 2.52 2.95 2.31 Modal overlap, 𝑀 1.03 3.31 8.37 Modal overlap factor, 𝑀𝑓 1.08 3.32 8.38 Table N.19: Predicted octave-band SPL statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 144.88 153.64 165.90 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 145.47 152.96 165.24 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 146.34 151.45 165.73 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 147.43 153.75 165.04 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 144.51 152.94 164.03 Predicted average SPL, ?̅?𝑝 (dB) 145.85 153.02 165.24 Standard deviation, 𝜎𝐿𝑝 (dB) 1.18 0.92 0.74 387 Table N.20: Predicted octave-band RT statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 5.08 3.99 2.99 Predicted RT @ Rec#1: 𝑇60 (s) 4.28 3.17 2.55 Predicted RT @ Rec#2: 𝑇60 (s) 4.46 3.60 2.74 Predicted RT @ Rec#3: 𝑇60 (s) 6.64 4.90 2.72 Predicted RT @ Rec#4: 𝑇60 (s) 4.91 3.73 3.07 Predicted RT @ Rec#5: 𝑇60 (s) 3.23 2.88 2.63 Predicted average RT: ?̅?60 (s) 4.46 3.54 2.73 Standard deviation, 𝜎𝑇60 (s) 1.28 0.79 0.20 Decay rate (dB/s) 13.46 16.95 21.97 Spatial uniformity of SPL: The predicted SPL deviations are shown in Table N.19. The overall deviations are greater than in the ‘without absorber’ case. Spatial uniformity of RT: The predicted RT deviations are shown in Table N.20. The overall deviations are greater than in the ‘without absorber’ case. Table N.21: Predicted octave-band decay-curve curvature statistics for ASTM C423 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 54.43 9.58 4.07 𝜅 @ Rec#2 74.13 10.61 3.19 𝜅 @ Rec#3 82.81 7.14 7.85 𝜅 @ Rec#4 36.47 6.22 23.49 𝜅 @ Rec#5 23.57 13.66 4.43 Average, 𝜅 54.28 9.44 8.61 Modal, 𝜅 1.19 2.00 6.18 Linearity of decay curves: The curvatures are shown in Table N.21. The overall curvatures are greater than in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix O.4. 388 Table N.22: Predicted octave-band sample absorber absorption coefficient statistics for ASTM C423 rectangular reverberation room with the longest z-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.22 0.34 0.54 Sabine error (%) 70.82 55.34 29.24 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.20 0.29 0.42 Eyring error (%) 73.83 62.13 45.27 Accuracy of the predicted sample absorption coefficients: As shown in Table N.22, the predicted values show greater deviation from the theoretical values for both Sabine and Eyring prediction cases. 389 Appendix: O O.1 Third-octave-band prediction for the ASTM C423 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber O.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 6.97 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 7.08 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 6.73 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 6.46 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.09 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 6.1 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 5.66 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.64 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 5.44 s390 O.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs in third-octave bands, at receiver #5 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 5.431 s0 1-10010Time (s)45 50 55-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.353 s0 1-40-2002040Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 6.326 s0 1-50050Time (s)70 80-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 5.659 s0 1-50050Time (s)90 100 110-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 6.267 s0 1-100-50050Time (s)120 130 140-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 5.281 s0 1-1000100Time (s)150 160 170-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 5.582 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.925 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 4.84 s0 1-400-2000200400Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)391 O.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 5.936 s0 1-1001020Time (s)45 50 55-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.437 s0 1-40-2002040Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 6.253 s0 1-40-2002040Time (s)70 80-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 5.477 s0 1-50050Time (s)90 100 110-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.839 s0 1-50050Time (s)120 130 140-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 5.664 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 5.069 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.911 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 4.899 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)392 O.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 5.804 s0 1-20020Time (s)45 50 55-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.557 s0 1-50050Time (s)60 65 700500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.964 s0 1-40-2002040Time (s)70 80-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 5.544 s0 1-50050Time (s)90 100 110-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.531 s0 1-100-50050Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 4.948 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 5.363 s0 1-2000200Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 5.243 s0 1-5000500Time (s)240 260 280-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 6.311 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)393 O.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 5.835 s0 1-20020Time (s)45 50 55-50005001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.115 s0 1-40-2002040Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 5.946 s0 1-40-2002040Time (s)70 80-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.353 s0 1-50050Time (s)90 100 110-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.192 s0 1-1000100Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.587 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 5.095 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.309 s0 1-2000200Time (s)240 260 280-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 5.3 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)394 O.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 5.302 s0 1-10010Time (s)45 50 55-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.952 s0 1-40-2002040Time (s)60 65 70-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 5.816 s0 1-40-2002040Time (s)70 80-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 5.867 s0 1-40-2002040Time (s)90 100 110-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.307 s0 1-50050Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 5.547 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 5.499 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.831 s0 1-400-2000200Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 4.67 s0 1-400-2000200400Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)395 O.2 Third-octave-band prediction for the ASTM C423 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber O.2.1 Theoretical modal decay curves and the corresponding RTs, in third-octave bands 0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 4.66 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 5.62 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 4.55 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 3.17 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 4.24 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 3.73 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.92 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 3.28 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.64 s396 O.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 3.506 s0 1-10010Time (s)45 50 55-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 2.31 s0 1-20020Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 3.078 s0 1-20020Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 2.831 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.656 s0 1-50050Time (s)120 130 140-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 3.045 s0 1-50050Time (s)150 160 170-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.402 s0 1-2000200Time (s)180 200 220-8000-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.508 s0 1-2000200Time (s)240 260 280-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 2.696 s0 1-2000200Time (s)300 350-15000-10000-500005000Frequency (Hz)Pressure (Pa)397 O.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 4.585 s0 1-10010Time (s)45 50 55-1500-1000-5000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 2.903 s0 1-40-2002040Time (s)60 65 70-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 3.8 s0 1-20020Time (s)70 80-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 3.649 s0 1-50050Time (s)90 100 110-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.955 s0 1-50050Time (s)120 130 140-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 4.324 s0 1-1000100Time (s)150 160 170020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.694 s0 1-1000100Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.557 s0 1-2000200Time (s)240 260 280-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 3.292 s0 1-400-2000200400Time (s)300 350-50000500010000Frequency (Hz)Pressure (Pa)398 O.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 3.649 s0 1-20020Time (s)45 50 55-2000200400Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 3.191 s0 1-50050Time (s)60 65 70020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 4.152 s0 1-20020Time (s)70 80-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 3.983 s0 1-20020Time (s)90 100 110010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.056 s0 1-50050Time (s)120 130 1400200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 4.402 s0 1-50050100Time (s)150 160 170-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 2.395 s0 1-2000200Time (s)180 200 220-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.825 s0 1-400-2000200400Time (s)240 260 280-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 3.073 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)399 O.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 2.822 s0 1-20020Time (s)45 50 55-2000200400600Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 2.816 s0 1-20020Time (s)60 65 70-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 3.192 s0 1-40-2002040Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 3.548 s0 1-50050Time (s)90 100 110020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 3.746 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 3.158 s0 1-1000100Time (s)150 160 170-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.518 s0 1-1000100Time (s)180 200 220-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.432 s0 1-400-2000200400Time (s)240 260 280-20000-100000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 3.101 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)400 O.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 4.302 s0 1-505Time (s)45 50 55-50005001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 2.771 s0 1-20020Time (s)60 65 70-1000010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 3.123 s0 1-20020Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 3.361 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.061 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.668 s0 1-1000100Time (s)150 160 170-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.613 s0 1-2000200Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.428 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 2.8 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)401 O.3 Octave-band prediction for the ASTM C423 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber O.3.1 Theoretical modal decay curves and the corresponding RTs, in octave bands O.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #5 O.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #4 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.91 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.18 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.54 s0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.426 s0 1-50050Time (s)60 80-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.789 s0 1-2000200Time (s)100 150-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.977 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.606 s0 1-50050Time (s)60 80-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 6.002 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.881 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)402 O.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #3 O.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #2 O.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #1 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.725 s0 1-50050Time (s)60 80-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.552 s0 1-1000100Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 5.45 s0 1-100001000Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.261 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.494 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.266 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 6.336 s0 1-50050Time (s)60 80-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.614 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.8 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)403 O.4 Octave-band prediction for the ASTM C423 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber O.4.1 Theoretical modal decay curves and the corresponding RTs, in octave bands O.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 O.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 5.08 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.99 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.99 s0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 3.232 s0 1-50050Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.884 s0 1-1000100Time (s)100 150-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.631 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 4.906 s0 1-50050Time (s)60 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.727 s0 1-1000100Time (s)100 150-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 3.068 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)404 O.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 O.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs in octave bands, at receiver #2 O.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.644 s0 1-50050100Time (s)60 80020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 4.904 s0 1-1000100Time (s)100 150-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.724 s0 1-100001000Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 4.457 s0 1-50050Time (s)60 80-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 3.596 s0 1-1000100Time (s)100 150-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.736 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 4.276 s0 1-40-2002040Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.17 s0 1-1000100Time (s)100 150-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.549 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)405 Appendix: P P.1 Third-octave-band prediction for the ASTM C423 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table P.1, the corresponding statistically-based cut-off frequency limit is 108 Hz, which suggests that prediction can be done from the 125-Hz band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 125-Hz third-octave band are 0.97, 1.02, 0.37 and 0.36, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 5.6 s for the RT, as shown in Table P.3, is 424 Hz. Table P.1: Predicted third-octave-band modal compositions for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 2 6 9 15 28 54 99 187 354 Modal density, 𝑛(𝑓) 0.17 0.43 0.47 0.65 0.97 1.46 2.15 3.22 4.85 Modal spacing, 𝛿𝑓 4.61 2.31 2.04 1.28 1.02 0.69 0.46 0.31 0.21 Modal width, 𝛿𝑤 0.36 0.34 0.34 0.36 0.37 0.39 0.40 0.41 0.42 Modal overlap, 𝑀 0.06 0.15 0.16 0.23 0.35 0.56 0.86 1.32 2.03 Modal overlap factor, 𝑀𝑓 0.08 0.15 0.17 0.28 0.36 0.56 0.86 1.34 2.03 Spatial uniformity of SPL: From Table P.2, the predicted standard deviations of SPL are less than 1.5 dB from the 160-Hz band, while the deviations are greater than 1.5 dB in the rest of the bands. Accuracy of the predicted SPLs: As presented in Table P.2, the errors associated with the predicted SPLs are always less than or around 1% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases. In the rest of the bands, the errors are greater than 1%. 406 Table P.2: Predicted third-octave-band SPL statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 149.9 143.1 151.1 150.3 159.5 160.3 162.7 166.1 170.1 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 138.3 147.8 150.3 148.0 156.9 160.4 165.0 166.9 171.1 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 143.7 145.7 147.6 153.6 155.8 159.5 164.2 166.7 170.6 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 153.2 146.8 147.4 152.8 155.2 160.1 163.9 166.1 169.5 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 146.9 142.0 150.5 153.1 159.2 159.4 162.8 166.8 170.4 Predicted average SPL, ?̅?𝑝 (dB) 148.9 145.6 149.6 152.0 157.7 160.0 163.8 166.5 170.4 Standard deviation, 𝜎𝐿𝑝 (dB) 6.4 2.5 1.7 2.4 2.0 0.5 1.0 0.4 0.6 Expected Sabine SPL, 𝐿𝑝 (dB) 152.1 148.1 151.6 154.0 159.2 161.1 165.3 167.5 171.0 Sabine error (%) 2.1 1.7 1.3 1.3 0.9 0.7 0.9 0.6 0.4 Expected Eyring SPL, 𝐿𝑝 (dB) 152.1 148.1 151.6 153.9 159.1 161.1 165.2 167.4 171.0 Eyring error (%) 2.12 1.66 1.29 1.24 0.91 0.70 0.85 0.54 0.36 Table P.3: Predicted third-octave-band RT statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 6.37 6.93 6.68 6.43 6.15 5.93 5.69 5.53 5.39 Predicted RT @ Rec#1: 𝑇60 (s) 5.70 5.54 6.44 5.61 5.76 5.23 5.29 5.48 5.49 Predicted RT @ Rec#2: 𝑇60 (s) 6.11 5.71 5.30 5.33 5.49 5.92 4.45 5.09 5.72 Predicted RT @ Rec#3: 𝑇60 (s) 5.88 6.13 6.14 5.61 6.12 5.64 5.35 5.22 5.10 Predicted RT @ Rec#4: 𝑇60 (s) 5.74 6.14 5.85 5.81 5.88 5.59 4.97 5.17 4.99 Predicted RT @ Rec#5: 𝑇60 (s) 6.04 7.78 7.30 5.73 5.43 6.05 4.97 5.58 5.83 Predicted average RT: ?̅?60 (s) 5.89 6.17 6.14 5.61 5.72 5.67 4.98 5.30 5.40 Standard deviation, 𝜎𝑇60 (s) 0.18 0.90 0.75 0.18 0.28 0.32 0.36 0.21 0.37 Expected Sabine RT: 𝑇60 (s) 4.80 4.80 4.80 4.80 4.80 4.80 4.80 4.80 4.80 Sabine error (%) 22.7 28.6 27.9 17.0 19.3 18.2 3.8 10.5 12.6 Expected Eyring RT, 𝑇60 (s) 4.75 4.75 4.75 4.75 4.75 4.75 4.75 4.75 4.75 Eyring error (%) 23.9 29.9 29.2 18.2 20.5 19.4 4.9 11.6 13.7 Decay rate (dB/s) 10.2 9.7 9.8 10.7 10.5 10.6 12.0 11.3 11.1 407 Table P.4: Predicted third-octave-band decay curve curvature statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 4.9 13.1 11.2 17.7 9.2 10.7 14.1 3.3 7.3 𝜅 @ Rec#2 1.9 9.2 50.4 21.4 4.6 9.3 9.6 5.5 5.1 𝜅 @ Rec#3 3.3 6.7 7.9 15.7 10.7 2.7 2.7 5.0 4.6 𝜅 @ Rec#4 3.9 23.0 11.6 25.2 9.6 8.4 5.4 9.4 7.2 𝜅 @ Rec#5 1.5 15.3 50.7 2.7 12.2 11.7 7.4 7.1 2.5 Average, 𝜅 3.1 13.5 26.4 16.5 9.2 8.6 7.8 6.1 5.4 Modal, 𝜅 0.03 0.11 0.02 0.05 0.02 0.03 0.02 0.02 0.02 Spatial uniformity of RT: From Table P.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.5 s from the 100-Hz third-octave band. Accuracy of the predicted RTs: As presented in Table P.3, the predicted RTs show errors less than or around 10% from the 200-Hz third-octave band for both Sabine and Eyring prediction cases; while the errors are greater than 15% in the rest of the bands. Linearity of temporal decay curves: From Table P.4, the predicted curvatures of the energy-decay curves vary within 5 to 17 from the 100-Hz band. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix Q.1. Accuracy of the predicted source-power levels: As presented in Table P.5, the predicted source-power-level errors are less than or around 10% from the 200-Hz third-octave band. Table P.5: Predicted third-octave-band source-power level statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 152.1 148.1 151.6 153.9 159.1 161.1 165.2 167.4 171.0 Sabine error (%) 16.9 20.2 19.8 13.2 14.6 13.9 3.5 8.6 10.1 Eyring error (%) 17.8 21.1 20.7 14.1 15.6 14.8 4.3 9.5 11.1 Predicted, ?̅?𝑤 (dB) (1 source) 149.1 145.1 148.6 150.9 156.1 158.1 162.2 164.4 168.0 408 Table P.6: Predicted third-octave-band room absorption statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 3.41 3.25 3.27 3.57 3.51 3.54 4.03 3.78 3.71 Sabine expected absorption, 𝐴 (m2) 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18 Sabine error (%) 18.5 22.2 21.8 14.5 16.1 15.4 3.7 9.5 11.2 Eyring expected absorption, 𝐴 (m2) 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 Eyring error (%) 19.3 23.0 22.6 15.4 17.0 16.2 4.7 10.4 12.1 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 18.5 22.2 21.8 14.5 16.1 15.4 3.7 9.5 11.2 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 19.2 22.8 22.4 15.2 16.8 16.1 4.6 10.3 12.0 Accuracy of the predicted room absorptions: As presented in Table P.6, the errors are less than or around 10% from the 200-Hz third-octave band, and greater than 15% in the rest of the bands. The same is true for the room absorption coefficient prediction case, as the absorption coefficient predictions depend on the accuracy of the absorption predictions. 409 P.2 Third-octave-band prediction for the ASTM C423 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table P.7. The cut-off frequency limit will remain be the same as before. Table P.7: Predicted third-octave-band modal compositions for ASTM C423 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 6 9 16 30 55 100 189 358 Modal density, 𝑛(𝑓) 0.25 0.43 0.47 0.70 1.03 1.49 2.17 3.26 4.90 Modal spacing, 𝛿𝑓 2.82 2.49 1.92 1.39 0.95 0.66 0.46 0.31 0.20 Modal width, 𝛿𝑤 9.98 0.87 4.39 2.95 3.08 2.81 2.46 2.32 2.28 Modal overlap, 𝑀 2.49 0.37 2.08 2.06 3.19 4.17 5.35 7.56 11.2 Modal overlap factor, 𝑀𝑓 3.54 0.35 2.29 2.12 3.23 4.23 5.38 7.60 11.2 Table P.8: Predicted third-octave-band SPL statistics for ASTM C423 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 147.8 140.9 145.9 146.6 152.5 155.5 159.1 162.2 166.5 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 138.0 145.0 145.4 148.1 156.1 155.3 160.3 163.5 167.2 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 141.3 138.2 141.3 150.5 150.4 156.9 158.4 163.8 165.2 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 151.0 144.3 144.1 151.2 151.1 156.8 161.3 162.9 164.6 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 144.9 140.6 145.5 151.1 150.9 153.5 157.4 163.0 165.6 Predicted average SPL, ?̅?𝑝 (dB) 146.8 142.5 144.7 149.9 152.8 155.8 159.5 163.1 165.9 Standard deviation, 𝜎𝐿𝑝 (dB) 5.67 2.94 1.90 2.09 2.38 1.39 1.55 0.63 1.04 410 Table P.9: Predicted third-octave-band RT statistics for ASTM C423 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 4.71 3.73 4.17 4.47 3.97 3.13 3.22 2.53 2.36 Predicted RT @ Rec#1: 𝑇60 (s) 4.75 4.16 4.32 3.26 3.20 2.67 3.14 2.57 2.30 Predicted RT @ Rec#2: 𝑇60 (s) 4.95 3.05 3.93 3.12 3.64 2.56 2.69 2.41 2.40 Predicted RT @ Rec#3: 𝑇60 (s) 4.62 3.86 3.76 3.72 3.51 2.62 2.75 2.21 2.99 Predicted RT @ Rec#4: 𝑇60 (s) 4.76 2.85 3.11 3.90 3.46 2.53 2.34 2.22 2.81 Predicted RT @ Rec#5: 𝑇60 (s) 4.88 6.01 4.43 4.56 4.32 3.49 3.42 2.95 4.60 Predicted average RT: ?̅?60 (s) 4.79 3.72 3.85 3.64 3.59 2.73 2.82 2.44 2.84 Standard deviation, 𝜎𝑇60 (s) 0.12 1.29 0.53 0.58 0.42 0.41 0.42 0.31 0.95 Decay rate (dB/s) 12.5 16.1 15.6 16.5 16.7 22.0 21.3 24.6 21.1 Spatial uniformity of SPL: The standard deviations of the SPL are presented in Table P.8. Overall, the deviations are greater than those obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT are presented in Table P.9. It is obvious that the deviations are greater than the previous ‘without absorber’ case in almost all third-octave bands. Table P.10: Predicted third-octave-band decay curve curvature statistics for ASTM C423 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 0.7 13.1 14.3 17.8 22.2 8.8 8.0 8.3 16.6 𝜅 @ Rec#2 17.8 20.5 52.0 11.3 5.8 5.2 11.5 4.3 16.8 𝜅 @ Rec#3 5.2 69.2 90.8 25.8 24.2 8.6 7.3 21.9 16.3 𝜅 @ Rec#4 0.4 15.1 3.3 45.4 7.7 10.1 17.0 2.8 11.2 𝜅 @ Rec#5 0.9 36.0 125.6 50.9 25.6 34.8 10.5 77.5 53.9 Average, 𝜅 5.0 30.8 57.2 30.2 17.1 13.5 10.9 23.0 23.0 Modal, 𝜅 0.10 0.42 1.42 2.86 0.93 2.50 3.28 2.10 2.38 Linearity of decay curves: The curvatures of the decay curves as presented in Table P.10 reveal that the curvatures are greater than the values obtained in the ‘without absorber’ case. The 411 modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix Q.2. Accuracy of the sample absorption coefficients: The predicted absorption coefficients of an absorber sample are presented in Table P.11. It is noticeable that the predicted values deviate from the theoretical values to a great extent. Table P.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ASTM C423 oblique reverberation room with the longest z-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.12 0.33 0.30 0.30 0.32 0.58 0.47 0.68 0.51 Sabine error (%) 84.3 56.7 60.8 61.1 58.0 23.1 37.7 10.6 32.5 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.11 0.28 0.26 0.26 0.27 0.44 0.38 0.49 0.40 Eyring error (%) 85.2 63.1 66.1 66.3 64.0 41.8 50.4 35.1 47.2 412 P.3 Octave-band prediction for the ASTM C423 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table P.12, the statistically-based cut-off frequency limit is 68 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz band are 1.07, 0.89, 0.38 and 0.42, respectively. On the other hand, the pure-tone-based Schroeder limit corresponding to a modal overlap factor of 3 and a band-averaged RT of 6.4 s, as shown in Table P.14, is 452 Hz. Table P.12: Predicted octave-band modal compositions for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 17 95 642 Modal density, 𝑛(𝑓) 0.39 1.07 3.61 Modal spacing, 𝛿𝑓 2.55 0.89 0.28 Modal width, 𝛿𝑤 0.34 0.38 0.41 Modal overlap, 𝑀 0.13 0.40 1.49 Modal overlap factor, 𝑀𝑓 0.13 0.42 1.49 Spatial uniformity of SPL: The predicted SPL standard deviations are less than or around 1.5 dB from the 125-Hz octave band, as shown in Table P.13. Accuracy of the predicted SPLs: As shown in Table P.13, the predicted SPLs show errors below 1% from the 125-Hz octave band. 413 Table P.13: Predicted octave-band SPL statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 150.04 159.27 168.23 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 148.49 158.19 169.36 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 146.85 157.82 168.86 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 151.06 157.97 168.01 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 148.62 158.94 168.57 Predicted average SPL, ?̅?𝑝 (dB) 149.25 158.48 168.63 Standard deviation, 𝜎𝐿𝑝 (dB) 1.63 0.64 0.53 Expected Sabine SPL, 𝐿𝑝 (dB) 150.91 159.96 169.57 Sabine error (%) 1.10 0.93 0.55 Expected Eyring SPL, 𝐿𝑝 (dB) 150.87 159.92 169.53 Eyring error (%) 1.08 0.90 0.53 Table P.14: Predicted octave-band RT statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 6.76 6.12 5.49 Predicted RT @ Rec#1: 𝑇60 (s) 5.96 5.49 5.49 Predicted RT @ Rec#2: 𝑇60 (s) 6.28 5.74 5.39 Predicted RT @ Rec#3: 𝑇60 (s) 9.95 5.90 5.11 Predicted RT @ Rec#4: 𝑇60 (s) 5.85 5.59 5.03 Predicted RT @ Rec#5: 𝑇60 (s) 14.10 5.97 5.60 Predicted average RT: ?̅?60 (s) 7.47 5.73 5.31 Standard deviation, 𝜎𝑇60 (s) 3.76 0.20 0.24 Expected Sabine RT: 𝑇60 (s) 4.80 4.80 4.80 Sabine error (%) 55.65 19.46 10.70 Expected Eyring RT, 𝑇60 (s) 4.75 4.75 4.75 Eyring error (%) 57.22 20.67 11.82 Decay rate (dB/s) 8.03 10.47 11.29 414 Table P.15: Predicted octave-band decay curve curvature statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 4.36 7.61 1.83 𝜅 @ Rec#2 6.23 4.91 4.02 𝜅 @ Rec#3 15.85 3.29 1.98 𝜅 @ Rec#4 5.64 3.35 3.36 𝜅 @ Rec#5 66.67 3.24 3.26 Average, 𝜅 19.75 4.48 2.89 Modal, 𝜅 0.06 0.04 0.02 Spatial uniformity of RT: The predicted RT standard deviations are smaller than 0.3 s from the 125-Hz band. These are shown in Table P.14. Accuracy of the predicted RTs: As shown in Table P.14, the predicted RTs show errors below or around 10% from the 250-Hz band, while the errors are greater than 15% in the rest of the bands. Degree of linearity of decay curves: The curvatures are within 2 to 5 from the 125-Hz octave band, as shown in Table P.15. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix Q.3. Table P.16: Predicted octave-band source-power-level statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 150.87 159.92 169.53 Sabine error (%) 32.37 14.77 8.76 Eyring error (%) 33.29 15.68 9.67 Predicted, ?̅?𝑤 (dB) (1 source) 147.86 156.91 166.52 415 Accuracy of the predicted source-power levels: The predicted source-power levels show errors below 10% from the 250-Hz band, as shown in Table P.16. Table P.17: Predicted octave-band room absorption statistics for ASTM C423 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 2.69 3.50 3.78 Sabine expected absorption, 𝐴 (m2) 4.18 4.18 4.18 Sabine error (%) 35.75 16.29 9.66 Eyring expected absorption, 𝐴 (m2) 4.22 4.22 4.22 Eyring error (%) 36.40 17.13 10.57 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Sabine error (%) 35.75 16.29 9.66 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 36.16 16.99 10.47 Accuracy of the predicted room absorptions: The predicted room absorptions, as well as the absorption coefficients, show errors below 10% from the 250-Hz band, while the errors are greater than 15% in the rest of the bands. These are shown in Table P.17. 416 P.4 Octave-band prediction for the ASTM C423 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit as there no change in the modal compositions except the modal-width and, hence, the modal overlap as shown in Table P.18. Table P.18: Predicted octave-band modal compositions for ASTM C423 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 18 99 649 Modal density, 𝑛(𝑓) 0.41 1.11 3.65 Modal spacing, 𝛿𝑓 2.44 0.89 0.27 Modal width, 𝛿𝑤 4.15 2.95 2.31 Modal overlap, 𝑀 1.70 3.28 8.43 Modal overlap factor, 𝑀𝑓 1.70 3.32 8.43 Table P.19: Predicted octave-band SPL statistics for ASTM C423 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 146.56 153.74 164.54 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 143.74 155.13 165.45 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 141.29 154.27 164.30 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 148.58 154.57 164.06 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 144.46 152.94 164.08 Predicted average SPL, ?̅?𝑝 (dB) 145.62 154.19 164.52 Standard deviation, 𝜎𝐿𝑝 (dB) 2.89 0.84 0.57 417 Table P.20: Predicted octave-band RT statistics for ASTM C423 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 4.26 3.99 2.83 Predicted RT @ Rec#1: 𝑇60 (s) 4.76 2.95 2.42 Predicted RT @ Rec#2: 𝑇60 (s) 3.52 3.21 2.41 Predicted RT @ Rec#3: 𝑇60 (s) 9.20 3.25 2.55 Predicted RT @ Rec#4: 𝑇60 (s) 4.37 3.08 2.47 Predicted RT @ Rec#5: 𝑇60 (s) 13.69 4.65 3.62 Predicted average RT: ?̅?60 (s) 5.52 3.34 2.63 Standard deviation, 𝜎𝑇60 (s) 4.64 0.70 0.53 Decay rate (dB/s) 10.87 17.98 22.83 Spatial uniformity of SPL: The predicted SPL deviations are shown in Table P.19. The overall deviations are greater than in the ‘without absorber’ case. Spatial uniformity of RT: The predicted RT deviations are shown in Table P.20. However, the overall deviations are greater than in the ‘without absorber’ case. Table P.21: Predicted octave-band decay curve curvature statistics for ASTM C423 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 2.27 3.77 9.36 𝜅 @ Rec#2 23.75 4.99 5.49 𝜅 @ Rec#3 99.76 9.11 9.75 𝜅 @ Rec#4 1.45 4.40 5.22 𝜅 @ Rec#5 93.77 42.60 44.83 Average, 𝜅 44.20 12.97 14.93 Modal, 𝜅 1.11 3.53 5.34 418 Linearity of decay curves: The curvatures are greater than in the ‘without absorber’ case as shown in table P.21. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix Q.4. Table P.22: Predicted octave-band sample absorber absorption coefficient statistics for ASTM C423 oblique reverberation room with the longest z-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.14 0.39 0.59 Sabine error (%) 80.95 49.32 22.10 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.13 0.32 0.45 Eyring error (%) 82.27 57.94 41.21 Accuracy of the predicted sample absorption coefficients: The predicted absorption coefficients of an absorber sample are presented in Table P.22. It is noticeable that the predicted values deviate from the theoretical values to a great extent. 419 Appendix: Q Q.1 Third-octave-band prediction for the ASTM C423 reverberation room – Case #3: oblique-shaped with the longest z-dimension and without absorber Q.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 6.37 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.93 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 6.68 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 6.43 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.15 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 5.93 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 5.69 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.53 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 5.39 s420 Q.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, IN third-octave bands, at receiver #5 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 6.039 s0 1-40-2002040Time (s)45 50 550200040006000Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 7.781 s0 1-20020Time (s)60 65 70-500050010001500Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 7.304 s0 1-20020Time (s)70 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 5.733 s0 1-50050Time (s)90 100 11002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.433 s0 1-50050Time (s)120 130 140-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 6.047 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 4.972 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.575 s0 1-2000200400Time (s)240 260 280-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 5.834 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)421 Q.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 5.742 s0 1-40-2002040Time (s)45 50 55-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.141 s0 1-20020Time (s)60 65 70-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 5.849 s0 1-20020Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 5.807 s0 1-50050Time (s)90 100 110-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.877 s0 1-50050Time (s)120 130 140-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 5.589 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 4.965 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.174 s0 1-2000200Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 4.986 s0 1-400-2000200400Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)422 Q.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 5.875 s0 1-20020Time (s)45 50 550100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.133 s0 1-20020Time (s)60 65 70-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 6.144 s0 1-20020Time (s)70 80-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 5.611 s0 1-50050Time (s)90 100 110-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 6.119 s0 1-50050Time (s)120 130 140-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 5.635 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 5.348 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 5.22 s0 1-2000200Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 5.095 s0 1-5000500Time (s)300 350024x 104Frequency (Hz)Pressure (Pa)423 Q.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 6.108 s0 1-20020Time (s)45 50 5505001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.707 s0 1-20020Time (s)60 65 70-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 5.301 s0 1-20020Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.33 s0 1-40-2002040Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.486 s0 1-1000100Time (s)120 130 140-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.924 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 4.447 s0 1-2000200Time (s)180 200 22001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.092 s0 1-400-2000200400Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 5.717 s0 1-5000500Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)424 Q.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 5.698 s0 1-20020Time (s)45 50 55-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.539 s0 1-10010Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 6.442 s0 1-50050Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 5.611 s0 1-40-2002040Time (s)90 100 110-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.756 s0 1-1000100Time (s)120 130 140-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 5.233 s0 1-1000100Time (s)150 160 170-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 5.29 s0 1-200-1000100Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.48 s0 1-2000200Time (s)240 260 280-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 5.492 s0 1-5000500Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)425 Q.2 Third-octave-band prediction for the ASTM C423 reverberation room – Case #3: oblique-shaped with the longest z-dimension and with absorber Q.2.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 4.71 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 3.73 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 4.17 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 4.47 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.97 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 3.13 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 3.22 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.53 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.36 s426 Q.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 4.878 s0 1-50050Time (s)45 50 551000200030004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.006 s0 1-20020Time (s)60 65 70500100015002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 4.432 s0 1-20020Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 4.563 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 4.316 s0 1-50050Time (s)120 130 140-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 3.486 s0 1-1000100Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 3.423 s0 1-1000100Time (s)180 200 220-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.951 s0 1-400-2000200400Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 4.596 s0 1-2000200Time (s)300 350-50000500010000Frequency (Hz)Pressure (Pa)427 Q.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 4.755 s0 1-40-2002040Time (s)45 50 55-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 2.851 s0 1-20020Time (s)60 65 70-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 3.111 s0 1-20020Time (s)70 80-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 3.896 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.461 s0 1-40-2002040Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 2.526 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.34 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.221 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 2.809 s0 1-2000200400Time (s)300 350-10000-500005000Frequency (Hz)Pressure (Pa)428 Q.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 4.624 s0 1-20020Time (s)45 50 55010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 3.862 s0 1-20020Time (s)60 65 70-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 3.763 s0 1-10010Time (s)70 80-50005001000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 3.715 s0 1-50050Time (s)90 100 110020004000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 3.51 s0 1-50050Time (s)120 130 140-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 2.623 s0 1-1000100Time (s)150 160 170-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 2.751 s0 1-2000200Time (s)180 200 220-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.21 s0 1-2000200Time (s)240 260 280-5000050001000015000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 2.994 s0 1-400-2000200400Time (s)300 350-50000500010000Frequency (Hz)Pressure (Pa)429 Q.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 4.945 s0 1-20020Time (s)45 50 5505001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 3.047 s0 1-2002040Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 3.928 s0 1-50050Time (s)70 80-1000010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 3.121 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 3.642 s0 1-50050100Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.557 s0 1-1000100Time (s)150 160 170-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.691 s0 1-2000200Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.411 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 2.403 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)430 Q.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 4.752 s0 1-20020Time (s)45 50 55-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 4.158 s0 1-10010Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 4.322 s0 1-40-2002040Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 3.261 s0 1-2002040Time (s)90 100 110-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.195 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.667 s0 1-1000100Time (s)150 160 170-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 3.14 s0 1-1000100Time (s)180 200 220-10000-50000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.567 s0 1-1000100200Time (s)240 260 280-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 2.297 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)431 Q.3 Octave-band prediction for the ASTM C423 reverberation room – Case #3: oblique-shaped with the longest z-dimension and without absorber Q.3.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands Q.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in bands, at receiver #5 Q.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.76 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 6.12 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.49 s0 5 10 15-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 14.102 s0 1-50050Time (s)60 80-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.973 s0 1-1000100200Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.596 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.852 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.593 s0 1-1000100Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.034 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)432 Q.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 Q.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 Q.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 9.949 s0 1-50050Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.9 s0 1-1000100Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 5.105 s0 1-5000500Time (s)200 300024x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.283 s0 1-50050Time (s)60 80-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.742 s0 1-2000200Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.386 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.956 s0 1-50050Time (s)60 80-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.486 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.486 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)433 Q.4 Octave-band prediction for the ASTM C423 reverberation room – Case #3: oblique-shaped with the longest z-dimension and with absorber Q.4.1 Theoretical modal decay curves and the corresponding RTs, in octave bands Q.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 Q.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.26 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.99 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.83 s0 5 10 15-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 13.688 s0 1-50050Time (s)60 80020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 4.653 s0 1-1000100Time (s)100 150-200002000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 3.618 s0 1-5000500Time (s)200 300-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 4.366 s0 1-50050Time (s)60 80-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.081 s0 1-1000100Time (s)100 150-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.465 s0 1-400-2000200400Time (s)200 300-10000-500005000Frequency (Hz)Pressure (Pa)434 Q.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 Q.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 Q.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 9.198 s0 1-40-2002040Time (s)60 80010002000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 3.252 s0 1-1000100Time (s)100 150-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.55 s0 1-5000500Time (s)200 300-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 3.515 s0 1-50050Time (s)60 80-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 3.205 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.414 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 4.758 s0 1-50050Time (s)60 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 2.948 s0 1-200-1000100Time (s)100 150-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.415 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)435 Appendix: R R.1 Third-octave-band prediction for the ASTM C423 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table R.1, the corresponding statistically-based cut-off frequency limit is 108 Hz, which suggests that prediction can be done from the 125-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 125-Hz third-octave are 0.97, 0.1.05, 0.37 and 0.36, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 6.4 s for the RT, as shown in Table R.3, is 452 Hz. Table R.1: Predicted third-octave-band modal compositions for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 5 10 15 28 56 101 188 354 Modal density, 𝑛(𝑓) 0.25 0.36 0.53 0.65 0.97 1.51 2.20 3.24 4.85 Modal spacing, 𝛿𝑓 5.43 3.06 1.72 1.37 1.05 0.67 0.45 0.31 0.21 Modal width, 𝛿𝑤 0.34 0.32 0.34 0.35 0.37 0.39 0.41 0.42 0.43 Modal overlap, 𝑀 0.09 0.11 0.18 0.23 0.36 0.59 0.90 1.36 2.08 Modal overlap factor, 𝑀𝑓 0.06 0.10 0.20 0.26 0.36 0.58 0.91 1.37 2.08 Spatial uniformity of SPL: From Table R.2, the predicted standard deviations of SPL are less than 1.5 dB from the 160-Hz band, while the deviations are greater than 1.5 dB in the rest of the bands. Accuracy of the predicted SPLs: As presented in Table R.2, the errors associated with the predicted SPLs are always less than or around 1% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases. 436 Table R.2: Predicted third-octave-band SPL statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 132.8 138.8 143.4 154.0 158.2 158.7 162.8 166.6 168.5 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 140.2 141.5 149.3 154.4 155.5 159.2 164.6 166.3 169.8 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 143.0 138.3 144.8 153.0 157.9 160.4 162.2 166.1 169.0 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 141.1 142.7 144.7 155.9 159.8 161.0 164.2 166.9 171.6 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 139.2 142.7 143.0 153.4 155.3 160.7 164.4 166.4 170.7 Predicted average SPL, ?̅?𝑝 (dB) 140.3 141.2 145.8 154.3 157.7 160.1 163.7 166.5 170.1 Standard deviation, 𝜎𝐿𝑝 (dB) 4.0 2.2 2.6 1.1 1.9 1.0 1.1 0.3 1.3 Expected Sabine SPL, 𝐿𝑝 (dB) 142.3 143.6 147.8 156.2 159.5 161.6 165.1 167.6 170.9 Sabine error (%) 1.4 1.7 1.4 1.2 1.2 0.9 0.8 0.7 0.5 Expected Eyring SPL, 𝐿𝑝 (dB) 142.2 143.6 147.8 156.1 159.5 161.6 165.1 167.5 170.9 Eyring error (%) 1.37 1.68 1.38 1.19 1.13 0.91 0.80 0.65 0.47 Table R.3: Predicted third-octave-band RT statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 6.55 7.27 6.85 6.32 5.99 5.76 5.43 5.30 5.19 Predicted RT @ Rec#1: 𝑇60 (s) 14.82 6.63 5.79 5.79 4.87 5.25 4.80 4.35 5.98 Predicted RT @ Rec#2: 𝑇60 (s) 6.23 5.90 5.66 5.41 4.88 5.45 5.05 5.60 5.05 Predicted RT @ Rec#3: 𝑇60 (s) 6.63 6.10 5.94 5.28 5.47 5.03 4.99 4.82 4.69 Predicted RT @ Rec#4: 𝑇60 (s) 7.83 6.26 6.50 5.89 5.45 5.30 4.78 5.12 5.00 Predicted RT @ Rec#5: 𝑇60 (s) 7.41 5.91 5.79 5.61 5.28 4.68 5.31 5.12 4.69 Predicted average RT: ?̅?60 (s) 7.80 6.15 5.92 5.58 5.17 5.13 4.98 4.96 5.04 Standard deviation, 𝜎𝑇60 (s) 3.65 0.31 0.33 0.25 0.30 0.30 0.22 0.46 0.53 Expected Sabine RT: 𝑇60 (s) 4.64 4.64 4.64 4.64 4.64 4.64 4.64 4.64 4.64 Sabine error (%) 68.1 32.6 27.7 20.4 11.6 10.6 7.3 7.1 8.7 Expected Eyring RT, 𝑇60 (s) 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.59 4.59 Eyring error (%) 69.8 33.9 29.0 21.7 12.7 11.7 8.4 8.2 9.8 Decay rate (dB/s) 7.70 9.76 10.1 10.7 11.6 11.7 12.1 12.1 11.9 437 Table R.4: Predicted third-octave-band decay-curve curvature statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 65.8 7.8 23.6 16.1 22.3 10.4 15.3 10.4 4.4 𝜅 @ Rec#2 7.6 10.2 13.0 10.1 4.5 10.5 5.5 7.9 3.6 𝜅 @ Rec#3 5.9 9.2 3.8 5.6 5.7 5.1 18.6 7.0 5.6 𝜅 @ Rec#4 12.1 10.0 4.0 6.6 24.5 12.1 5.1 4.6 9.4 𝜅 @ Rec#5 4.7 6.7 7.6 11.4 5.8 6.3 5.9 10.0 6.3 Average, 𝜅 19.20 8.76 10.40 9.96 12.54 8.86 10.09 8.00 5.85 Modal, 𝜅 0.01 0.06 0.07 0.00 0.01 0.01 0.01 0.01 0.00 Spatial uniformity of RT: From Table R.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.5 s from the 63-Hz third-octave band. Accuracy of the predicted RTs: As presented in Table R.3, the predicted RTs show errors less than or around 10% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases, while the errors are greater than 15% in the rest of the bands. Linearity of temporal decay curves: From Table R.4, the predicted curvatures of the energy-decay curves vary within 5 to 13 from the 100-Hz band. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix S.1. Accuracy of the predicted source-power levels: As presented in Table R.5, the source-power-level errors are less than or around 10% from the 125-Hz third-octave band. Table R.5: Predicted third-octave-band source-power level statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 142.4 143.8 148.0 156.3 159.6 161.7 165.2 167.7 171.0 Sabine error (%) 38.0 23.1 20.4 15.9 9.8 9.0 6.5 6.2 7.5 Eyring error (%) 38.9 24.0 21.3 16.9 10.7 10.0 7.4 7.2 8.5 Predicted, ?̅?𝑤 (dB) (1 source) 139.4 140.8 144.9 153.3 156.6 158.7 162.2 164.7 168.0 438 Table R.6: Predicted third-octave-band room absorption statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 2.58 3.27 3.40 3.60 3.89 3.92 4.04 4.05 3.99 Sabine expected absorption, 𝐴 (m2) 4.34 4.34 4.34 4.34 4.34 4.34 4.34 4.34 4.34 Sabine error (%) 40.5 24.6 21.7 17.0 10.4 9.6 6.8 6.6 8.0 Eyring expected absorption, 𝐴 (m2) 4.38 4.38 4.38 4.38 4.38 4.38 4.38 4.38 4.38 Eyring error (%) 41.1 25.33 22.48 17.80 11.2 10.48 7.78 7.54 8.95 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 40.5 24.6 21.7 17.0 10.4 9.6 6.8 6.6 8.0 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 40.9 25.1 22.3 17.7 11.2 10.4 7.7 7.5 8.9 Accuracy of the predicted room absorptions: As presented in Table R.6, the errors are less than or around 10% from the 125-Hz third-octave band, and greater than 15% in the rest of the bands. The same is true for the room absorption coefficient prediction case, as the absorption coefficient predictions depend on the accuracy of the absorption predictions. 439 R.2 Third-octave-band prediction for the ASTM C423 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table R.7. The cut-off frequency limit will remain be the same as before. Table R.7: Predicted third-octave-band modal compositions for ASTM C423 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 5 10 15 28 56 101 188 354 Modal density, 𝑛(𝑓) 0.25 0.36 0.53 0.65 0.97 1.51 2.20 3.24 4.85 Modal spacing, 𝛿𝑓 5.43 3.06 1.72 1.37 1.05 0.67 0.45 0.31 0.21 Modal width, 𝛿𝑤 0.34 0.32 0.34 0.35 0.37 0.39 0.41 0.42 0.43 Modal overlap, 𝑀 0.09 0.11 0.18 0.23 0.36 0.59 0.90 1.36 2.08 Modal overlap factor, 𝑀𝑓 0.06 0.10 0.20 0.26 0.36 0.58 0.91 1.37 2.08 Table R.8: Predicted third-octave-band SPL statistics for ASTM C423 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 131.4 130.0 140.7 148.5 155.1 152.3 161.2 161.1 165.7 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 134.3 136.3 146.7 149.6 148.9 156.6 160.1 163.2 166.2 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 138.5 136.1 141.6 142.6 151.9 154.2 157.7 160.6 164.3 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 137.8 140.5 139.2 149.0 155.6 158.6 158.5 164.6 167.1 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 136.8 135.2 137.1 146.2 154.9 156.1 159.6 162.8 165.9 Predicted average SPL, ?̅?𝑝 (dB) 136.5 136.8 142.4 147.8 153.9 156.1 159.6 162.7 165.9 Standard deviation, 𝜎𝐿𝑝 (dB) 3.02 3.95 3.85 2.94 2.93 2.46 1.36 1.67 1.02 440 Table R.9: Predicted third-octave-band RT statistics for ASTM C423 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 6.55 7.27 6.85 6.32 5.99 5.76 5.43 5.30 5.19 Predicted RT @ Rec#1: 𝑇60 (s) 25.8 5.2 4.1 3.0 3.2 3.1 2.5 2.4 3.2 Predicted RT @ Rec#2: 𝑇60 (s) 3.86 3.93 3.52 2.75 2.81 2.75 2.65 2.20 2.63 Predicted RT @ Rec#3: 𝑇60 (s) 4.74 4.47 3.04 2.50 2.50 2.50 2.63 2.19 2.74 Predicted RT @ Rec#4: 𝑇60 (s) 6.49 5.32 3.17 2.38 3.08 2.79 2.66 2.00 2.08 Predicted RT @ Rec#5: 𝑇60 (s) 11.01 4.83 2.98 2.15 3.20 2.58 2.53 2.03 2.57 Predicted average RT: ?̅?60 (s) 6.63 4.69 3.31 2.52 2.93 2.73 2.60 2.16 2.59 Stnd. deviation, 𝜎𝑇60 (s) 9.99 0.57 0.45 0.34 0.30 0.24 0.06 0.17 0.41 Decay rate (dB/s) 9.0 12.8 18.1 23.8 20.5 22.0 23.1 27.8 23.1 Spatial uniformity of SPL: The standard deviations of the SPL are presented in Table R.8. It is noticeable that the deviations are greater than those obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT are presented in Table R.9. It is obvious that the deviations are greater than the ‘without absorber’ case in almost all third-octave bands. Table R.10: Predicted third-octave-band decay-curve curvature statistics for ASTM C423 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 555 204 5.0 12.3 48.5 7.9 11.7 10.8 53.9 𝜅 @ Rec#2 73.8 24.2 1.6 26.8 19.7 4.6 52.9 27.7 23.5 𝜅 @ Rec#3 118 7.9 8.6 10.9 20.6 20.2 6.2 6.1 20.3 𝜅 @ Rec#4 85.6 9.3 25.2 27.6 61.0 5.9 14.4 19.7 3.4 𝜅 @ Rec#5 308 148 10.9 53.5 24.4 21.9 21.8 2.8 14.2 Average, 𝜅 228 78.7 10.3 26.2 34.8 12.1 21.4 13.4 23.0 Modal, 𝜅 0.01 0.06 0.07 0.00 0.01 0.01 0.01 0.01 0.00 441 Table L.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ISO 354 oblique reverberation room with the longest x-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.06 0.13 0.35 0.58 0.39 0.46 0.49 0.70 0.50 Sabine error (%) 92.3 82.5 53.5 24.2 48.2 40.1 35.9 8.2 34.6 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.06 0.12 0.30 0.44 0.33 0.37 0.39 0.50 0.39 Eyring error (%) 92.5 83.6 60.8 42.4 57.2 51.9 49.3 33.9 48.5 Linearity of decay curves: The curvatures of the energy-decay curves as presented in Table L.10 reveal that the curvatures are greater than the values obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix S.2. Accuracy of the sample absorption coefficients: The predicted absorption coefficients of an absorber sample are presented in Table R.11. It is noticeable that the predicted values deviate from the theoretical values to a great extent. 442 R.3 Octave-band prediction for the ASTM C423 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table R.12, the statistically-based cut-off frequency limit is 68 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz band are 1.28, 0.78, 0.36 and 0.47, respectively. On the other hand, the pure-tone-based Schroeder limit, corresponding to a modal overlap factor of 3 and a band-averaged RT of 5.8 s, as shown in Table R.14, is 430 Hz. Table R.12: Predicted octave-band modal compositions for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 17 98 645 Modal density, 𝑛(𝑓) 0.39 1.10 3.62 Modal spacing, 𝛿𝑓 2.74 0.90 0.28 Modal width, 𝛿𝑤 0.33 0.38 0.42 Modal overlap, 𝑀 0.13 0.42 1.53 Modal overlap factor, 𝑀𝑓 0.12 0.42 1.54 Spatial uniformity of SPL: The predicted SPL standard deviations are less than or around 1.5 dB from the 125-Hz octave band, as shown in Table R.13. Accuracy of the predicted SPLs: As shown in Table R.13, the predicted SPLs show errors below or around 1.1% from the 125-H in all octave-bands. Spatial uniformity of RT: The predicted RT standard deviations are less than 0.3 s from the 125-Hz band, as shown in Table R.14. 443 Table R.13: Predicted octave-band SPL statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 140.92 158.02 167.46 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 146.55 157.16 168.36 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 143.68 158.59 167.48 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 144.00 160.21 169.55 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 142.82 158.21 168.87 Predicted average SPL, ?̅?𝑝 (dB) 143.98 158.56 168.42 Standard deviation, 𝜎𝐿𝑝 (dB) 2.09 1.13 0.91 Expected Sabine SPL, 𝐿𝑝 (dB) 145.53 160.36 169.49 Sabine error (%) 1.06 1.13 0.63 Expected Eyring SPL, 𝐿𝑝 (dB) 145.49 160.32 169.45 Eyring error (%) 1.03 1.10 0.61 Table R.14: Predicted octave-band RT statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.00 5.96 5.27 Predicted RT @ Rec#1: 𝑇60 (s) 12.69 5.30 5.39 Predicted RT @ Rec#2: 𝑇60 (s) 5.97 5.04 5.13 Predicted RT @ Rec#3: 𝑇60 (s) 7.23 5.18 4.76 Predicted RT @ Rec#4: 𝑇60 (s) 8.82 5.61 5.11 Predicted RT @ Rec#5: 𝑇60 (s) 6.19 5.03 4.81 Predicted average RT: ?̅?60 (s) 7.58 5.22 5.03 Standard deviation, 𝜎𝑇60 (s) 2.84 0.24 0.26 Expected Sabine RT: 𝑇60 (s) 4.64 4.64 4.64 Sabine error (%) 63.47 12.62 8.46 Expected Eyring RT, 𝑇60 (s) 4.59 4.59 4.59 Eyring error (%) 65.13 13.76 9.56 Decay rate (dB/s) 7.92 11.49 11.93 444 Table R.15: Predicted octave-band decay-curve curvature statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 25.05 5.72 1.92 𝜅 @ Rec#2 6.57 1.88 1.29 𝜅 @ Rec#3 3.95 4.99 2.93 𝜅 @ Rec#4 11.51 7.58 4.43 𝜅 @ Rec#5 3.19 5.66 3.93 Average, 𝜅 10.05 5.17 2.90 Modal, 𝜅 0.07 0.02 0.01 Accuracy of the predicted RTs: As shown in Table R.14, the predicted RTs show errors below or around 13% from the 125-Hz band, while the errors are greater than 15% in the rest of the bands. Degree of linearity of decay curves: The curvature values are within 2 to 6 from the 125-Hz octave-band, as shown in Table R.15. The modal-decay curves and the energy-decay curves predicted in octave-bands are presented in Appendix S.3. Accuracy of the predicted source-power levels: The predicted source-power levels show errors below or around 10% from the 125-Hz band, as shown in Table R.16. Table R.16: Predicted octave-band source-power level statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 145.64 160.48 169.61 Sabine error (%) 36.38 10.56 7.33 Eyring error (%) 37.33 11.50 8.28 Predicted, ?̅?𝑤 (dB) (1 source) 142.63 157.47 166.60 445 Table R.17: Predicted octave-band room absorption statistics for ASTM C423 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 2.65 3.85 4.00 Sabine expected absorption, 𝐴 (m2) 4.34 4.34 4.34 Sabine error (%) 38.83 11.21 7.80 Eyring expected absorption, 𝐴 (m2) 4.38 4.38 4.38 Eyring error (%) 39.44 12.10 8.72 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Sabine error (%) 38.83 11.21 7.80 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 39.20 11.99 8.64 Accuracy of the predicted room absorptions: The predicted room absorptions, as well as the absorption coefficients, show errors below or around 10% from the 125-Hz octave band for both Sabine and Eyring prediction cases as shown in Table R.17. 446 R.4 Octave-band prediction for the ASTM C423 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit as there no change in the modal compositions except the modal-width and, hence, the modal-overlap as shown in Table R.18. Table R.18: Predicted octave-band modal compositions for ASTM C423 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 17 98 645 Modal density, 𝑛(𝑓) 0.39 1.10 3.62 Modal spacing, 𝛿𝑓 2.74 0.90 0.28 Modal width, 𝛿𝑤 0.33 0.38 0.42 Modal overlap, 𝑀 0.13 0.42 1.53 Modal overlap factor, 𝑀𝑓 0.12 0.42 1.54 Table R.19: Predicted octave-band SPL statistics for ASTM C423 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 137.41 153.58 164.09 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 143.38 154.05 164.79 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 140.20 152.43 162.61 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 140.14 156.69 165.56 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 137.25 154.78 164.45 Predicted average SPL, ?̅?𝑝 (dB) 140.29 154.55 164.41 Standard deviation, 𝜎𝐿𝑝 (dB) 2.60 1.60 1.10 447 Table R.20: Predicted octave-band RT statistics for ASTM C423 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.00 5.96 5.27 Predicted RT @ Rec#1: 𝑇60 (s) 9.28 3.21 2.98 Predicted RT @ Rec#2: 𝑇60 (s) 3.98 2.86 2.56 Predicted RT @ Rec#3: 𝑇60 (s) 4.53 2.51 2.60 Predicted RT @ Rec#4: 𝑇60 (s) 5.11 2.87 2.12 Predicted RT @ Rec#5: 𝑇60 (s) 4.87 2.92 2.37 Predicted average RT: ?̅?60 (s) 5.10 2.85 2.50 Standard deviation, 𝜎𝑇60 (s) 2.19 0.25 0.32 Decay rate (dB/s) 11.77 21.02 24.04 Spatial uniformity of SPL: The predicted SPL deviations are greater than in the ‘without absorber’ case as shown in Table R.19. Spatial uniformity of RT: The predicted RT deviations are greater than in the ‘without absorber’ case as shown in Table R.20. Linearity of temporal decay curves: The curvatures are greater than in the ‘without absorber’ case as shown in Table R.21. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix S.4. Table R.21: Predicted octave-band decay-curve curvature statistics for ASTM C423 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 71.24 16.58 35.23 𝜅 @ Rec#2 10.18 2.61 25.66 𝜅 @ Rec#3 31.35 5.87 5.70 𝜅 @ Rec#4 30.46 11.57 6.58 𝜅 @ Rec#5 95.55 14.64 3.72 Average, 𝜅 47.75 10.25 15.38 Modal, 𝜅 0.07 0.02 0.01 448 Table R.22: Predicted octave-band sample absorber absorption coefficient statistics for ASTM C423 oblique reverberation room with the longest x-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.12 0.30 0.38 Sabine error (%) 84.25 61.02 50.40 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.11 0.26 0.31 Eyring error (%) 85.16 66.26 58.68 Accuracy of the predicted sample absorption coefficients: The predicted absorption coefficients of an absorber sample are presented in Table R.22. It is noticeable that the predicted values deviate from the theoretical values to a great extent. 449 Appendix: S S.1 Third-octave-band prediction for the ASTM C423 reverberation room – Case #4: oblique-shaped with the longest x-dimension and without absorber S.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 6.55 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 7.27 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 6.85 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 6.32 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 5.99 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 5.76 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 5.43 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.3 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 5.19 s450 S.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 7.414 s0 1-10010Time (s)45 50 55-500050010001500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 5.906 s0 1-10010Time (s)60 65 70-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 5.793 s0 1-10010Time (s)70 80010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 5.608 s0 1-40-20020Time (s)90 100 110-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.277 s0 1-50050Time (s)120 130 140-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 4.681 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 5.311 s0 1-2000200Time (s)180 200 220-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.12 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 4.687 s0 1-400-2000200400Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)451 S.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 7.83 s0 1-2002040Time (s)45 50 55-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.263 s0 1-20-10010Time (s)60 65 70-100001000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 6.499 s0 1-20020Time (s)70 80-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 5.888 s0 1-50050Time (s)90 100 110050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.445 s0 1-1000100Time (s)120 130 140-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 5.295 s0 1-200-1000100Time (s)150 160 170-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 4.78 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.117 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 5 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)452 S.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 6.632 s0 1-10010Time (s)45 50 55-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.099 s0 1-10010Time (s)60 65 70-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.941 s0 1-20020Time (s)70 80-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 5.278 s0 1-40-2002040Time (s)90 100 110-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.467 s0 1-1000100Time (s)120 130 140-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 5.032 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 4.989 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.815 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 4.687 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)453 S.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 6.231 s0 1-10010Time (s)45 50 55-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.897 s0 1-10010Time (s)60 65 70-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 5.658 s0 1-2002040Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.407 s0 1-50050Time (s)90 100 110-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 4.882 s0 1-50050Time (s)120 130 140-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.452 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 5.048 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.596 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 5.054 s0 1-5000500Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)454 S.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 5 10 15-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 14.816 s0 1-1001020Time (s)45 50 55-600-400-200Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 6.633 s0 1-10-505Time (s)60 65 70-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 5.786 s0 1-20020Time (s)70 80-500050010001500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 5.788 s0 1-50050Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 4.867 s0 1-50050Time (s)120 130 140-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 5.252 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 4.798 s0 1-200-1000100Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.349 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 5.984 s0 1-400-2000200400Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)455 S.2 Third-octave-band prediction for the ASTM C423 reverberation room – Case #4: oblique-shaped with the longest x-dimension and with absorber S.2.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 2.69 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.67 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 3.13 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 3.14 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.55 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 3.11 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.74 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.52 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.22 s456 S.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 2 4 6 8 10 12-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 11.014 s0 1-20020Time (s)45 50 55-400-2000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 4.834 s0 1-20020Time (s)60 65 70-400-2000200400Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 2.977 s0 1-50510Time (s)70 80-2000200400600800Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 2.15 s0 1-20020Time (s)90 100 110-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 3.195 s0 1-50050Time (s)120 130 140-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 2.578 s0 1-1000100Time (s)150 160 170-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.531 s0 1-1000100Time (s)180 200 220-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.028 s0 1-2000200Time (s)240 260 280-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 2.565 s0 1-400-2000200400Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)457 S.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 6.491 s0 1-40-2002040Time (s)45 50 55-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.318 s0 1-20020Time (s)60 65 70-50005001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 3.168 s0 1-1001020Time (s)70 80-1000-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 2.383 s0 1-50050Time (s)90 100 110020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.083 s0 1-1000100Time (s)120 130 140-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 2.791 s0 1-1000100Time (s)150 160 170-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.658 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 1.999 s0 1-2000200Time (s)240 260 280-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 2.076 s0 1-5000500Time (s)300 350-10000-500005000Frequency (Hz)Pressure (Pa)458 S.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 4.74 s0 1-505Time (s)45 50 55-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 4.473 s0 1-505Time (s)60 65 70-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 3.041 s0 1-10010Time (s)70 80-500050010001500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 2.496 s0 1-20020Time (s)90 100 110-1000-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.496 s0 1-50050Time (s)120 130 140-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 2.496 s0 1-1000100Time (s)150 160 170-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 2.626 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.187 s0 1-1000100200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 2.741 s0 1-2000200Time (s)300 350-50000500010000Frequency (Hz)Pressure (Pa)459 S.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 3.859 s0 1-10010Time (s)45 50 55-400-2000200400600800Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 3.932 s0 1-505Time (s)60 65 70-5000500Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 3.522 s0 1-20020Time (s)70 80-10000100020003000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 2.746 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.809 s0 1-40-2002040Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.745 s0 1-1000100Time (s)150 160 170-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.647 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.196 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 2.633 s0 1-2000200400Time (s)300 350-15000-10000-500005000Frequency (Hz)Pressure (Pa)460 S.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 5 10 15 20 25-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 25.829 s0 1-20020Time (s)45 50 55-200-150-100-50Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.188 s0 1-10010Time (s)60 65 70-2000200Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 4.059 s0 1-20020Time (s)70 80050010001500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 3.021 s0 1-40-2002040Time (s)90 100 110-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.182 s0 1-50050Time (s)120 130 140-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 3.108 s0 1-50050Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.547 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.417 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 3.209 s0 1-2000200Time (s)300 350-10000-500005000Frequency (Hz)Pressure (Pa)461 S.3 Octave-band prediction for the ASTM C423 reverberation room – Case #4: oblique-shaped with the longest x-dimension and without absorber S.3.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands S.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 S.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 7 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 5.96 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.27 s0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.189 s0 1-20020Time (s)60 80-1000010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.034 s0 1-1000100200Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.812 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 8.824 s0 1-50050Time (s)60 80-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.606 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.108 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)462 S.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 S.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 S.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 7.232 s0 1-20020Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.177 s0 1-1000100200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.76 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.966 s0 1-40-2002040Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.039 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 5.127 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8 10 12-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 12.689 s0 1-40-20020Time (s)60 80-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.297 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.392 s0 1-5000500Time (s)200 300-2-101x 104Frequency (Hz)Pressure (Pa)463 S.4 Octave-band prediction for the ASTM C423 reverberation room – Case #4: oblique-shaped with the longest x-dimension and with absorber S.4.1 Theoretical modal decay curves and the corresponding RTs, in octave bands S.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 S.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.27 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.29 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.49 s0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 4.87 s0 1-20-10010Time (s)60 80-400-2000200400600800Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.918 s0 1-1000100Time (s)100 150-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.373 s0 1-400-2000200400Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.106 s0 1-40-2002040Time (s)60 80-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.867 s0 1-2000200Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.124 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)464 S.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 S.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 S.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 4.533 s0 1-10010Time (s)60 80-500050010001500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.51 s0 1-1000100Time (s)100 150-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.597 s0 1-400-2000200400Time (s)200 300-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 3.975 s0 1-40-2002040Time (s)60 80-10000100020003000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.858 s0 1-1000100200Time (s)100 150-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.561 s0 1-5000500Time (s)200 300-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 9.276 s0 1-20020Time (s)60 80050010001500Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.21 s0 1-1000100Time (s)100 150-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.977 s0 1-5000500Time (s)200 300-10000-500005000Frequency (Hz)Pressure (Pa)465 Appendix: T T.1 Third-octave-band prediction for the ASTM E90 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table T.1, the corresponding statistically-based cut-off frequency limit is 125 Hz, which suggests that prediction can be done from the 160-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 160-Hz third-octave are 1, 0.95, 0.43 and 0.46, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 5.1 s for the RT, as shown in Table T.3, is 498 Hz. Table T.1: Predicted third-octave-band modal compositions for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 2 5 6 11 20 37 69 123 235 Modal density, 𝑛(𝑓) 0.17 0.36 0.32 0.48 0.69 1.00 1.50 2.12 3.22 Modal spacing, 𝛿𝑓 1.36 2.98 3.28 1.95 1.40 0.95 0.67 0.47 0.31 Modal width, 𝛿𝑤 0.38 0.38 0.39 0.41 0.41 0.43 0.46 0.47 0.48 Modal overlap, 𝑀 0.06 0.14 0.12 0.19 0.29 0.43 0.70 0.99 1.55 Modal overlap factor, 𝑀𝑓 0.28 0.13 0.12 0.21 0.29 0.46 0.69 1.00 1.55 Spatial uniformity of SPL: From Table T.2, the predicted standard deviations of SPL, a measure of spatial variation, reveal the deviations are less than 1.5 dB from the 160-Hz third-octave band. The deviations are greater than 2.5 dB in the rest of the bands. 466 Table T.2: Predicted third-octave-band SPL statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 152.7 149.0 139.3 158.7 150.2 158.7 166.4 169.6 172.0 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 153.1 148.0 145.4 161.2 151.4 157.7 165.5 165.7 172.0 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 146.3 155.6 145.8 156.1 156.2 160.1 162.6 167.4 171.0 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 148.6 156.8 147.5 156.0 158.8 159.0 163.7 169.5 169.9 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 152.7 146.8 141.3 160.3 151.0 158.1 164.3 168.7 171.4 Predicted average SPL, ?̅?𝑝 (dB) 151.4 153.2 144.8 159.0 154.9 158.8 164.7 168.4 171.3 Standard deviation, 𝜎𝐿𝑝 (dB) 3.2 5.1 3.6 2.4 4.1 0.9 1.5 1.6 0.9 Expected Sabine SPL, 𝐿𝑝 (dB) 154.5 155.5 146.4 161.0 156.3 160.2 166.3 169.8 171.9 Sabine error (%) 2.0 1.5 1.1 1.2 0.9 0.9 1.0 0.8 0.3 Expected Eyring SPL, 𝐿𝑝 (dB) 154.5 155.4 146.4 160.9 156.2 160.2 166.3 169.8 171.9 Eyring error (%) 2.00 1.47 1.07 1.22 0.85 0.88 0.95 0.81 0.32 Table T.3: Predicted third-octave-band RT statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 5.85 6.90 5.95 5.61 5.49 5.24 4.91 4.84 4.67 Predicted RT @ Rec#1: 𝑇60 (s) 5.25 5.52 8.90 6.04 6.25 4.66 4.11 4.08 5.15 Predicted RT @ Rec#2: 𝑇60 (s) 5.33 5.70 5.86 4.32 5.14 5.21 4.46 4.39 4.36 Predicted RT @ Rec#3: 𝑇60 (s) 6.62 6.14 5.04 5.30 4.99 4.49 4.05 4.08 4.94 Predicted RT @ Rec#4: 𝑇60 (s) 5.84 5.79 5.21 4.73 5.15 4.37 3.96 3.99 4.75 Predicted RT @ Rec#5: 𝑇60 (s) 5.33 6.45 6.94 4.96 4.98 4.99 4.65 4.35 4.73 Predicted average RT: ?̅?60 (s) 5.63 5.90 6.11 5.01 5.26 4.72 4.23 4.17 4.77 Standard deviation, 𝜎𝑇60 (s) 0.58 0.37 1.62 0.65 0.54 0.35 0.29 0.18 0.29 Expected Sabine RT: 𝑇60 (s) 4.13 4.13 4.13 4.13 4.13 4.13 4.13 4.13 4.13 Sabine error (%) 36.4 42.9 48.1 21.2 27.4 14.4 2.5 1.0 15.6 Expected Eyring RT, 𝑇60 (s) 4.09 4.09 4.09 4.09 4.09 4.09 4.09 4.09 4.09 Eyring error (%) 37.7 44.3 49.6 22.5 28.7 15.5 3.5 2.0 16.8 Decay rate (dB/s) 10.7 10.2 9.8 12.0 11.4 12.7 14.2 14.4 12.6 467 Accuracy of the predicted SPLs: The accuracy of the predicted steady-state SPLs is checked against the Sabine and Eyring versions of the diffuse-field formulae as presented in Table T.2. The errors associated with the predicted SPL are always less than 1% from the 125-Hz band for both Sabine and Eyring prediction cases. Spatial uniformity of RT: From Table T.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.5 s from the 125-Hz third-octave band. Accuracy of the predicted RTs: As presented in Table T.3, the predicted RTs show errors less than, or around, 15% from the 160-Hz third-octave band for both Sabine and Eyring prediction cases, while the errors are greater than 15% in the rest of the bands. Linearity of temporal decay curves: From Table T.4, the predicted curvatures of the energy-decay curves vary within 4 to 13 in al third-octave bands. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix U.1. Accuracy of the predicted source-power levels: As presented in Table T.5, the source-power-level errors are less than, or around, 15% from the 160-Hz third-octave band. Table T.4: Predicted third-octave-band decay curve curvature statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 4.4 2.9 39.8 25.5 11.8 9.4 11.5 4.4 5.0 𝜅 @ Rec#2 3.3 3.0 6.8 5.5 2.6 20.9 5.0 9.2 4.5 𝜅 @ Rec#3 3.9 4.6 5.0 7.0 6.7 9.9 7.6 8.0 1.7 𝜅 @ Rec#4 6.3 2.6 2.2 1.7 3.4 11.5 9.2 8.5 8.4 𝜅 @ Rec#5 5.1 0.8 10.8 4.2 3.7 5.7 7.5 18.9 5.1 Average, 𝜅 4.6 2.8 12.9 8.8 5.7 11.5 8.2 9.8 4.9 Modal, 𝜅 0.00 0.76 0.04 0.04 0.02 0.03 0.04 0.03 0.01 468 Table T.5: Predicted third-octave-band source-power level statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 153.3 154.3 145.2 159.8 155.1 159.0 165.1 168.6 170.7 Sabine error (%) 28.1 31.6 34.2 18.5 22.7 13.3 2.7 1.3 14.2 Eyring error (%) 29.2 32.7 35.2 19.5 23.7 14.3 3.7 2.3 15.3 Predicted, ?̅?𝑤 (dB) (1 source) 150.3 151.3 142.2 156.8 152.1 156.0 162.1 165.6 167.7 Accuracy of the predicted room absorptions: Table T.6 shows that the predicted room-averaged absorption errors are less than 15% from the 160-Hz third-octave band, and greater than 20% in the rest of the bands. The same is true for the room-averaged absorption-coefficient prediction case, as the absorption-coefficient predictions depend on the accuracy of the absorption predictions. The room-averaged absorption-coefficient prediction statistics are also presented in Table T.6. Table T.6: Predicted third-octave-band room absorption statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 2.31 2.20 2.12 2.59 2.47 2.75 3.07 3.11 2.72 Sabine expected absorption, 𝐴 (m2) 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 Sabine error (%) 26.7 30.0 32.5 17.5 21.5 12.6 2.4 1.0 13.5 Eyring expected absorption, 𝐴 (m2) 3.18 3.18 3.18 3.18 3.18 3.18 3.18 3.18 3.18 Eyring error (%) 27.4 30.7 33.1 18.4 22.3 13.5 3.4 2.0 14.4 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 26.7 30.0 32.5 17.5 21.5 12.6 2.4 1.0 13.5 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 27.2 30.5 32.9 18.2 22.1 13.3 3.4 2.0 14.2 469 T.2 Third-octave-band prediction for the ASTM E90 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table T.7. The cut-off frequency limit will remain be the same as before. Table T.7: Predicted third-octave-band modal compositions for ASTM E90 rectangular reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 2 5 7 12 22 36 70 125 238 Modal density, 𝑛(𝑓) 0.17 0.36 0.37 0.52 0.76 0.97 1.52 2.16 3.26 Modal spacing, 𝛿𝑓 3.83 2.94 2.77 1.66 1.27 0.97 0.65 0.46 0.31 Modal width, 𝛿𝑤 1.51 1.16 5.66 3.87 4.39 3.99 3.02 3.82 3.15 Modal overlap, 𝑀 0.25 0.41 2.09 2.02 3.33 3.88 4.60 8.24 10.3 Modal overlap factor, 𝑀𝑓 0.40 0.39 2.05 2.34 3.45 4.11 4.63 8.29 10.3 Table T.8: Predicted third-octave-band SPL statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 152.2 144.3 137.3 156.2 147.5 158.0 159.8 166.1 167.5 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 152.7 143.5 140.5 158.4 150.4 154.9 159.6 162.1 165.1 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 144.0 148.3 143.1 152.5 151.2 152.1 159.2 162.7 163.9 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 147.5 149.8 141.5 152.6 151.9 156.8 159.9 162.6 165.5 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 152.2 140.1 135.8 157.2 149.0 158.3 156.3 164.8 167.0 Predicted average SPL, ?̅?𝑝 (dB) 150.8 146.5 140.4 156.0 150.3 156.6 159.1 163.9 166.0 Standard deviation, 𝜎𝐿𝑝 (dB) 4.01 4.19 3.14 2.77 1.78 2.65 1.53 1.74 1.47 470 Table T.9: Predicted third-octave-band RT statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 4.22 2.12 2.78 2.84 2.65 2.59 2.72 2.09 1.91 Predicted RT @ Rec#1: 𝑇60 (s) 4.22 2.74 7.35 2.89 3.96 2.70 2.18 2.07 2.76 Predicted RT @ Rec#2: 𝑇60 (s) 4.24 2.45 4.04 2.79 2.65 2.37 2.66 1.90 2.16 Predicted RT @ Rec#3: 𝑇60 (s) 4.24 2.25 2.40 3.04 2.26 2.20 1.98 1.88 3.01 Predicted RT @ Rec#4: 𝑇60 (s) 4.19 2.35 2.40 2.90 2.37 2.37 1.80 1.78 2.13 Predicted RT @ Rec#5: 𝑇60 (s) 4.22 3.90 5.56 2.82 2.83 2.47 2.36 1.91 2.51 Predicted average RT: ?̅?60 (s) 4.22 2.63 3.58 2.89 2.71 2.41 2.15 1.90 2.47 Standard deviation, 𝜎𝑇60 (s) 0.02 0.68 2.30 0.10 0.69 0.18 0.34 0.11 0.38 Decay rate (dB/s) 14.2 22.8 16.8 20.8 22.2 24.9 27.8 31.5 24.3 Spatial uniformity of SPL: The standard deviations of the SPL for this ‘with absorber’ case, as a measure of the spatial variation are presented in Table T.8. Overall, the deviations are greater than the values obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT for this ‘with absorber’ case, as a measure of the spatial variation are presented in Table T.9. The overall deviations are greater than in the previous ‘without absorber’ case in almost all third-octave bands. Table T.10: Predicted third-octave-band decay-curve curvature statistics for ASTM E90 rectangular reverberation room with the longest dimension in the x-direction and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 0.28 4.1 208.2 20.6 51.3 17.5 14.3 34.9 68.9 𝜅 @ Rec#2 0.03 5.8 38.8 7.6 4.9 5.2 14.0 4.5 28.0 𝜅 @ Rec#3 24.5 5.3 23.7 6.9 6.1 8.3 6.4 6.9 41.0 𝜅 @ Rec#4 4.26 10.7 22.0 4.9 10.7 16.9 20.3 11.6 2.4 𝜅 @ Rec#5 0.07 54.5 42.1 14.3 8.9 12.0 20.3 19.0 63.4 Average, 𝜅 5.84 16.1 67.0 10.9 16.4 12.0 15.0 15.4 40.7 Modal, 𝜅 0.00 0.03 0.28 0.62 1.23 1.68 7.05 2.19 1.98 471 Linearity of decay curves: The curvatures of the energy-decay curves as presented in Table T.10 reveal that the curvatures are greater than those obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix U.2. Accuracy of the sample absorption coefficients: The accuracy of the absorption coefficients of an absorber sample predicted using the Sabine and Eyring versions of the diffuse-field formulae is checked against the theoretical random absorption-coefficient values predicted using Eq. (3.17); these are presented in Table T.11. It is noticeable that predicted values deviate from the theoretical values to a great extent. The Sabine predictions show errors around 50% from the 160-Hz third-octave band, while the Eyring predictions show errors around 55% in the same bands. Table T.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ASTM E90 rectangular reverberation room with the longest x-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.12 0.42 0.23 0.29 0.36 0.40 0.45 0.57 0.39 Sabine error (%) 84.7 45.1 69.9 61.8 53.2 47.1 40.6 25.5 49.0 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.11 0.34 0.20 0.25 0.30 0.33 0.36 0.43 0.32 Eyring error (%) 85.5 55.1 73.1 66.8 60.7 56.4 52.2 43.1 57.7 472 T.3 Octave-band prediction for the ASTM E90 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table T.12, the statistically-based cut-off frequency limit is 78 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz band are 0.76, 1.32, 0.42 and 0.32, respectively. On the other hand, the pure-tone-based Schroeder limit, corresponding to a modal overlap factor of 3 and a band-averaged RT of 5.5 s, as shown in Table T.14, is 517 Hz. Spatial uniformity of SPL: The predicted SPL standard deviations are smaller than 1.5 dB from the 125-Hz band, as shown in Table T.13. Table T.12: Predicted octave-band modal compositions for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 12 68 428 Modal density, 𝑛(𝑓) 0.27 0.76 2.40 Modal spacing, 𝛿𝑓 3.44 1.32 0.42 Modal width, 𝛿𝑤 0.39 0.42 0.47 Modal overlap, 𝑀 0.11 0.32 1.14 Modal overlap factor, 𝑀𝑓 0.11 0.32 1.14 Accuracy of the predicted SPLs: The accuracy of the predicted steady-state SPLs is checked against the Sabine and Eyring versions of the diffuse-field as shown in Table T.13. The predicted results show errors below 1% from the 125-Hz octave band, while the error is around 1.3% in the 63-Hz band for both the Sabine and Eyring cases. 473 Table T.13: Predicted octave-band SPL statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 150.53 158.12 170.76 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 150.89 159.16 169.79 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 152.50 158.79 169.06 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 153.92 158.97 169.23 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 150.04 158.77 169.82 Predicted average SPL, ?̅?𝑝 (dB) 151.82 158.78 169.77 Standard deviation, 𝜎𝐿𝑝 (dB) 1.63 0.39 0.67 Expected Sabine SPL, 𝐿𝑝 (dB) 153.78 160.40 170.90 Sabine error (%) 1.28 1.01 0.66 Expected Eyring SPL, 𝐿𝑝 (dB) 153.74 160.36 170.86 Eyring error (%) 1.25 0.99 0.63 Table T.14: Predicted octave-band RT statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 6.49 5.46 4.77 Predicted RT @ Rec#1: 𝑇60 (s) 8.09 5.61 4.53 Predicted RT @ Rec#2: 𝑇60 (s) 5.53 4.73 4.39 Predicted RT @ Rec#3: 𝑇60 (s) 6.22 4.79 4.59 Predicted RT @ Rec#4: 𝑇60 (s) 5.89 4.78 4.23 Predicted RT @ Rec#5: 𝑇60 (s) 6.91 5.06 4.56 Predicted average RT: ?̅?60 (s) 6.41 4.97 4.46 Standard deviation, 𝜎𝑇60 (s) 1.02 0.37 0.15 Expected Sabine RT: 𝑇60 (s) 4.13 4.13 4.13 Sabine error (%) 55.28 20.39 7.93 Expected Eyring RT, 𝑇60 (s) 4.09 4.09 4.09 Eyring error (%) 56.86 21.61 9.02 Decay rate (dB/s) 9.36 12.07 13.46 474 Table T.15: Predicted octave-band decay-curve curvature statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 14.98 3.95 1.82 𝜅 @ Rec#2 2.12 2.62 3.19 𝜅 @ Rec#3 4.99 4.19 0.87 𝜅 @ Rec#4 2.62 3.73 7.77 𝜅 @ Rec#5 3.04 2.42 5.25 Average, 𝜅 5.55 3.38 3.78 Modal, 𝜅 0.65 0.07 0.03 Spatial uniformity of RT: The predicted RT standard deviations are less than 0.5 s from the 125-Hz band. These are shown in Table T.14. Accuracy of the predicted RTs: The accuracy of the predicted RTs is checked against the Sabine and Eyring versions of the diffuse-field formulae as shown in Table T.14. The predicted results show errors below 10% from the 250-Hz band, while the errors are greater than 20% in the rest of the bands. Linearity of decay curves: The curvatures are within 3 to 6 in all octave bands as shown in Table T.15. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix U.3. Table T.16: Predicted octave-band source-power level statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 152.59 159.21 169.70 Sabine error (%) 37.47 17.84 7.75 Eyring error (%) 38.53 18.90 8.82 Predicted, ?̅?𝑤 (dB) (1 source) 149.58 156.20 166.69 475 Table T.17: Predicted octave-band room absorption statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 2.03 2.61 2.91 Sabine expected absorption, 𝐴 (m2) 3.15 3.15 3.15 Sabine error (%) 35.60 16.93 7.35 Eyring expected absorption, 𝐴 (m2) 3.18 3.18 3.18 Eyring error (%) 36.25 17.77 8.28 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Sabine error (%) 35.60 16.93 7.35 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 36.01 17.62 8.20 Accuracy of the predicted source-power levels: The predicted source-power levels show errors below 10% from the 250-Hz band; the errors are greater than 15% in the rest of the bands as shown in Table T.16. Accuracy of the predicted room absorptions: The predicted room-averaged absorptions, as well as the absorption coefficients, show errors below 10% from the 250-Hz band, while errors are greater than 15% in the rest of the bands. These are shown in Table T.17. 476 T.4 Octave-band prediction for the ASTM E90 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit. There is also no change in the modal compositions except the modal-width and, hence, the modal-overlap as shown in Table T.18. The same reasoning is applicable here what is discussed in the ‘third-octave-band prediction section with absorber’ case. Table T.18: Predicted octave-band modal compositions for ASTM E90 rectangular reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 12 71 434 Modal density, 𝑛(𝑓) 0.27 0.80 2.44 Modal spacing, 𝛿𝑓 3.45 1.26 0.41 Modal width, 𝛿𝑤 3.67 4.06 3.32 Modal overlap, 𝑀 1.00 3.24 8.09 Modal overlap factor, 𝑀𝑓 1.07 3.23 8.11 Spatial uniformity of SPL: The predicted SPL deviations are shown in Table T.19. The overall deviations are higher than in the ‘without absorber’ case. Spatial uniformity of RT: The predicted RT deviations are shown in Table T.20. The overall deviations are greater than in the ‘without absorber’ case. Linearity of decay curves: The curvatures are shown in Table T.21. The overall curvatures are greater than in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix U.4. 477 Table T.19: Predicted octave-band SPL statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 149.12 156.53 166.41 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 149.49 156.47 163.77 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 146.43 152.77 163.29 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 148.28 155.23 164.17 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 148.71 157.13 165.42 Predicted average SPL, ?̅?𝑝 (dB) 148.53 155.87 164.77 Standard deviation, 𝜎𝐿𝑝 (dB) 1.20 1.76 1.29 Accuracy of the sample absorption coefficients: The accuracy of the predicted absorption coefficients of an absorber sample is checked against the theoretical results as shown in Table T.22. The Sabine predictions show errors around 35% from the 250-Hz band, while the Eyring predictions yield errors around 50% from the same band. Table T.20: Predicted octave-band RT statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 3.75 2.70 2.32 Predicted RT @ Rec#1: 𝑇60 (s) 5.79 3.00 2.23 Predicted RT @ Rec#2: 𝑇60 (s) 4.15 2.70 2.13 Predicted RT @ Rec#3: 𝑇60 (s) 2.54 2.67 2.13 Predicted RT @ Rec#4: 𝑇60 (s) 2.90 2.62 1.86 Predicted RT @ Rec#5: 𝑇60 (s) 4.70 2.70 2.21 Predicted average RT: ?̅?60 (s) 3.66 2.73 2.10 Stnd. deviation, 𝜎𝑇60 (s) 1.39 0.15 0.15 Decay rate (dB/s) 16.38 21.97 28.53 478 Table T.21: Predicted octave-band decay-curve curvature statistics for ASTM E90 rectangular reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 16.28 16.61 31.68 𝜅 @ Rec#2 0.59 4.41 14.49 𝜅 @ Rec#3 5.31 3.84 6.54 𝜅 @ Rec#4 2.70 7.18 2.38 𝜅 @ Rec#5 2.07 7.72 24.18 Average, 𝜅 5.39 7.95 15.85 Modal, 𝜅 3.53 1.15 6.28 Table T.22: Predicted octave-band sample absorber absorption coefficient statistics for ASTM E90 rectangular reverberation room with the longest x-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.23 0.33 0.50 Sabine error (%) 69.53 56.99 34.49 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.21 0.28 0.39 Eyring error (%) 72.80 63.32 48.40 479 Appendix: U U.1 Third-octave-band prediction for the ASTM E90 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and without absorber U.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 5.85 s0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.9 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 5.95 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 5.61 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 5.49 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 5.24 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 4.91 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 4.84 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 4.67 s480 U.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 5.325 s0 1-40-2002040Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.449 s0 1-40-2002040Time (s)60 65 70-4000-20000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 6.935 s0 1-40-2002040Time (s)70 80-500050010001500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 4.96 s0 1-1000100Time (s)90 100 110-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 4.976 s0 1-50050Time (s)120 130 140-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 4.993 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 4.65 s0 1-2000200Time (s)180 200 220-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.345 s0 1-5000500Time (s)240 260 280-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 4.727 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)481 U.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 5.842 s0 1-40-2002040Time (s)45 50 55-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.786 s0 1-50050Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 5.206 s0 1-20020Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 4.734 s0 1-50050Time (s)90 100 1100500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.149 s0 1-1000100Time (s)120 130 140-8000-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 4.367 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 3.963 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 3.992 s0 1-400-2000200400Time (s)240 260 280-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 4.753 s0 1-400-2000200400Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)482 U.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 6.621 s0 1-40-2002040Time (s)45 50 55010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.135 s0 1-40-2002040Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.036 s0 1-20020Time (s)70 80-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 5.299 s0 1-50050Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 4.987 s0 1-50050Time (s)120 130 140-8000-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 4.487 s0 1-1000100Time (s)150 160 170-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 4.047 s0 1-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.077 s0 1-400-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 4.937 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)483 U.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 5.332 s0 1-40-2002040Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.703 s0 1-20020Time (s)60 65 70-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 5.858 s0 1-60-40-2002040Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 4.316 s0 1-1000100Time (s)90 100 110-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.14 s0 1-50050Time (s)120 130 140-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.206 s0 1-1000100Time (s)150 160 170-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 4.459 s0 1-2000200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.387 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 4.364 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)484 U.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 5.247 s0 1-40-2002040Time (s)45 50 55020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.519 s0 1-20020Time (s)60 65 70-200002000Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 8.9 s0 1-40-2002040Time (s)70 80-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 6.042 s0 1-50050100Time (s)90 100 110-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 6.249 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 4.662 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 4.114 s0 1-2000200Time (s)180 200 220-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.08 s0 1-400-2000200400Time (s)240 260 280-3-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 5.153 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)485 U.2 Third-octave-band prediction for the ASTM E90 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and with absorber U.2.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 4.22 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 2.12 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 2.78 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 2.84 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 2.65 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 2.59 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.72 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.09 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 1.91 s486 U.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 4.218 s0 1-40-2002040Time (s)45 50 55-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 3.898 s0 1-40-2002040Time (s)60 65 70-1500-1000-5000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 5.558 s0 1-20020Time (s)70 80-2000200400Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 2.823 s0 1-1000100Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.832 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 2.473 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.357 s0 1-1000100Time (s)180 200 220-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 1.91 s0 1-2000200Time (s)240 260 280-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 2.513 s0 1-5000500Time (s)300 350-5000050001000015000Frequency (Hz)Pressure (Pa)487 U.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 4.191 s0 1-40-2002040Time (s)45 50 55020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 2.348 s0 1-40-2002040Time (s)60 65 70-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 2.404 s0 1-10010Time (s)70 80-1000-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 2.903 s0 1-50050Time (s)90 100 110020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.371 s0 1-1000100Time (s)120 130 140-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 2.368 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 1.802 s0 1-2000200Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 1.779 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 2.13 s0 1-2000200Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)488 U.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 4.24 s0 1-40-2002040Time (s)45 50 55-500050010001500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 2.254 s0 1-20020Time (s)60 65 70-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 2.397 s0 1-20020Time (s)70 80-1000-5000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 3.042 s0 1-50050Time (s)90 100 110-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.26 s0 1-50050Time (s)120 130 140-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 2.203 s0 1-50050Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 1.977 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 1.88 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 3.005 s0 1-2000200Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)489 U.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 4.235 s0 1-40-2002040Time (s)45 50 55-10000-50000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 2.45 s0 1-10010Time (s)60 65 70-3000-2000-1000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 4.04 s0 1-50050Time (s)70 80-1000-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 2.785 s0 1-1000100Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.654 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.366 s0 1-1000100Time (s)150 160 170-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.66 s0 1-2000200Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 1.898 s0 1-2000200Time (s)240 260 280-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 2.161 s0 1-400-2000200400Time (s)300 350-500005000Frequency (Hz)Pressure (Pa)490 U.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 4.22 s0 1-40-2002040Time (s)45 50 5502000400060008000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 2.739 s0 1-10010Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 2.888 s0 1-1000100Time (s)90 100 110-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.96 s0 1-50050Time (s)120 130 140-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.699 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 7.347 s0 1-20020Time (s)70 80-1000-800-600-400-200Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.176 s0 1-1000100Time (s)180 200 220-10000-50000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.072 s0 1-2000200400Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 2.758 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)491 U.3 Octave-band prediction for the ASTM E90 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and without absorber U.3.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands U.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 U.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.49 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 5.46 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 4.77 s0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 6.911 s0 1-100-50050Time (s)60 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.056 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.558 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.889 s0 1-1000100Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 4.778 s0 1-2000200Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.233 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)492 U.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 U.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 U.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.215 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 4.786 s0 1-200-1000100Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.585 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.531 s0 1-50050Time (s)60 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 4.727 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.392 s0 1-100001000Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 8.087 s0 1-50050Time (s)60 80-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.607 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.533 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)493 U.4 Octave-band prediction for the ASTM E90 reverberation room – Case #1: rectangular-shaped with the longest x-dimension and with absorber U.4.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands U.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 U.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 3.75 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 2.7 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.32 s0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 4.697 s0 1-50050Time (s)60 80-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.697 s0 1-2000200Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.21 s0 1-5000500Time (s)200 300-5000050001000015000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 2.898 s0 1-50050Time (s)60 80-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.615 s0 1-2000200Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 1.864 s0 1-400-2000200400Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)494 U.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 U.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 U.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 2.543 s0 1-50050Time (s)60 80-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.673 s0 1-1000100Time (s)100 150-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.125 s0 1-5000500Time (s)200 300-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 4.15 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.701 s0 1-1000100200Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.132 s0 1-5000500Time (s)200 300-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.794 s0 1-50050Time (s)60 80-200002000400060008000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 2.999 s0 1-2000200Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.229 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)495 Appendix: V V.1 Third-octave-band prediction for the ASTM E90 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table V.1, the corresponding statistically-based cut-off frequency limit is 125 Hz, which suggests that prediction can be done from the 160-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 160-Hz third-octave are 1, 0.97, 0.44 and 0.46, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 5.7 s for the RT, as shown in Table V.3, is 527 Hz. Table V.1: Predicted third-octave-band modal compositions for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 2 5 6 12 20 37 68 125 238 Modal density, 𝑛(𝑓) 0.17 0.36 0.32 0.52 0.69 1.00 1.48 2.16 3.26 Modal spacing, 𝛿𝑓 0.44 2.96 3.05 1.64 1.48 0.97 0.68 0.46 0.30 Modal width, 𝛿𝑤 0.38 0.40 0.40 0.41 0.42 0.44 0.45 0.47 0.49 Modal overlap, 𝑀 0.06 0.14 0.13 0.21 0.29 0.44 0.66 1.02 1.59 Modal overlap factor, 𝑀𝑓 0.87 0.13 0.13 0.25 0.29 0.46 0.66 1.03 1.60 Spatial uniformity of SPL: From Table V.2, the predicted standard deviations of SPL, a measure of spatial variation, reveal that the deviations are smaller than 1.5 dB from the 160-Hz third-octave band. 496 Table V.2: Predicted third-octave-band SPL statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 142.8 146.4 149.4 158.6 155.7 158.1 163.6 167.3 171.8 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 140.1 146.6 153.3 157.3 159.6 159.8 165.6 168.9 170.7 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 140.5 150.3 146.8 158.7 155.0 158.4 161.2 168.3 171.5 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 141.5 153.7 147.3 160.2 158.8 158.8 164.2 169.3 171.9 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 144.2 148.0 152.2 158.2 157.7 158.1 166.6 168.8 172.7 Predicted average SPL, ?̅?𝑝 (dB) 142.1 149.9 150.5 158.7 157.7 158.7 164.6 168.6 171.8 Standard deviation, 𝜎𝐿𝑝 (dB) 1.7 3.2 3.0 1.1 2.0 0.7 2.1 0.8 0.7 Expected Sabine SPL, 𝐿𝑝 (dB) 144.3 152.5 152.0 160.1 159.0 159.7 165.4 169.4 172.5 Sabine error (%) 1.6 1.7 0.9 0.9 0.8 0.7 0.5 0.5 0.4 Expected Eyring SPL, 𝐿𝑝 (dB) 144.3 152.5 151.9 160.1 159.0 159.7 165.4 169.4 172.4 Eyring error (%) 1.54 1.70 0.91 0.84 0.80 0.63 0.46 0.49 0.39 Table V.3: Predicted third-octave-band RT statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 5.75 6.09 5.84 5.68 5.57 5.33 5.25 5.02 4.88 Predicted RT @ Rec#1: 𝑇60 (s) 7.23 5.98 7.21 5.56 4.89 4.91 5.19 4.73 4.59 Predicted RT @ Rec#2: 𝑇60 (s) 9.36 6.10 6.98 5.37 5.59 5.02 4.90 4.61 4.95 Predicted RT @ Rec#3: 𝑇60 (s) 4.35 4.94 5.63 5.70 4.77 5.31 4.92 4.84 4.40 Predicted RT @ Rec#4: 𝑇60 (s) 7.80 4.79 5.56 5.69 5.36 4.82 5.09 4.52 4.13 Predicted RT @ Rec#5: 𝑇60 (s) 7.62 5.68 6.49 6.24 5.96 5.51 5.02 4.57 4.88 Predicted average RT: ?̅?60 (s) 6.80 5.44 6.30 5.70 5.27 5.10 5.02 4.65 4.57 Standard deviation, 𝜎𝑇60 (s) 1.90 0.60 0.76 0.33 0.50 0.29 0.12 0.13 0.34 Expected Sabine RT: 𝑇60 (s) 4.03 4.03 4.03 4.03 4.03 4.03 4.03 4.03 4.03 Sabine error (%) 69.0 35.2 56.5 41.5 31.0 26.7 24.8 15.5 13.5 Expected Eyring RT, 𝑇60 (s) 3.99 3.99 3.99 3.99 3.99 3.99 3.99 3.99 3.99 Eyring error (%) 70.7 36.6 58.1 42.9 32.3 28.0 26.0 16.7 14.6 Decay rate (dB/s) 8.8 11.0 9.5 10.5 11.4 11.8 11.9 12.9 13.1 497 Table V.4: Predicted third-octave-band decay-curve curvature statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 7.9 6.0 61.1 3.8 6.8 5.3 11.5 8.3 4.2 𝜅 @ Rec#2 74.3 8.7 60.0 4.9 5.0 13.1 8.2 4.9 6.3 𝜅 @ Rec#3 2.1 3.7 7.4 5.6 7.3 3.0 10.3 7.2 2.7 𝜅 @ Rec#4 62.3 3.4 21.7 4.4 2.5 8.9 10.6 9.6 3.1 𝜅 @ Rec#5 8.4 6.4 3.7 4.8 2.0 3.6 7.9 6.9 13.7 Average, 𝜅 31.0 5.6 30.8 4.7 4.7 6.8 9.7 7.4 6.0 Modal, 𝜅 0.00 0.33 0.06 0.11 0.15 0.16 0.17 0.26 0.25 Accuracy of the predicted SPLs: The accuracy of the predicted steady-state SPLs is checked against the Sabine and Eyring versions of the diffuse-field formulae as presented in Table V.2. The errors associated with the predicted SPLs are always less than 1% from the 80-Hz third-octave band for both Sabine and Eyring prediction cases. Spatial uniformity of RT: From Table V.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.5 s from the 100-Hz third-octave band. Accuracy of the predicted RTs: The accuracy of the predicted RTs is checked against the expected values of the RTs calculated using the Sabine and Eyring versions of the diffuse-field formulae as presented in Table V.3. The predicted errors are less than, or around, 15% from the 250-Hz third-octave band for both Sabine and Eyring prediction cases. Table V.5: Predicted third-octave-band source-power level statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 143.3 151.5 150.9 159.0 157.9 158.7 164.3 168.3 171.4 Sabine error (%) 44.1 28.1 39.0 31.6 25.5 22.7 21.4 14.5 12.8 Eyring error (%) 45.2 29.2 40.0 32.7 26.6 23.8 22.5 15.6 13.9 Predicted, ?̅?𝑤 (dB) (1 source) 140.3 148.5 147.9 156.0 154.9 155.6 161.3 165.3 168.4 498 Linearity of temporal decay curves: From Table HV.4, the predicted curvatures of the energy-decay curves vary within 4 to 10 from the 100-Hz third-octave band. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix W.1. Accuracy of the predicted source-power levels: The predicted source-power levels and the corresponding errors in third-octave bands are presented in Table V.5. The errors associated with the predicted source-power levels are less than 15% from the 250-Hz third-octave band. The errors are greater than 20% in the rest of the bands. Accuracy of the predicted room absorptions: It is seen from Table V.6 that the errors are less than or around 15% from the 250-Hz third-octave band, and greater than 20% in the rest of the bands. The same is true for the room absorption-coefficient prediction case, as the absorption-coefficient predictions depend on the accuracy of the absorption predictions. Table V.6: Predicted third-octave-band room absorption statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 1.92 2.40 2.07 2.29 2.47 2.56 2.60 2.81 2.85 Sabine expected absorption, 𝐴 (m2) 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24 Sabine error (%) 40.8 26.0 36.1 29.3 23.7 21.1 19.9 13.4 11.9 Eyring expected absorption, 𝐴 (m2) 3.27 3.27 3.27 3.27 3.27 3.27 3.27 3.27 3.27 Eyring error (%) 41.4 26.8 36.7 30.0 24.4 21.9 20.7 14.3 12.8 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 40.8 26.0 36.1 29.3 23.7 21.1 19.9 13.4 11.9 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 41.2 26.6 36.5 29.8 24.2 21.7 20.5 14.2 12.7 499 V.2 Third-octave-band prediction for the ASTM E90 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table V.7. The cut-off limit will remain be the same as before. Table V.7: Predicted third-octave-band modal compositions for ASTM E90 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 4 7 12 21 37 69 128 237 Modal density, 𝑛(𝑓) 0.25 0.29 0.37 0.52 0.72 1.00 1.50 2.21 3.25 Modal spacing, 𝛿𝑓 4.27 2.74 2.47 1.49 1.35 0.99 0.67 0.45 0.31 Modal width, 𝛿𝑤 1.26 0.89 5.41 3.66 4.14 2.17 3.27 2.82 2.46 Modal overlap, 𝑀 0.32 0.25 1.99 1.91 3.00 2.17 4.90 6.22 7.99 Modal overlap factor, 𝑀𝑓 0.30 0.32 2.19 2.45 3.06 2.19 4.91 6.22 7.99 Table V.8: Predicted third-octave-band SPL statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 139.6 142.0 143.6 152.7 152.7 154.9 157.4 165.1 168.3 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 139.0 144.0 145.0 151.3 152.7 153.0 159.2 164.6 167.5 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 137.0 146.9 141.7 153.7 149.8 153.7 157.8 166.5 168.5 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 139.9 150.5 141.4 155.9 155.5 156.3 160.2 165.8 168.2 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 139.2 144.0 142.7 150.7 152.7 152.2 161.6 163.7 166.5 Predicted average SPL, ?̅?𝑝 (dB) 139.0 146.6 143.1 153.3 153.0 154.3 159.5 165.2 167.9 Standard deviation, 𝜎𝐿𝑝 (dB) 1.16 3.54 1.51 2.14 2.06 1.64 1.76 1.10 0.81 500 Table V.9: Predicted third-octave-band RT statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 2.54 4.13 3.17 3.60 3.50 3.28 2.96 2.37 2.28 Predicted RT @ Rec#1: 𝑇60 (s) 6.27 4.52 4.18 3.18 3.17 2.58 2.70 2.36 2.49 Predicted RT @ Rec#2: 𝑇60 (s) 9.57 3.59 3.93 2.80 3.43 3.28 2.45 2.49 3.72 Predicted RT @ Rec#3: 𝑇60 (s) 3.92 3.29 4.07 3.53 3.65 2.99 2.45 2.06 2.18 Predicted RT @ Rec#4: 𝑇60 (s) 7.78 3.29 3.24 3.43 3.27 2.32 2.45 2.00 2.10 Predicted RT @ Rec#5: 𝑇60 (s) 5.99 4.08 7.34 3.49 3.78 3.83 2.73 2.83 3.03 Predicted average RT: ?̅?60 (s) 6.14 3.69 4.22 3.26 3.44 2.91 2.55 2.31 2.58 Standard deviation, 𝜎𝑇60 (s) 2.21 0.54 1.64 0.31 0.26 0.60 0.15 0.34 0.69 Decay rate (dB/s) 9.8 16.2 14.2 18.4 17.4 20.6 23.5 26.0 23.2 Spatial uniformity of SPL: The standard deviations of the SPL for this ‘with absorber’ case, as a measure of spatial variation, are presented in Table V.8. Overall, the deviations are greater than the values obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT for this ‘with absorber’ case, as a measure of spatial variation, are presented in Table V.9. It is obvious that the deviations are greater than in the previous ‘without absorber’ case in almost all third-octave bands. Table V.10: Predicted third-octave-band decay-curve curvature statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 182 18.6 127.2 12.9 29.8 10.8 9.6 5.4 26.2 𝜅 @ Rec#2 397 40.8 144.7 19.1 74.6 51.2 10.2 12.9 50.6 𝜅 @ Rec#3 68.3 10.8 65.1 14.1 7.2 50.3 18.5 7.2 42.5 𝜅 @ Rec#4 183 5.3 9.3 8.0 1.8 20.1 10.5 6.7 12.1 𝜅 @ Rec#5 180 35.1 114.0 29.1 80.4 35.7 16.9 19.8 57.2 Average, 𝜅 202 22.1 92.0 16.6 38.8 33.6 13.1 10.4 37.7 Modal, 𝜅 0.62 0.41 0.16 3.89 0.67 1.18 3.14 2.00 2.56 501 Linearity of decay curves: The curvatures of the energy-decay curves as presented in Table V.10 reveal that the curvatures are greater than those obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave-bands are presented in Appendix W.2. Table V.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ASTM E90 rectangular reverberation room with the longest z-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.04 0.22 0.20 0.33 0.26 0.38 0.49 0.56 0.43 Sabine error (%) 94.8 70.9 73.8 56.0 66.2 50.4 35.0 26.5 43.6 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.04 0.20 0.18 0.28 0.23 0.31 0.39 0.43 0.35 Eyring error (%) 94.9 73.9 76.2 62.6 70.2 58.7 48.7 43.7 54.1 Accuracy of the sample absorption coefficients: The accuracy of the predicted absorption coefficients of an absorber sample are presented in Table V.11. It is noticeable that the predicted values deviate from the theoretical values to a great extent. 502 V.3 Octave-band prediction for the ASTM E90 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: The modal compositions data shown in Table V.12 suggests that prediction can be done from the 125-Hz band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz octave band are 0.79, 1.29, 0.43 and 0.34, respectively. On the other hand, the pure-tone-based Schroeder limit corresponding to a modal overlap factor of 3 and a band-averaged RT of 6.4 s, as shown in Table V.14, is 558 Hz. Table V.12: Predicted octave-band modal compositions for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 12 70 431 Modal density, 𝑛(𝑓) 0.27 0.79 2.42 Modal spacing, 𝛿𝑓 3.25 1.29 0.41 Modal width, 𝛿𝑤 0.39 0.43 0.48 Modal overlap, 𝑀 0.11 0.34 1.15 Modal overlap factor, 𝑀𝑓 0.12 0.34 1.17 Spatial uniformity of SPL: The predicted SPL standard deviations are smaller than 1.5 dB in all octave bands, as shown in Table V.13. Accuracy of the predicted SPLs: As shown in Table V.13, the predicted SPLs show errors below 1% in all octave bands for both Sabine and Eyring prediction cases. Spatial uniformity of RT: The predicted RT standard deviations are less than 0.5 s from the 125-Hz band. These are shown in Table V.14. Accuracy of the predicted RTs: As shown in Table V.14, the predicted RTs show errors below or around 15% from the 250-Hz octave band. 503 Table V.13: Predicted octave-band SPL statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 147.78 158.45 169.73 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 150.32 159.87 169.76 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 148.23 158.36 169.58 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 150.84 159.82 170.38 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 150.01 158.87 170.99 Predicted average SPL, ?̅?𝑝 (dB) 149.60 159.12 170.12 Standard deviation, 𝜎𝐿𝑝 (dB) 1.36 0.73 0.59 Expected Sabine SPL, 𝐿𝑝 (dB) 150.54 160.22 170.98 Sabine error (%) 0.62 0.68 0.51 Expected Eyring SPL, 𝐿𝑝 (dB) 150.49 160.18 170.94 Eyring error (%) 0.60 0.66 0.48 Table V.14: Predicted octave-band RT statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 5.97 5.48 5.01 Predicted RT @ Rec#1: 𝑇60 (s) 8.04 5.21 4.81 Predicted RT @ Rec#2: 𝑇60 (s) 9.84 5.54 4.80 Predicted RT @ Rec#3: 𝑇60 (s) 6.46 5.31 4.47 Predicted RT @ Rec#4: 𝑇60 (s) 6.14 5.30 4.43 Predicted RT @ Rec#5: 𝑇60 (s) 12.78 6.52 4.78 Predicted average RT: ?̅?60 (s) 8.04 5.54 4.65 Standard deviation, 𝜎𝑇60 (s) 2.82 0.54 0.19 Expected Sabine RT: 𝑇60 (s) 4.03 4.03 4.03 Sabine error (%) 99.68 37.54 15.53 Expected Eyring RT, 𝑇60 (s) 3.99 3.99 3.99 Eyring error (%) 101.71 38.94 16.70 Decay rate (dB/s) 7.46 10.84 12.90 504 Table V.15: Predicted octave-band curvature statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 13.60 2.72 2.12 𝜅 @ Rec#2 57.16 2.69 2.40 𝜅 @ Rec#3 6.72 2.14 1.30 𝜅 @ Rec#4 6.90 1.59 2.64 𝜅 @ Rec#5 53.75 2.50 3.47 Average, 𝜅 27.63 2.33 2.39 Modal, 𝜅 0.12 0.17 0.27 Linearity of decay curves: The curvatures are within 2 to 3 from the 125-Hz band, as shown in Table V.15. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix W.3. Accuracy of the predicted source-power levels: The predicted source-power levels show errors below 15% from the 250-Hz band, while the errors are greater than 20% in the rest of the bands, as shown in Table V.16. Accuracy of the predicted room absorptions: The predicted room-averaged absorptions, as well as the absorption coefficients, show errors below 15% from the 250-Hz band, while the errors are greater than 20% in the rest of the bands. These are shown in Table V.17. Table V.16: Predicted octave-band source-power level statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 149.46 159.15 169.91 Sabine error (%) 53.85 29.45 14.51 Eyring error (%) 54.95 30.54 15.60 Predicted, ?̅?𝑤 (dB) (1 source) 146.45 156.14 166.90 505 Table V.17: Predicted octave-band room absorption statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 1.62 2.36 2.80 Sabine expected absorption, 𝐴 (m2) 3.24 3.24 3.24 Sabine error (%) 49.92 27.30 13.45 Eyring expected absorption, 𝐴 (m2) 3.27 3.27 3.27 Eyring error (%) 50.42 28.02 14.31 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.02 Sabine error (%) 49.92 27.30 13.45 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.02 Eyring error (%) 50.17 27.82 14.19 506 V.4 Octave-band prediction for the ASTM E90 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit as there no change in the modal compositions except the modal width and, hence, the modal overlap as shown in Table V.18. Table V.18: Predicted octave-band modal compositions for ASTM E90 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 14 69 435 Modal density, 𝑛(𝑓) 0.32 0.78 2.44 Modal spacing, 𝛿𝑓 3.14 1.22 0.41 Modal width, 𝛿𝑤 3.23 3.04 2.69 Modal overlap, 𝑀 1.03 2.36 6.58 Modal overlap factor, 𝑀𝑓 1.03 2.50 6.57 Table V.19: Predicted octave-band SPL statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 142.84 154.20 166.39 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 144.17 153.08 165.85 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 144.33 153.27 167.00 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 147.41 156.23 166.68 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 143.13 152.75 165.26 Predicted average SPL, ?̅?𝑝 (dB) 144.71 154.10 166.28 Standard deviation, 𝜎𝐿𝑝 (dB) 1.85 1.42 0.69 507 Table V.20: Predicted octave-band RT statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 3.82 3.44 2.58 Predicted RT @ Rec#1: 𝑇60 (s) 8.73 3.14 2.52 Predicted RT @ Rec#2: 𝑇60 (s) 13.54 3.86 3.26 Predicted RT @ Rec#3: 𝑇60 (s) 5.91 3.63 2.10 Predicted RT @ Rec#4: 𝑇60 (s) 3.77 3.15 2.07 Predicted RT @ Rec#5: 𝑇60 (s) 25.13 4.50 2.89 Predicted average RT: ?̅?60 (s) 7.55 3.59 2.49 Stnd. deviation, 𝜎𝑇60 (s) 9.53 0.57 0.52 Decay rate (dB/s) 7.95 16.71 24.11 Spatial uniformity of SPL: The predicted SPL deviations are shown in Table V.19. The overall deviations are greater than in the ‘without absorber’ case. Spatial uniformity of RT: The predicted RT deviations are shown in Table V.20. The overall deviations are greater than in the ‘without absorber’ case. Table V.21: Predicted octave-band decay-curve curvature statistics for ASTM E90 rectangular reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 117.11 12.71 18.47 𝜅 @ Rec#2 243.86 48.11 30.58 𝜅 @ Rec#3 74.25 12.12 8.70 𝜅 @ Rec#4 11.97 3.77 3.73 𝜅 @ Rec#5 179.14 53.45 32.09 Average, 𝜅 125.27 26.03 18.72 Modal, 𝜅 1.58 1.43 4.10 Linearity of decay curves: The curvatures are shown in Table V.21. The overall curvatures are greater than in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix W.4. 508 Table V.22: Predicted octave-band sample absorber absorption coefficient statistics for ASTM E90 rectangular reverberation room with the longest z-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.25 0.48 Sabine error (%) 97.40 67.16 37.25 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.02 0.22 0.38 Eyring error (%) 97.42 70.94 50.09 Accuracy of the predicted sample absorption coefficients: As shown in Table V.22, the predicted Sabin absorption coefficients show errors around 40% from the 250-Hz band, while the Eyring predictions yield errors around 50% in the same bands. 509 Appendix: W W.1 Third-octave-band prediction for the ASTM E90 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber W.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 5.75 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.09 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 5.84 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 5.68 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 5.57 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 5.33 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 5.25 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.02 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 4.88 s510 W.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs in third-octave bands, at receiver #5 0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 7.617 s0 1-40-2002040Time (s)45 50 55-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 5.675 s0 1-20020Time (s)60 65 70020004000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 6.492 s0 1-50050Time (s)70 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 6.243 s0 1-50050Time (s)90 100 11001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.963 s0 1-1000100Time (s)120 130 140-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 5.505 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 5.021 s0 1-2000200400Time (s)180 200 220-3-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.574 s0 1-400-2000200400Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 4.88 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)511 W.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 7.795 s0 1-50050Time (s)45 50 55-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 4.786 s0 1-50050Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 5.561 s0 1-20020Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 5.685 s0 1-1000100Time (s)90 100 110-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.355 s0 1-50050Time (s)120 130 140-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 4.816 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 5.092 s0 1-2000200Time (s)180 200 220-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.521 s0 1-400-2000200400Time (s)240 260 280-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 4.129 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)512 W.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 4.346 s0 1-10010Time (s)45 50 55-1500-1000-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 4.943 s0 1-40-2002040Time (s)60 65 70-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.625 s0 1-40-2002040Time (s)70 80-10000100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 5.696 s0 1-1000100Time (s)90 100 110-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 4.766 s0 1-50050Time (s)120 130 140-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 5.314 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 4.919 s0 1-1000100Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.836 s0 1-2000200400Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 4.401 s0 1-5000500Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)513 W.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 9.362 s0 1-40-2002040Time (s)45 50 5505001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.102 s0 1-50050Time (s)60 65 70-4000-20000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 6.98 s0 1-50050Time (s)70 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.368 s0 1-50050Time (s)90 100 110050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.585 s0 1-1000100Time (s)120 130 140-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.019 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 4.903 s0 1-2000200Time (s)180 200 220-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.608 s0 1-400-2000200400Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 4.951 s0 1-5000500Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)514 W.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 7.225 s0 1-40-2002040Time (s)45 50 55010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.984 s0 1-2002040Time (s)60 65 700100020003000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 7.212 s0 1-50050Time (s)70 80-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 5.563 s0 1-50050Time (s)90 100 11001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 4.886 s0 1-50050Time (s)120 130 140-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 4.91 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 5.192 s0 1-1000100Time (s)180 200 220-20000-100000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.725 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 4.589 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)515 W.2 Third-octave-band prediction for the ASTM E90 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber W.2.1 Theoretical modal decay curves and the corresponding RTs, in third-octave bands 0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 2.54 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.13 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 3.17 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 3.6 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.5 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 3.28 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.96 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.37 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.28 s516 W.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 6.598 s0 1-40-2002040Time (s)45 50 55-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 5.793 s0 1-20020Time (s)60 65 70020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 5.873 s0 1-50050Time (s)70 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 6.879 s0 1-50050Time (s)90 100 11001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.986 s0 1-1000100Time (s)120 130 140-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 5.183 s0 1-1000100Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 5.747 s0 1-2000200400Time (s)180 200 220-3-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.435 s0 1-400-2000200400Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 3.972 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)517 W.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 5.549 s0 1-50050Time (s)45 50 55-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.097 s0 1-50050Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 5.527 s0 1-20020Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 6.138 s0 1-1000100Time (s)90 100 110-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.04 s0 1-50050Time (s)120 130 140-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 4.874 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 6.026 s0 1-2000200Time (s)180 200 220-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.517 s0 1-400-2000200400Time (s)240 260 280-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 4.26 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)518 W.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 4.551 s0 1-10010Time (s)45 50 55-1500-1000-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.24 s0 1-40-2002040Time (s)60 65 70-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.739 s0 1-40-2002040Time (s)70 80-10000100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 6.03 s0 1-1000100Time (s)90 100 110-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.579 s0 1-50050Time (s)120 130 140-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 5.288 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 5.645 s0 1-1000100Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.554 s0 1-2000200400Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 4.487 s0 1-5000500Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)519 W.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 6.32 s0 1-40-2002040Time (s)45 50 5505001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.308 s0 1-50050Time (s)60 65 70-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 5.736 s0 1-50050Time (s)70 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.594 s0 1-50050Time (s)90 100 110050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 6.08 s0 1-1000100Time (s)120 130 140-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 5.491 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 4.401 s0 1-2000200Time (s)180 200 220-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.639 s0 1-400-2000200400Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 4.854 s0 1-5000500Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)520 W.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 6.346 s0 1-40-2002040Time (s)45 50 55010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.913 s0 1-2002040Time (s)60 65 700100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 6.188 s0 1-50050Time (s)70 80-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 5.924 s0 1-50050Time (s)90 100 11001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.68 s0 1-50050Time (s)120 130 140-4000-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 5.052 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 4.636 s0 1-1000100Time (s)180 200 220-20000-100000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.737 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 4.823 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)521 W.3 Octave-band prediction for the ASTM E90 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and without absorber W.3.1 Theoretical modal decay curves and the corresponding RTs, in octave bands W.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #5 W.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #4 0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 5.97 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 5.48 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 5.01 s0 5 10-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 12.778 s0 1-1000100Time (s)60 80-500005000Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 6.519 s0 1-2000200Time (s)100 15001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.784 s0 1-100001000Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.144 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.3 s0 1-2000200Time (s)100 150-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.429 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)522 W.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #3 W.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #2 W.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands at receiver #1 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 6.459 s0 1-50050Time (s)60 80-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.307 s0 1-1000100Time (s)100 150-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.474 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 9.835 s0 1-1000100Time (s)60 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.537 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.796 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 8.037 s0 1-50050Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.213 s0 1-1000100Time (s)100 15001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.805 s0 1-100001000Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)523 W.4 Octave-band prediction for the ASTM E90 reverberation room – Case #2: rectangular-shaped with the longest z-dimension and with absorber W.4.1 Theoretical modal decay curves and the corresponding RTs, in octave bands W.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 W.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 3.82 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.44 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.58 s0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 8.927 s0 1-1000100Time (s)60 80-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 6.106 s0 1-2000200Time (s)100 15001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.383 s0 1-100001000Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.601 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.153 s0 1-2000200Time (s)100 150-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.58 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)524 W.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 W.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs in octave bands, at receiver #2 W.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.876 s0 1-50050Time (s)60 80-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.64 s0 1-1000100Time (s)100 150-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.535 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.854 s0 1-1000100Time (s)60 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.857 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.65 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 6.657 s0 1-50050Time (s)60 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 5.739 s0 1-1000100Time (s)100 15001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 5.113 s0 1-100001000Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)525 Appendix: X X.1 Third-octave-band prediction for the ASTM E90 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table X.1, the corresponding statistically-based cut-off frequency limit is 125 Hz, which suggests that prediction can be done from the 160-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 160-Hz third-octave band are 1, 0.97, 0.43 and 0.44, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 5.9 s for the RT, as shown in Table X.3, is 536 Hz. Table X.1: Predicted third-octave-band modal compositions for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 2 4 7 11 19 37 67 125 241 Modal density, 𝑛(𝑓) 0.17 0.29 0.37 0.48 0.66 1.00 1.46 2.16 3.30 Modal spacing, 𝛿𝑓 3.42 3.58 2.67 2.03 1.51 0.97 0.68 0.46 0.30 Modal width, 𝛿𝑤 0.43 0.39 0.39 0.41 0.42 0.43 0.46 0.48 0.49 Modal overlap, 𝑀 0.07 0.11 0.14 0.19 0.28 0.43 0.66 1.02 1.61 Modal overlap factor, 𝑀𝑓 0.13 0.11 0.14 0.20 0.28 0.44 0.67 1.04 1.61 Spatial uniformity of SPL: From Table X.2, the predicted standard deviations of SPL are less than 1.5 dB from the 160-Hz band, while the deviations are greater than 1.5 dB in the rest of the bands. Accuracy of the predicted SPLs: As presented in Table X.2, the errors associated with the predicted SPLs are always less than or around 1% from the 125-Hz third-octave band for both Sabine and Eyring prediction cases. 526 Table X.2: Predicted third-octave-band SPL statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 141.9 157.5 143.5 155.3 155.9 160.1 163.3 165.7 172.5 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 137.5 146.1 144.0 158.3 158.5 161.0 164.7 165.7 170.7 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 137.7 147.3 149.3 154.5 160.8 161.1 165.2 166.2 172.9 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 138.8 153.3 149.6 156.1 157.6 159.7 160.5 168.2 172.1 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 142.9 157.1 154.0 155.5 154.5 161.1 162.6 165.1 170.9 Predicted average SPL, ?̅?𝑝 (dB) 140.3 154.4 149.8 156.1 158.0 160.6 163.5 166.3 171.9 Standard deviation, 𝜎𝐿𝑝 (dB) 2.5 5.9 4.8 1.4 2.5 0.7 1.9 1.2 1.0 Expected Sabine SPL, 𝐿𝑝 (dB) 142.5 156.7 151.9 158.2 159.7 161.9 164.6 167.1 172.1 Sabine error (%) 1.5 1.5 1.4 1.3 1.0 0.8 0.7 0.5 0.1 Expected Eyring SPL, 𝐿𝑝 (dB) 142.4 156.7 151.9 158.1 159.6 161.9 164.6 167.1 172.0 Eyring error (%) 1.45 1.44 1.39 1.26 1.02 0.75 0.64 0.45 0.05 Table X.3: Predicted third-octave-band RT statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 5.10 5.85 5.74 5.50 5.40 5.29 4.95 4.74 4.62 Predicted RT @ Rec#1: 𝑇60 (s) 8.40 5.39 5.12 5.10 4.83 4.86 4.46 4.65 4.21 Predicted RT @ Rec#2: 𝑇60 (s) 13.69 9.31 5.70 5.18 4.59 4.87 4.61 4.87 5.75 Predicted RT @ Rec#3: 𝑇60 (s) 5.75 5.64 5.66 5.03 5.09 4.68 4.66 4.83 4.82 Predicted RT @ Rec#4: 𝑇60 (s) 5.45 5.36 5.35 4.77 4.79 4.71 5.06 4.56 6.88 Predicted RT @ Rec#5: 𝑇60 (s) 6.47 5.37 5.10 4.80 5.32 5.54 5.03 5.07 5.52 Predicted average RT: ?̅?60 (s) 7.10 5.93 5.37 4.97 4.91 4.91 4.75 4.79 5.29 Standard deviation, 𝜎𝑇60 (s) 3.54 1.76 0.28 0.18 0.28 0.35 0.27 0.20 1.02 Expected Sabine RT: 𝑇60 (s) 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 4.10 Sabine error (%) 73.1 44.6 31.0 21.2 19.7 19.8 15.8 16.7 28.9 Expected Eyring RT, 𝑇60 (s) 4.06 4.06 4.06 4.06 4.06 4.06 4.06 4.06 4.06 Eyring error (%) 74.9 46.1 32.3 22.4 20.9 21.0 17.0 17.9 30.2 Decay rate (dB/s) 8.45 10.12 11.17 12.0 12.2 12.2 12.6 12.5 11.35 527 Table X.4: Predicted third-octave-band decay curve curvature statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 31.8 2.9 2.4 14.9 7.5 6.4 14.0 4.4 4.5 𝜅 @ Rec#2 177 15.7 3.2 5.7 6.4 4.6 16.8 5.5 3.1 𝜅 @ Rec#3 3.8 5.5 18.6 14.7 7.9 4.2 2.2 6.6 13.0 𝜅 @ Rec#4 1.8 2.7 4.3 9.7 8.0 9.7 9.5 13.6 20.2 𝜅 @ Rec#5 8.8 3.0 4.0 7.7 5.6 6.2 4.6 6.5 1.9 Average, 𝜅 44.7 5.96 6.50 10.52 7.08 6.21 9.43 7.32 8.53 Modal, 𝜅 0.00 0.01 0.00 0.00 0.04 0.02 0.02 0.02 0.02 Spatial uniformity of RT: From Table X.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.5 s from the 180-Hz third-octave band. Accuracy of the predicted RTs: As presented in Table X.3, the predicted RTs show errors less than or around 15% from the 200-Hz third-octave band for both Sabine and Eyring prediction cases; while the errors are greater than 15% in the rest of the bands. Linearity of temporal decay curves: From Table X.4, the predicted curvatures of the energy-decay curves vary within 6 to 11 in all third-octave bands. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix Y.1. Accuracy of the predicted source-power levels: As presented in Table X.5, the predicted source-power-level errors are less than or around 15% from the 200-Hz third-octave band. Table X.5: Predicted third-octave-band source-power level statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 141.3 155.6 150.8 157.1 158.6 160.8 163.5 166.0 171.0 Sabine error (%) 44.8 32.7 25.1 18.6 17.5 17.5 14.5 15.2 23.7 Eyring error (%) 45.8 33.8 26.2 19.6 18.6 18.6 15.5 16.3 24.8 Predicted, ?̅?𝑤 (dB) (1 source) 138.3 152.6 147.8 154.1 155.5 157.8 160.5 163.0 167.9 528 Table X.6: Predicted third-octave-band room absorption statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 1.86 2.22 2.45 2.65 2.68 2.68 2.77 2.75 2.49 Sabine expected absorption, 𝐴 (m2) 3.21 3.21 3.21 3.21 3.21 3.21 3.21 3.21 3.21 Sabine error (%) 42.2 30.8 23.7 17.5 16.5 16.5 13.7 14.3 22.4 Eyring expected absorption, 𝐴 (m2) 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24 Eyring error (%) 42.8 31.5 24.4 18.3 17.3 17.4 14.5 15.2 23.2 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine error (%) 42.2 30.8 23.7 17.5 16.5 16.5 13.7 14.3 22.4 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Eyring error (%) 42.6 31.3 24.2 18.2 17.2 17.2 14.4 15.1 23.0 Accuracy of the predicted room absorptions: As presented in Table X.6, the errors are less than or around 15% from the 200-Hz third-octave band, and greater than 15% in the rest of the bands. The same is true for the room absorption coefficient prediction case, as the absorption coefficient predictions depend on the accuracy of the absorption predictions. 529 X.2 Third-octave-band prediction for the ASTM E90 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table X.7. The cut-off frequency limit will remain be the same as before. Table X.7: Predicted third-octave-band modal compositions for ASTM E90 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 2 4 7 13 19 37 68 129 240 Modal density, 𝑛(𝑓) 0.17 0.29 0.37 0.57 0.66 1.00 1.48 2.22 3.29 Modal spacing, 𝛿𝑓 3.29 3.77 2.50 1.71 1.41 0.97 0.67 0.45 0.30 Modal width, 𝛿𝑤 1.09 0.67 5.30 3.48 4.42 2.21 3.24 2.79 2.43 Modal overlap, 𝑀 0.18 0.19 1.95 1.97 2.89 2.21 4.79 6.21 8.00 Modal overlap factor, 𝑀𝑓 0.33 0.18 2.12 2.03 3.14 2.28 4.81 6.21 8.03 Spatial uniformity of SPL: The standard deviations of the SPL are presented in Table X.8. The deviations are greater than those obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT are presented in Table X.9. It is obvious that the deviations are greater than in the previous ‘without absorber’ case in almost all third-octave bands. Linearity of decay curves: The curvatures of the decay curves as presented in Table X.10 reveal that the curvatures are greater than the values obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix Y.2. 530 Table X.8: Predicted third-octave-band SPL statistics for ASTM E90 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 142.0 155.0 140.7 151.9 153.2 157.5 160.0 162.0 169.7 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 137.7 146.3 141.7 153.8 157.2 158.3 159.0 161.9 168.4 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 137.5 145.5 140.7 150.0 153.2 157.6 160.2 162.9 168.9 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 137.1 150.4 142.4 152.5 159.0 156.0 156.0 163.4 168.5 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 143.1 154.3 148.3 153.7 154.7 155.3 157.5 164.3 169.0 Predicted average SPL, ?̅?𝑝 (dB) 140.2 151.9 143.9 152.6 156.1 157.1 158.8 163.0 168.9 Standard deviation, 𝜎𝐿𝑝 (dB) 2.95 4.72 3.44 1.58 2.68 1.27 1.79 1.01 0.51 Table X.9: Predicted third-octave-band RT statistics for ASTM E90 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 2.15 3.42 4.52 3.60 3.51 3.23 2.72 2.46 2.19 Predicted RT @ Rec#1: 𝑇60 (s) 5.46 3.61 3.79 3.46 3.27 2.81 2.71 2.31 2.28 Predicted RT @ Rec#2: 𝑇60 (s) 10.53 4.18 2.66 2.42 3.29 2.84 2.55 2.38 2.96 Predicted RT @ Rec#3: 𝑇60 (s) 4.32 3.65 3.07 2.57 2.75 2.61 2.49 2.53 3.35 Predicted RT @ Rec#4: 𝑇60 (s) 3.80 3.58 3.88 3.60 3.78 3.15 2.21 2.62 5.28 Predicted RT @ Rec#5: 𝑇60 (s) 4.36 3.58 3.05 3.74 3.06 2.68 2.70 2.52 2.80 Predicted average RT: ?̅?60 (s) 4.99 3.70 3.22 3.05 3.19 2.81 2.52 2.47 3.09 Standard deviation, 𝜎𝑇60 (s) 2.88 0.26 0.53 0.62 0.38 0.21 0.20 0.13 1.19 Decay rate (dB/s) 12.0 16.2 18.6 19.6 18.8 21.4 23.8 24.3 19.4 Accuracy of the sample absorption coefficients: The predicted absorption coefficients of an absorber sample are presented in Table X.11. It is noticeable that the predicted values deviate from the theoretical values to a great extent. 531 Table X.10: Predicted third-octave-band decay curve curvature statistics for ASTM E90 oblique reverberation room with the longest z-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 77.3 0.8 2.7 4.2 6.2 8.3 6.4 9.0 14.1 𝜅 @ Rec#2 485.3 6.4 22.8 11.3 8.9 17.3 6.4 10.1 42.7 𝜅 @ Rec#3 74.0 5.7 40.4 60.7 14.6 7.4 14.2 10.3 66.5 𝜅 @ Rec#4 28.2 0.4 12.8 11.4 1.9 5.2 10.8 17.4 95.3 𝜅 @ Rec#5 40.6 0.6 25.2 5.2 5.3 28.5 23.2 24.0 28.5 Average, 𝜅 141.1 2.8 20.8 18.6 7.4 13.3 12.2 14.2 49.4 Modal, 𝜅 0.07 0.02 1.17 0.42 1.39 1.01 1.99 2.63 2.64 Table X.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ASTM E90 oblique reverberation room with the longest z-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.16 0.26 0.32 0.33 0.29 0.40 0.49 0.51 0.35 Sabine error (%) 79.6 65.2 57.3 56.7 62.4 47.5 35.8 32.4 53.6 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.14 0.23 0.28 0.28 0.25 0.33 0.39 0.40 0.30 Eyring error (%) 81.1 69.4 63.5 63.1 67.3 56.7 49.2 47.1 60.9 532 X.3 Octave-band prediction for the ASTM E90 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table X.12, the statistically-based cut-off frequency limit is 78 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz band are 0.75, 1.3, 0.42 and 0.33, respectively. On the other hand, the pure-tone-based Schroeder limit corresponding to a modal overlap factor of 3 and a band-averaged RT of 5.9 s, as shown in Table X.14, is 536 Hz. Table X.12: Predicted octave-band modal compositions for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 13 67 433 Modal density, 𝑛(𝑓) 0.30 0.75 2.43 Modal spacing, 𝛿𝑓 3.18 1.30 0.41 Modal width, 𝛿𝑤 0.39 0.42 0.48 Modal overlap, 𝑀 0.12 0.32 1.17 Modal overlap factor, 𝑀𝑓 0.12 0.33 1.17 Spatial uniformity of SPL: The predicted SPL standard deviations are less than 1 dB from the 125-Hz octave band, as shown in Table X.13. Accuracy of the predicted SPLs: As shown in Table X.13, the predicted SPLs show errors below 1% from the 125-Hz octave band, while the error is around 1.5% in the 63-Hz band for both Sabine and Eyring cases. 533 Table X.13: Predicted octave-band SPL statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 153.93 158.42 169.88 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 144.52 160.21 168.78 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 147.47 160.53 170.47 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 151.01 158.85 169.92 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 155.08 158.95 168.56 Predicted average SPL, ?̅?𝑝 (dB) 151.93 159.47 169.58 Standard deviation, 𝜎𝐿𝑝 (dB) 4.73 0.92 0.82 Expected Sabine SPL, 𝐿𝑝 (dB) 154.26 161.09 170.25 Sabine error (%) 1.51 1.01 0.39 Expected Eyring SPL, 𝐿𝑝 (dB) 154.22 161.05 170.21 Eyring error (%) 1.48 0.98 0.37 Table X.14: Predicted octave-band RT statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 5.73 5.37 4.73 Predicted RT @ Rec#1: 𝑇60 (s) 5.37 4.82 4.28 Predicted RT @ Rec#2: 𝑇60 (s) 6.65 4.89 4.99 Predicted RT @ Rec#3: 𝑇60 (s) 5.87 4.93 4.47 Predicted RT @ Rec#4: 𝑇60 (s) 5.90 4.79 5.75 Predicted RT @ Rec#5: 𝑇60 (s) 5.74 5.40 5.49 Predicted average RT: ?̅?60 (s) 5.88 4.96 4.93 Standard deviation, 𝜎𝑇60 (s) 0.47 0.25 0.64 Expected Sabine RT: 𝑇60 (s) 4.10 4.10 4.10 Sabine error (%) 43.34 20.87 20.23 Expected Eyring RT, 𝑇60 (s) 4.06 4.06 4.06 Eyring error (%) 44.80 22.10 21.45 Decay rate (dB/s) 10.21 12.10 12.17 534 Table X.15: Predicted octave-band decay curve curvature statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 2.34 2.87 3.03 𝜅 @ Rec#2 9.07 2.84 0.61 𝜅 @ Rec#3 4.57 3.85 2.12 𝜅 @ Rec#4 0.35 4.73 6.85 𝜅 @ Rec#5 0.39 1.36 1.00 Average, 𝜅 3.34 3.13 2.72 Modal, 𝜅 0.01 0.03 0.03 Spatial uniformity of RT: The predicted RT standard deviations are around 0.6 s in all octave bands. These are shown in Table X.14. Accuracy of the predicted RTs: As shown in Table X.14, the predicted RTs show errors below or around 20% from the 125-Hz band, while the errors are greater than 20% in the rest of the bands. Degree of linearity of decay curves: The curvatures are within 2 to 4 in all octave bands, as shown in Table X.15. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix Y.3. Table X.16: Predicted octave-band source-power level statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 153.15 159.98 169.14 Sabine error (%) 32.06 18.31 17.82 Eyring error (%) 33.13 19.39 18.90 Predicted, ?̅?𝑤 (dB) (1 source) 150.14 156.97 166.13 535 Accuracy of the predicted source-power levels: The predicted source-power levels show errors below 18% from the 125-Hz band, as shown in Table X.16. Table X.17: Predicted octave-band room absorption statistics for ASTM E90 oblique reverberation room with the longest z-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 2.24 2.66 2.67 Sabine expected absorption, 𝐴 (m2) 3.21 3.21 3.21 Sabine error (%) 30.24 17.27 16.83 Eyring expected absorption, 𝐴 (m2) 3.24 3.24 3.24 Eyring error (%) 30.94 18.10 17.66 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Sabine error (%) 30.24 17.27 16.83 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.02 0.02 Eyring error (%) 30.72 17.95 17.51 Accuracy of the predicted room absorptions: The predicted room-averaged absorptions, as well as the absorption coefficients, show errors below 18% from the 125-Hz band, while the errors are greater than 20% in the rest of the bands. These are shown in Table X.17. 536 X.4 Octave-band prediction for the ASTM E90 reverberation room – Case #3: Oblique-shaped with the longest z-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit as there no change in the modal compositions except the modal-width and, hence, the modal overlap, as shown in Table X.18. Table X.18: Predicted octave-band modal compositions for ASTM E90 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 13 69 437 Modal density, 𝑛(𝑓) 0.30 0.78 2.46 Modal spacing, 𝛿𝑓 3.12 1.25 0.40 Modal width, 𝛿𝑤 3.23 3.06 2.67 Modal overlap, 𝑀 0.95 2.37 6.54 Modal overlap factor, 𝑀𝑓 1.03 2.44 6.60 Table X.19: Predicted octave-band SPL statistics for ASTM E90 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 151.45 155.76 166.87 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 144.06 157.72 165.76 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 143.08 155.57 166.45 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 147.31 157.49 166.02 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 151.61 155.52 166.65 Predicted average SPL, ?̅?𝑝 (dB) 148.85 156.52 166.37 Standard deviation, 𝜎𝐿𝑝 (dB) 4.27 1.10 0.46 537 Table X.20: Predicted octave-band RT statistics for ASTM E90 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 4.26 3.43 2.45 Predicted RT @ Rec#1: 𝑇60 (s) 3.55 3.11 2.26 Predicted RT @ Rec#2: 𝑇60 (s) 4.15 3.06 2.59 Predicted RT @ Rec#3: 𝑇60 (s) 4.45 2.83 2.78 Predicted RT @ Rec#4: 𝑇60 (s) 3.70 3.61 4.00 Predicted RT @ Rec#5: 𝑇60 (s) 3.90 3.21 2.72 Predicted average RT: ?̅?60 (s) 3.92 3.15 2.77 Standard deviation, 𝜎𝑇60 (s) 0.36 0.29 0.67 Decay rate (dB/s) 15.29 19.07 21.68 Spatial uniformity of SPL: The predicted SPL deviations are shown in Table X.19. The overall deviations are greater than in the ‘without absorber’ case. Spatial uniformity of RT: The predicted RT deviations are shown in Table X.20. The overall deviations are greater than in the ‘without absorber’ case. Table X.21: Predicted octave-band decay curve curvature statistics for ASTM E90 oblique reverberation room with the longest z-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 0.57 3.66 8.19 𝜅 @ Rec#2 9.61 6.89 23.85 𝜅 @ Rec#3 12.51 11.56 26.45 𝜅 @ Rec#4 1.17 0.71 35.26 𝜅 @ Rec#5 4.20 6.77 28.59 Average, 𝜅 5.61 5.92 24.47 Modal, 𝜅 2.59 0.99 3.43 538 Linearity of decay curves: The curvatures are shown in Table X.21. The overall curvatures are greater than in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix Y.4. Table X.22: Predicted octave-band sample absorber absorption coefficient statistics for ASTM E90 oblique reverberation room with the longest z-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.22 0.30 0.41 Sabine error (%) 70.94 60.16 45.51 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.20 0.26 0.34 Eyring error (%) 73.92 65.63 55.38 Accuracy of the predicted sample absorption coefficients: As shown in Table X.22, the predicted Sabine absorption coefficients show errors around 45% from the 250-Hz band, while the Eyring predictions yield errors around 55% from the same band. 539 Appendix: Y Y.1 Third-octave-band prediction for the ASTM E90 reverberation room – Case #3: oblique-shaped with the longest z-dimension and without absorber Y.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 2.54 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.13 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 3.17 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 3.6 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.5 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 3.28 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.96 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.37 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.28 s540 Y.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, IN third-octave bands, at receiver #5 0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 6.474 s0 1-50050Time (s)45 50 55050010001500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 5.372 s0 1-50050Time (s)60 65 70-15000-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 5.102 s0 1-50050Time (s)70 80-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 4.802 s0 1-50050Time (s)90 100 110-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.315 s0 1-50050Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 5.544 s0 1-1000100Time (s)150 160 170012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 5.03 s0 1-1000100Time (s)180 200 220-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.069 s0 1-2000200Time (s)240 260 280-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 5.52 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)541 Y.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 5.448 s0 1-40-20020Time (s)45 50 55-1000-5000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.36 s0 1-20020Time (s)60 65 700500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 5.351 s0 1-40-2002040Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 4.77 s0 1-50050Time (s)90 100 110-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 4.787 s0 1-1000100Time (s)120 130 140-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 4.714 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 5.055 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.555 s0 1-400-2000200400Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 6.882 s0 1-5000500Time (s)300 350-10123x 104Frequency (Hz)Pressure (Pa)542 Y.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 5.748 s0 1-20020Time (s)45 50 55-800-600-400-2000200Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.635 s0 1-20020Time (s)60 65 70-3000-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 5.656 s0 1-20020Time (s)70 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 5.031 s0 1-50050Time (s)90 100 1100500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 5.087 s0 1-1000100Time (s)120 130 140-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 4.675 s0 1-1000100Time (s)150 160 170-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 4.656 s0 1-2000200Time (s)180 200 2200123x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.827 s0 1-2000200Time (s)240 260 280-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 4.818 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)543 Y.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 5 10 15-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 13.687 s0 1-50050Time (s)45 50 550200400Frequency (Hz)Pressure (Pa)0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 9.307 s0 1-50050Time (s)60 65 70-3000-2000-1000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 5.697 s0 1-40-20020Time (s)70 80-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.175 s0 1-50050Time (s)90 100 110050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 4.593 s0 1-1000100Time (s)120 130 140-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 4.871 s0 1-1000100Time (s)150 160 170-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 4.608 s0 1-2000200Time (s)180 200 220-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.868 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 5.747 s0 1-5000500Time (s)300 350-2-101x 104Frequency (Hz)Pressure (Pa)544 Y.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 8.395 s0 1-50050Time (s)45 50 55-1000-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.392 s0 1-50050Time (s)60 65 70-15000-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 5.118 s0 1-20020Time (s)70 80010002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 5.097 s0 1-50050Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 4.834 s0 1-50050Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 4.857 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 4.458 s0 1-1000100200Time (s)180 200 220-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.651 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 4.206 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)545 Y.2 Third-octave-band prediction for the ASTM E90 reverberation room – Case #3: oblique-shaped with the longest z-dimension and with absorber Y.2.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 2.15 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 3.42 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 4.52 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 3.6 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.51 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 3.23 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.72 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.46 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 2.19 s546 Y.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 4.364 s0 1-50050Time (s)45 50 550200400600Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 3.575 s0 1-50050Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 3.05 s0 1-50050Time (s)70 800100020003000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 3.737 s0 1-50050Time (s)90 100 110-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 3.056 s0 1-50050Time (s)120 130 140-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 2.683 s0 1-1000100Time (s)150 160 170-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.696 s0 1-1000100Time (s)180 200 220-20000200040006000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.523 s0 1-2000200Time (s)240 260 280-5000050001000015000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 2.802 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)547 Y.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 3.795 s0 1-20020Time (s)45 50 55-600-400-2000200Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 3.575 s0 1-20020Time (s)60 65 700200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 3.876 s0 1-20020Time (s)70 80-500050010001500Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 3.6 s0 1-50050Time (s)90 100 110-20000200040006000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.782 s0 1-1000100Time (s)120 130 140-5000050001000015000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 3.149 s0 1-50050Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.208 s0 1-1000100Time (s)180 200 220-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.617 s0 1-2000200Time (s)240 260 280-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 5.28 s0 1-5000500Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)548 Y.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 4.316 s0 1-1001020Time (s)45 50 55-2000200Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 3.645 s0 1-20020Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 3.065 s0 1-20020Time (s)70 80-5000500Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 2.57 s0 1-50050Time (s)90 100 110-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.746 s0 1-50050Time (s)120 130 140-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 2.609 s0 1-1000100Time (s)150 160 170-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 2.493 s0 1-2000200Time (s)180 200 2200500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.53 s0 1-2000200Time (s)240 260 280-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 3.353 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)549 Y.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 2 4 6 8 10-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 10.526 s0 1-50050Time (s)45 50 550100200Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 4.182 s0 1-50050Time (s)60 65 70-3000-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 2.661 s0 1-20020Time (s)70 80-1500-1000-5000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 2.421 s0 1-50050Time (s)90 100 110-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 3.294 s0 1-1000100Time (s)120 130 140-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.842 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.548 s0 1-1000100200Time (s)180 200 220-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.376 s0 1-2000200Time (s)240 260 280-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 2.956 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)550 Y.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 5.459 s0 1-50050Time (s)45 50 55-2000200400Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 3.613 s0 1-50050Time (s)60 65 70-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 3.786 s0 1-20020Time (s)70 80050010001500Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 3.458 s0 1-50050Time (s)90 100 110-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.266 s0 1-50050Time (s)120 130 140-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.807 s0 1-1000100Time (s)150 160 170-200002000400060008000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.708 s0 1-1000100Time (s)180 200 220-10000-50000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.31 s0 1-2000200Time (s)240 260 280-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 2.284 s0 1-5000500Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)551 Y.3 Octave-band prediction for the ASTM E90 reverberation room – Case #3: oblique-shaped with the longest z-dimension and without absorber Y.3.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands Y.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in bands, at receiver #5 Y.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 3.82 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.44 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.58 s0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 5.744 s0 1-1000100Time (s)60 80-15000-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.404 s0 1-1000100200Time (s)100 150-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 5.493 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.895 s0 1-50050Time (s)60 800500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 4.789 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 5.747 s0 1-50005001000Time (s)200 300-10123x 104Frequency (Hz)Pressure (Pa)552 Y.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 Y.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 Y.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.872 s0 1-50050Time (s)60 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 4.93 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.468 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 2 4 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 6.649 s0 1-50050Time (s)60 80-3000-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 4.887 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.993 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.374 s0 1-1000100Time (s)60 80-15000-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 4.824 s0 1-1000100Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.276 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)553 Y.4 Octave-band prediction for the ASTM E90 reverberation room – Case #3: oblique-shaped with the longest z-dimension and with absorber Y.4.1 Theoretical modal decay curves and the corresponding RTs, in octave bands Y.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 Y.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 4.26 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.43 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.45 s0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 3.903 s0 1-1000100Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 3.212 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.718 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 3.701 s0 1-50050Time (s)60 800200040006000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 3.61 s0 1-2000200Time (s)100 150-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 3.996 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)554 Y.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 Y.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 Y.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 4.448 s0 1-2002040Time (s)60 80-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.833 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 2.78 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 4.145 s0 1-50050Time (s)60 80-3000-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 3.063 s0 1-2000200Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.591 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 3.549 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 3.113 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.258 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)555 Appendix: Z Z.1 Third-octave-band prediction for the ASTM E90 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table Z.1, the corresponding statistically-based cut-off frequency limit is 125 Hz, which suggests that prediction can be done from the 160-Hz third-octave band. The modal density, modal spacing, modal-width and modal-overlap factor values corresponding to the 160-Hz third-octave are 1.51, 0.67, 0.39 and 0.58, respectively. The Schroeder low-frequency limit, considering to a band-averaged value of 5.8 s for the RT, as shown in Table Z.3, is 531 Hz. Table Z.1: Predicted third-octave-band modal compositions for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 4 7 10 22 37 67 127 243 Modal density, 𝑛(𝑓) 0.25 0.29 0.37 0.43 0.76 1.00 1.46 2.19 3.33 Modal spacing, 𝛿𝑓 1.52 2.65 2.58 2.29 1.35 1.00 0.68 0.46 0.30 Modal width, 𝛿𝑤 0.36 0.40 0.39 0.41 0.43 0.45 0.47 0.48 0.50 Modal overlap, 𝑀 0.09 0.11 0.14 0.18 0.32 0.45 0.68 1.05 1.65 Modal overlap factor, 𝑀𝑓 0.24 0.15 0.15 0.18 0.32 0.45 0.69 1.05 1.65 Spatial uniformity of SPL: From Table Z.2, the predicted standard deviations of SPL are less than 1.5 dB from the 100-Hz third-octave band, while the deviations are greater than 1.5 dB in the rest of the bands. Accuracy of the predicted SPLs: As presented in Table Z.2, the errors associated with the predicted SPLs are always less than 1% from the 80-Hz third-octave band for both Sabine and Eyring prediction cases. 556 Table Z.2: Predicted third-octave-band SPL statistics for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 149.5 143.7 147.2 154.1 160.4 157.9 161.8 167.9 169.5 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 150.4 137.9 149.2 153.5 158.9 158.0 164.7 168.3 171.2 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 153.5 146.6 139.7 155.9 158.7 158.0 166.3 166.3 170.0 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 153.6 148.7 154.4 153.6 158.3 159.0 166.2 165.8 173.0 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 151.4 149.8 153.5 154.7 159.1 159.8 163.5 168.0 173.0 Predicted average SPL, ?̅?𝑝 (dB) 152.0 146.9 151.1 154.5 159.1 158.6 164.8 167.4 171.6 Standard deviation, 𝜎𝐿𝑝 (dB) 1.9 5.1 6.4 1.0 0.8 0.8 1.9 1.1 1.7 Expected Sabine SPL, 𝐿𝑝 (dB) 154.1 149.3 152.6 156.0 160.4 159.6 165.7 168.0 172.0 Sabine error (%) 1.4 1.6 1.0 1.0 0.8 0.6 0.6 0.4 0.2 Expected Eyring SPL, 𝐿𝑝 (dB) 154.0 149.3 152.5 156.0 160.4 159.6 165.7 168.0 171.9 Eyring error (%) 1.33 1.58 0.96 0.97 0.78 0.61 0.54 0.34 0.21 Table Z.3: Predicted third-octave-band RT statistics for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 6.55 7.27 6.85 6.32 5.99 5.76 5.43 5.30 5.19 Predicted RT @ Rec#1: 𝑇60 (s) 6.21 5.73 5.82 6.95 4.86 4.74 5.56 4.59 4.88 Predicted RT @ Rec#2: 𝑇60 (s) 6.73 6.17 6.60 5.78 5.49 5.27 5.14 4.19 4.13 Predicted RT @ Rec#3: 𝑇60 (s) 7.69 5.43 5.98 4.95 4.58 4.60 5.10 5.17 4.33 Predicted RT @ Rec#4: 𝑇60 (s) 6.58 5.40 5.88 4.73 4.91 4.94 4.17 4.69 5.38 Predicted RT @ Rec#5: 𝑇60 (s) 7.47 5.14 5.62 4.52 5.53 4.89 3.58 4.59 4.44 Predicted average RT: ?̅?60 (s) 6.89 5.55 5.96 5.25 5.05 4.87 4.59 4.63 4.59 Standard deviation, 𝜎𝑇60 (s) 0.62 0.39 0.37 1.01 0.42 0.25 0.82 0.35 0.50 Expected Sabine RT: 𝑇60 (s) 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 3.67 Sabine error (%) 87.8 51.3 62.4 43.1 37.5 32.8 24.9 26.0 25.1 Expected Eyring RT, 𝑇60 (s) 3.63 3.63 3.63 3.63 3.63 3.63 3.63 3.63 3.63 Eyring error (%) 89.7 52.8 64.1 44.6 38.9 34.2 26.2 27.3 26.3 Decay rate (dB/s) 8.70 10.81 10.06 11.42 11.89 12.3 13.0 12.9 13.07 557 Table Z.4: Predicted third-octave-band decay-curve curvature statistics for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 19.5 24.1 10.2 62.1 10.5 16.3 12.8 19.2 9.6 𝜅 @ Rec#2 29.7 39.0 7.1 22.2 32.3 24.6 17.4 13.2 13.7 𝜅 @ Rec#3 15.7 9.6 8.0 7.2 26.8 4.4 15.8 17.6 4.0 𝜅 @ Rec#4 0.6 17.8 5.6 18.8 16.6 8.2 18.5 18.3 18.1 𝜅 @ Rec#5 36.5 3.6 17.0 23.2 14.2 9.1 15.0 25.7 9.6 Average, 𝜅 20.4 18.82 9.59 26.69 20.08 12.52 15.91 18.80 11.00 Modal, 𝜅 0.01 0.06 0.07 0.00 0.01 0.01 0.01 0.01 0.00 Spatial uniformity of RT: From Table Z.3, it is clear that the standard deviations of RT, a measure of spatial variation, are less than 0.5 s from the 125-Hz third-octave band with the exception in the 200-Hz band. Accuracy of the predicted RTs: As presented in Table Z.3, the predicted RTs show errors less than, or around, 25% from the 200-Hz third-octave band for both Sabine and Eyring prediction cases, while the errors are greater than 25% in the rest of the bands. Linearity of temporal decay curves: From Table Z.4, the predicted curvatures of the energy-decay curves vary within 10 to 20 from the 125-Hz band. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix AA.1. Table Z.5: Predicted third-octave-band source-power level statistics for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 153.4 148.6 151.9 155.3 159.8 159.0 165.1 167.3 171.3 Sabine error (%) 55.3 40.1 45.5 35.7 32.3 29.2 23.6 24.4 23.7 Eyring error (%) 56.5 41.3 46.7 36.9 33.5 30.4 24.8 25.6 24.9 Predicted, ?̅?𝑤 (dB) (1 source) 150.4 145.6 148.9 152.3 156.8 156.0 162.1 164.3 168.3 558 Table Z.6: Predicted third-octave-band room absorption statistics for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 1.90 2.36 2.20 2.50 2.60 2.69 2.86 2.84 2.86 Sabine expected absorption, 𝐴 (m2) 3.58 3.58 3.58 3.58 3.58 3.58 3.58 3.58 3.58 Sabine error (%) 46.8 33.9 38.4 30.1 27.3 24.7 20.0 20.6 20.0 Eyring expected absorption, 𝐴 (m2) 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 3.61 Eyring error (%) 47.3 34.6 39.0 30.8 28.0 25.5 20.8 21.4 20.8 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 Sabine error (%) 46.8 33.9 38.4 30.1 27.3 24.7 20.0 20.6 20.0 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 Eyring error (%) 47.0 34.3 38.8 30.6 27.8 25.3 20.6 21.3 20.7 Accuracy of the predicted source-power levels: As presented in Table Z.5, the source-power-level errors are less than, or around, 25% from the 200-Hz third-octave band. Accuracy of the predicted room absorptions: As presented in Table Z.6, the errors are less than, or around, 25% from the 160-Hz third-octave band, and greater than 25% in the rest of the bands. The same is true for the room absorption coefficient prediction case, as the absorption coefficient predictions depend on the accuracy of the absorption predictions. 559 Z.2 Third-octave-band prediction for the ASTM E90 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the modal compositions except the modal-width and, hence, the modal-overlap factor as shown in Table Z.7. The cut-off frequency limit will remain be the same as before. Table Z.7: Predicted third-octave-band modal compositions for ASTM E90 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 3 3 8 11 23 38 69 130 245 Modal density, 𝑛(𝑓) 0.25 0.21 0.42 0.48 0.79 1.03 1.50 2.24 3.36 Modal spacing, 𝛿𝑓 2.43 1.61 2.49 2.14 1.21 0.98 0.67 0.44 0.30 Modal width, 𝛿𝑤 1.05 1.29 4.96 4.19 3.98 4.06 3.04 3.74 3.08 Modal overlap, 𝑀 0.26 0.28 2.09 2.00 3.16 4.17 4.56 8.38 10.35 Modal overlap factor, 𝑀𝑓 0.43 0.80 1.99 1.96 3.28 4.16 4.56 8.42 10.34 Table Z.8: Predicted third-octave-band SPL statistics for ASTM E90 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 148.1 143.6 145.2 153.2 155.4 153.3 159.9 165.4 164.9 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 149.3 134.7 145.9 150.5 155.8 155.6 159.7 165.0 167.5 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 151.4 143.9 139.9 148.6 152.9 150.2 160.7 162.3 164.8 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 151.5 145.6 149.2 146.8 154.3 155.4 163.0 163.3 168.8 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 149.7 145.5 145.6 146.7 154.1 151.4 158.6 163.2 169.5 Predicted average SPL, ?̅?𝑝 (dB) 150.2 143.9 146.1 149.9 154.6 153.7 160.6 164.0 167.5 Standard deviation, 𝜎𝐿𝑝 (dB) 1.43 4.73 3.46 2.87 1.16 2.44 1.66 1.32 2.22 560 Table Z.9: Predicted third-octave-band RT statistics for ASTM E90 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 3.96 1.90 2.17 3.58 2.83 2.76 2.74 2.02 1.93 Predicted RT @ Rec#1: 𝑇60 (s) 3.75 2.62 2.19 3.67 2.28 2.39 2.39 2.02 2.17 Predicted RT @ Rec#2: 𝑇60 (s) 4.45 5.64 2.68 3.73 2.42 2.33 2.53 1.83 2.30 Predicted RT @ Rec#3: 𝑇60 (s) 4.25 2.10 3.17 3.40 2.02 2.17 2.45 1.83 1.85 Predicted RT @ Rec#4: 𝑇60 (s) 3.79 2.22 1.83 3.45 2.35 2.18 2.11 1.67 2.02 Predicted RT @ Rec#5: 𝑇60 (s) 4.37 2.29 2.15 3.30 2.25 2.36 2.19 2.40 2.46 Predicted average RT: ?̅?60 (s) 4.10 2.60 2.32 3.50 2.25 2.28 2.32 1.92 2.14 Stnd. deviation, 𝜎𝑇60 (s) 0.33 1.56 0.53 0.18 0.15 0.10 0.18 0.28 0.24 Decay rate (dB/s) 14.6 23.0 25.9 17.1 26.6 26.3 25.8 31.3 28.1 Spatial uniformity of SPL: The standard deviations of the SPL are presented in Table Z.8. It is noticeable that the deviations are greater than those obtained in the ‘without absorber’ case. Spatial uniformity of RT: The standard deviations of the RT are presented in Table Z.9. It is obvious that the deviations are greater than the ‘without absorber’ case in almost all third-octave bands. Table Z.10: Predicted third-octave-band decay-curve curvature statistics for ASTM E90 oblique reverberation room with the longest x-dimension and with absorber. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) 𝜅 @ Rec#1 26.0 89.1 31.9 1.0 41.4 31.1 14.0 19.5 27.2 𝜅 @ Rec#2 26.1 91.9 35.9 1.9 4.6 20.5 13.2 10.4 47.7 𝜅 @ Rec#3 45.9 44.3 82.1 1.5 35.8 5.9 13.2 6.1 8.4 𝜅 @ Rec#4 0.2 89.1 16.4 0.7 20.8 5.9 9.7 8.7 21.3 𝜅 @ Rec#5 48.0 33.6 49.0 1.2 28.8 13.8 13.0 23.4 82.1 Average, 𝜅 29.3 69.6 43.1 1.3 26.3 15.5 12.6 13.6 37.3 Modal, 𝜅 0.01 0.16 0.32 0.21 3.26 2.02 3.86 2.55 2.10 561 Table Z.11: Predicted third-octave-band sample absorber absorption coefficient statistics for ASTM E90 oblique reverberation room with the longest x-dimension. 50 63 80 100 125 160 200 250 315⏞ Third−octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.20 0.41 0.53 0.19 0.50 0.47 0.43 0.62 0.50 Sabine error (%) 73.7 45.8 30.0 74.7 34.8 38.1 43.6 18.9 33.6 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.18 0.34 0.41 0.17 0.39 0.38 0.35 0.46 0.40 Eyring error (%) 76.2 55.6 45.7 77.0 48.6 50.6 54.1 39.5 47.9 Linearity of decay curves: The curvatures of the energy-decay curves as presented in Table Z.10 reveal that the curvatures are greater than the values obtained in the ‘without absorber’ case. The modal-decay curves and the energy-decay curves predicted in third-octave bands are presented in Appendix AA.2. Accuracy of the sample absorption coefficients: The predicted absorption coefficients of an absorber sample are presented in Table Z.11. It is noticeable that the predicted values deviate from the theoretical values to a great extent. 562 Z.3 Octave-band prediction for the ASTM E90 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and without absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: Based on the modal compositions as shown in Table Z.12, the statistically-based cut-off frequency is 78 Hz, which suggests that prediction can be done from the 125-Hz octave band. The modal density, modal spacing, modal-width and modal-overlap factor corresponding to the 125-Hz band are 1.1, 0.9, 0.38 and 0.42, respectively. On the other hand, the pure-tone-based Schroeder limit, corresponding to a modal overlap factor of 3 and a band-averaged RT of 5.5 s, as shown in Table Z.14, is 517 Hz. Table Z.12: Predicted octave-band modal compositions for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 13 68 439 Modal density, 𝑛(𝑓) 0.30 0.76 2.47 Modal spacing, 𝛿𝑓 3.32 1.32 0.40 Modal width, 𝛿𝑤 0.38 0.43 0.49 Modal overlap, 𝑀 0.11 0.33 1.20 Modal overlap factor, 𝑀𝑓 0.12 0.33 1.20 Spatial uniformity of SPL: The predicted SPL standard deviations are less than or around 1.5 dB from the 125-Hz band, as shown in Table Z.13. Accuracy of the predicted SPLs: As shown in Table Z.13, the predicted SPLs show errors below or around 1% from the 125-Hz octave-band. Spatial uniformity of RT: The predicted RT standard deviations are less than 0.5 s in all octave bands, as shown in Table Z.14. 563 Table Z.13: Predicted octave-band SPL statistics for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 148.19 159.12 168.35 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 149.13 158.24 169.74 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 150.57 158.57 168.83 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 153.74 158.33 170.61 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 152.71 159.24 170.68 Predicted average SPL, ?̅?𝑝 (dB) 151.37 158.72 169.74 Standard deviation, 𝜎𝐿𝑝 (dB) 2.41 0.46 1.05 Expected Sabine SPL, 𝐿𝑝 (dB) 153.12 159.89 170.37 Sabine error (%) 1.14 0.73 0.37 Expected Eyring SPL, 𝐿𝑝 (dB) 153.08 159.85 170.33 Eyring error (%) 1.12 0.71 0.34 Table Z.14: Predicted octave-band RT statistics for ASTM E90 oblique reverberation room with the longest x-dimension tion and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 7.00 5.96 5.27 Predicted RT @ Rec#1: 𝑇60 (s) 6.23 4.97 4.77 Predicted RT @ Rec#2: 𝑇60 (s) 6.55 5.87 4.29 Predicted RT @ Rec#3: 𝑇60 (s) 7.19 4.77 4.61 Predicted RT @ Rec#4: 𝑇60 (s) 6.29 5.01 5.04 Predicted RT @ Rec#5: 𝑇60 (s) 6.14 5.44 4.46 Predicted average RT: ?̅?60 (s) 6.46 5.18 4.62 Standard deviation, 𝜎𝑇60 (s) 0.43 0.44 0.29 Expected Sabine RT: 𝑇60 (s) 3.67 3.67 3.67 Sabine error (%) 75.90 41.20 25.79 Expected Eyring RT, 𝑇60 (s) 3.63 3.63 3.63 Eyring error (%) 77.68 42.63 27.06 Decay rate (dB/s) 9.29 11.58 12.99 564 Table Z.15: Predicted octave-band decay-curve curvature statistics for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 10.41 7.01 3.28 𝜅 @ Rec#2 6.35 12.91 7.02 𝜅 @ Rec#3 11.86 7.12 3.59 𝜅 @ Rec#4 1.87 5.47 6.57 𝜅 @ Rec#5 10.22 6.90 5.04 Average, 𝜅 8.14 7.88 5.10 Modal, 𝜅 0.07 0.02 0.01 Accuracy of the predicted RTs: As shown in Table Z.14, the predicted RTs show errors below or around 25% from the 250-Hz band, while the errors are greater than 25% in the rest of the bands. Degree of linearity of decay curves: The curvature values are within 5 to 10 in all octave-bands, as shown in Table Z.15. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix AA.3. Accuracy of the predicted source-power levels: The predicted source-power levels show errors below or around 25% from the 250-Hz band, as shown in Table Z.16. Table Z.16: Predicted octave-band source-power level statistics for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted, ?̅?𝑤 (dB) (2 sources) 152.45 159.22 169.70 Sabine error (%) 51.07 34.53 24.26 Eyring error (%) 52.26 35.73 25.46 Predicted, ?̅?𝑤 (dB) (1 source) 149.44 156.21 166.69 565 Table Z.17: Predicted octave-band room absorption statistics for ASTM E90 oblique reverberation room with the longest x-dimension and without absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Average predicted absorption, 𝐴 (m2) 2.03 2.53 2.84 Sabine expected absorption, 𝐴 (m2) 3.58 3.58 3.58 Sabine error (%) 43.15 29.18 20.50 Eyring expected absorption, 𝐴 (m2) 3.61 3.61 3.61 Eyring error (%) 43.72 29.89 21.30 Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.02 0.02 0.02 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.02 Sabine error (%) 43.15 29.18 20.50 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.01 0.01 0.02 Eyring error (%) 43.47 29.68 21.13 Accuracy of the predicted room absorptions: The predicted room-averaged absorptions, as well as the absorption coefficients, show errors below or around 20% from the 250-Hz band for both Sabine and Eyring prediction cases as shown in Table Z.17. 566 Z.4 Octave-band prediction for the ASTM E90 reverberation room – Case #4: Oblique-shaped with the longest x-dimension and with absorber The results are discussed in terms of the descriptors described in chapter 3 (see section 3.2) using the calculation procedure as described also in chapter 3 (see section 3.3). Cut-off frequency: There is no change in the statistically-based limit as there no change in the modal compositions except the modal-width and, hence, the modal-overlap as shown in Table Z.18. Table Z.18: Predicted octave-band modal compositions for ASTM E90 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Number of modes, 𝑁(𝑓) 14 70 446 Modal density, 𝑛(𝑓) 0.32 0.79 2.51 Modal spacing, 𝛿𝑓 3.13 1.25 0.40 Modal width, 𝛿𝑤 3.33 4.11 3.26 Modal overlap, 𝑀 1.06 3.23 8.18 Modal overlap factor, 𝑀𝑓 1.06 3.30 8.19 Table Z.19: Predicted octave-band SPL statistics for ASTM E90 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Predicted SPL@ Rec#1: 𝐿𝑝 (dB) 146.87 154.97 164.92 Predicted SPL@ Rec#2: 𝐿𝑝 (dB) 147.19 155.32 166.03 Predicted SPL@ Rec#3: 𝐿𝑝 (dB) 148.47 151.83 163.83 Predicted SPL@ Rec#4: 𝐿𝑝 (dB) 150.22 154.38 166.79 Predicted SPL@ Rec#5: 𝐿𝑝 (dB) 148.31 152.54 166.87 Predicted average SPL, ?̅?𝑝 (dB) 148.38 154.02 165.84 Standard deviation, 𝜎𝐿𝑝 (dB) 1.33 1.56 1.31 567 Table Z.20: Predicted octave-band RT statistics for ASTM E90 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) Modal RT, 𝑇60 (s) 3.64 3.16 2.35 Predicted RT @ Rec#1: 𝑇60 (s) 3.14 2.95 2.19 Predicted RT @ Rec#2: 𝑇60 (s) 4.05 2.81 2.10 Predicted RT @ Rec#3: 𝑇60 (s) 4.08 2.44 1.95 Predicted RT @ Rec#4: 𝑇60 (s) 3.54 2.41 2.03 Predicted RT @ Rec#5: 𝑇60 (s) 4.37 2.53 2.41 Predicted average RT: ?̅?60 (s) 3.78 2.61 2.12 Standard deviation, 𝜎𝑇60 (s) 0.49 0.24 0.18 Decay rate (dB/s) 15.86 22.99 28.27 Spatial uniformity of SPL: The predicted SPL deviations are greater than in the ‘without absorber’ case, as shown in Table Z.19. Spatial uniformity of RT: The predicted RT deviations are greater than in the ‘without absorber’ case, as shown in Table Z.20. Linearity of temporal decay curves: The curvatures are greater than in the ‘without absorber’ case, as shown in Table Z.21. The modal-decay curves and the energy-decay curves predicted in octave bands are presented in Appendix AA.4. Table Z.21: Predicted octave-band decay-curve curvature statistics for ASTM E90 oblique reverberation room with the longest x-dimension and with absorber. 63 125 250 ⏞ Octave−band center frequency (Hz) 𝜅 @ Rec#1 16.83 10.82 20.82 𝜅 @ Rec#2 12.70 3.80 19.31 𝜅 @ Rec#3 38.57 15.02 3.51 𝜅 @ Rec#4 12.16 6.18 12.78 𝜅 @ Rec#5 36.43 30.54 48.45 Average, 𝜅 23.34 13.27 20.97 Modal, 𝜅 1.70 2.71 6.84 568 Table Z.22: Predicted octave-band sample absorber absorption coefficient statistics for ASTM E90 oblique reverberation room with the longest x-dimension. 63 125 250 ⏞ Octave−band center frequency (Hz) Random expected absorption coeff, 𝛼𝑟𝑎𝑛 0.76 0.76 0.76 Sabine predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.13 0.23 0.31 Sabine error (%) 82.35 69.31 58.94 Eyring predicted absorption coeff, 𝛼𝑟𝑒𝑣 0.13 0.21 0.27 Eyring error (%) 83.49 72.63 64.73 Accuracy of the predicted sample absorption coefficients: As shown in Table Z.22, the predicted Sabine absorption coefficients show errors around 60% from the 250-Hz band, while the Eyring predictions yield errors around 65% from the same band. 569 Appendix: AA AA.1 Third-octave-band prediction for the ASTM E90 reverberation room – Case #4: oblique-shaped with the longest x-dimension and without absorber AA.1.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 2 4 6-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 6.51 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 5.53 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 6.14 s0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 5.55 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 5.24 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 5 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 4.78 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 4.64 s0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 4.49 s570 AA.1.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 5.726 s0 1-40-2002040Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 4.959 s0 1-20020Time (s)60 65 70020004000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 5.371 s0 1-50050Time (s)70 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 4.755 s0 1-50050Time (s)90 100 110-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.146 s0 1-1000100Time (s)120 130 1400500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 4.869 s0 1-1000100Time (s)150 160 170-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 4.191 s0 1-200-1000100Time (s)180 200 220-5000050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.765 s0 1-5000500Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 4.19 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)571 AA.1.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 5.921 s0 1-2002040Time (s)45 50 55-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 5.214 s0 1-40-2002040Time (s)60 65 70-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 5.327 s0 1-50050Time (s)70 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 4.94 s0 1-50050Time (s)90 100 1100500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 4.822 s0 1-1000100Time (s)120 130 140-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 4.822 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 4.34 s0 1-2000200Time (s)180 200 220-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.338 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 4.919 s0 1-100001000Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)572 AA.1.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 5.797 s0 1-50050Time (s)45 50 55-6000-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.111 s0 1-40-2002040Time (s)60 65 70-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 6.078 s0 1-40-20020Time (s)70 80050010001500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 4.827 s0 1-50050Time (s)90 100 110050001000015000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 4.475 s0 1-1000100Time (s)120 130 140-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 4.5 s0 1-1000100Time (s)150 160 170-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 4.618 s0 1-2000200Time (s)180 200 220-3-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.455 s0 1-2000200Time (s)240 260 280-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 4.32 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)573 AA.1.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 5.773 s0 1-50050Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 2 4 6 8-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 7.342 s0 1-20020Time (s)60 65 70-5000500Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 5.932 s0 1-50050Time (s)70 80-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 5.235 s0 1-50050Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.319 s0 1-1000100Time (s)120 130 140-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 4.861 s0 1-1000100Time (s)150 160 170-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 4.399 s0 1-2000200Time (s)180 200 220-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.296 s0 1-2000200Time (s)240 260 280-1012x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 4.484 s0 1-5000500Time (s)300 350-202x 104Frequency (Hz)Pressure (Pa)574 AA.1.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 5.847 s0 1-20020Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 5.685 s0 1-20020Time (s)60 65 70-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 5.468 s0 1-20020Time (s)70 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 5.608 s0 1-50050Time (s)90 100 110-200002000400060008000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 4.56 s0 1-1000100Time (s)120 130 140-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 4.904 s0 1-1000100Time (s)150 160 170-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 4.807 s0 1-2000200Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 3.929 s0 1-2000200Time (s)240 260 28001000020000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 4.483 s0 1-5000500Time (s)300 350-1012x 104Frequency (Hz)Pressure (Pa)575 AA.2 Third-octave-band prediction for the ASTM E90 reverberation room – Case #4: oblique-shaped with the longest x-dimension and with absorber AA.2.1 Theoretical modal-decay curves and the corresponding RTs, in third-octave bands 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 50-Hz band Time (s)Level (dB)T60 = 3.96 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 1.9 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 80-Hz band Time (s)Level (dB)T60 = 2.17 s0 1 2 3-70-60-50-40-30-20-100Modal RT for the 100-Hz band Time (s)Level (dB)T60 = 3.58 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 2.83 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 160-Hz band Time (s)Level (dB)T60 = 2.76 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 200-Hz band Time (s)Level (dB)T60 = 2.74 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.02 s0 0.5 1 1.5 2-70-60-50-40-30-20-100Modal RT for the 315-Hz band Time (s)Level (dB)T60 = 1.93 s576 AA.2.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #5 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #5Time (s)Level (dB) T60 = 4.372 s0 1-40-2002040Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 2.289 s0 1-40-2002040Time (s)60 65 70010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #5Time (s)Level (dB) T60 = 2.153 s0 1-20020Time (s)70 80-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #5Time (s)Level (dB) T60 = 3.299 s0 1-20020Time (s)90 100 110-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.252 s0 1-1000100Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #5Time (s)Level (dB) T60 = 2.363 s0 1-50050Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #5Time (s)Level (dB) T60 = 2.191 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.397 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #5Time (s)Level (dB) T60 = 2.457 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)577 AA.2.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #4 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #4Time (s)Level (dB) T60 = 3.786 s0 1-20020Time (s)45 50 55-8000-6000-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 2.224 s0 1-20020Time (s)60 65 70010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #4Time (s)Level (dB) T60 = 1.829 s0 1-40-2002040Time (s)70 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #4Time (s)Level (dB) T60 = 3.449 s0 1-40-2002040Time (s)90 100 110-10000100020003000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.349 s0 1-1000100Time (s)120 130 140-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #4Time (s)Level (dB) T60 = 2.183 s0 1-50050Time (s)150 160 170-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #4Time (s)Level (dB) T60 = 2.112 s0 1-400-2000200Time (s)180 200 220-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 1.667 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #4Time (s)Level (dB) T60 = 2.018 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)578 AA.2.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #3 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #3Time (s)Level (dB) T60 = 4.247 s0 1-40-2002040Time (s)45 50 55-6000-4000-2000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 2.104 s0 1-20020Time (s)60 65 70-100001000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #3Time (s)Level (dB) T60 = 3.171 s0 1-20020Time (s)70 80-5000500Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #3Time (s)Level (dB) T60 = 3.403 s0 1-20020Time (s)90 100 110020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.015 s0 1-50050Time (s)120 130 140-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #3Time (s)Level (dB) T60 = 2.17 s0 1-50050Time (s)150 160 170-1000010002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #3Time (s)Level (dB) T60 = 2.45 s0 1-2000200Time (s)180 200 220-8000-6000-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 1.829 s0 1-2000200Time (s)240 260 280-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #3Time (s)Level (dB) T60 = 1.853 s0 1-400-2000200400Time (s)300 350-10000-500005000Frequency (Hz)Pressure (Pa)579 AA.2.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #2 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #2Time (s)Level (dB) T60 = 4.449 s0 1-50050Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.636 s0 1-20020Time (s)60 65 70-400-2000200Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #2Time (s)Level (dB) T60 = 2.682 s0 1-50050Time (s)70 80-2000-100001000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #2Time (s)Level (dB) T60 = 3.732 s0 1-50050Time (s)90 100 110-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.416 s0 1-100-50050Time (s)120 130 140-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #2Time (s)Level (dB) T60 = 2.331 s0 1-1000100Time (s)150 160 170-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #2Time (s)Level (dB) T60 = 2.532 s0 1-1000100200Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 1.825 s0 1-2000200Time (s)240 260 280-10000-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #2Time (s)Level (dB) T60 = 2.297 s0 1-5000500Time (s)300 350-101x 104Frequency (Hz)Pressure (Pa)580 AA.2.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in third-octave bands, at receiver #1 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 50-Hz band at receiver #1Time (s)Level (dB) T60 = 3.745 s0 1-2002040Time (s)45 50 55-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 2.622 s0 1-40-2002040Time (s)60 65 70-2000-10000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 80-Hz band at receiver #1Time (s)Level (dB) T60 = 2.188 s0 1-20020Time (s)70 80-2000-10000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 100-Hz band at receiver #1Time (s)Level (dB) T60 = 3.668 s0 1-50050Time (s)90 100 11002000400060008000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 2.281 s0 1-1000100Time (s)120 130 140-4000-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 160-Hz band at receiver #1Time (s)Level (dB) T60 = 2.386 s0 1-1000100Time (s)150 160 170-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 200-Hz band at receiver #1Time (s)Level (dB) T60 = 2.385 s0 1-1000100Time (s)180 200 220-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.017 s0 1-2000200Time (s)240 260 280-50000500010000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 315-Hz band at receiver #1Time (s)Level (dB) T60 = 2.171 s0 1-2000200Time (s)300 350-10000-500005000Frequency (Hz)Pressure (Pa)581 AA.3 Octave-band prediction for the ASTM E90 reverberation room – Case #4: oblique-shaped with the longest x-dimension and without absorber M.3.1 Theoretical modal-decay curves and the corresponding RTs, in octave bands AA.3.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 AA.3.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4 5 6-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 6.19 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 5.23 s0 1 2 3 4 5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 4.6 s0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 5.953 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 5.171 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 4.31 s0 1-100001000Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 6.275 s0 1-50050Time (s)60 80-10000-50000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 5.003 s0 1-1000100200Time (s)100 150-50000500010000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 4.608 s0 1-100001000Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)582 AA.3.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 AA.3.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 AA.3.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 5.825 s0 1-50050Time (s)60 80-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 4.585 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 4.371 s0 1-5000500Time (s)200 300-3-2-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 5.821 s0 1-50050Time (s)60 80-4000-200002000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 5.532 s0 1-2000200Time (s)100 150-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 4.356 s0 1-5000500Time (s)200 300-202x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4 5 6-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 6.26 s0 1-40-200204060Time (s)60 80-4000-20000Frequency (Hz)Pressure (Pa)0 1 2 3 4 5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 4.782 s0 1-2000200Time (s)100 150-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 4.282 s0 1-5000500Time (s)200 300-1012x 104Frequency (Hz)Pressure (Pa)583 AA.4 Octave-band prediction for the ASTM E90 reverberation room – Case #4: oblique-shaped with the longest x-dimension and with absorber AA.4.1 Theoretical modal decay curves and the corresponding RTs, in octave bands AA.4.2 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #5 AA.4.3 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #4 0 1 2 3 4-70-60-50-40-30-20-100Modal RT for the 63-Hz band Time (s)Level (dB)T60 = 3.64 s0 0.5 1 1.5 2 2.5 3-70-60-50-40-30-20-100Modal RT for the 125-Hz band Time (s)Level (dB)T60 = 3.16 s0 0.5 1 1.5 2 2.5-70-60-50-40-30-20-100Modal RT for the 250-Hz band Time (s)Level (dB)T60 = 2.35 s0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #5Time (s)Level (dB) T60 = 4.374 s0 1-50050Time (s)60 80-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #5Time (s)Level (dB) T60 = 2.527 s0 1-1000100Time (s)100 150-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #5Time (s)Level (dB) T60 = 2.41 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 1 2 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #4Time (s)Level (dB) T60 = 3.543 s0 1-50050Time (s)60 80-8000-6000-4000-200002000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #4Time (s)Level (dB) T60 = 2.414 s0 1-1000100Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #4Time (s)Level (dB) T60 = 2.028 s0 1-100001000Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)584 AA.4.4 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #3 AA.4.5 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #2 AA.4.6 Predicted frequency responses (FR), impulse responses (IR) and energy-decay curves (EDC), as well as the corresponding RTs, in octave bands, at receiver #1 0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #3Time (s)Level (dB) T60 = 4.08 s0 1-50050Time (s)60 80-6000-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #3Time (s)Level (dB) T60 = 2.436 s0 1-1000100Time (s)100 150-2000020004000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #3Time (s)Level (dB) T60 = 1.945 s0 1-5000500Time (s)200 300-10000-500005000Frequency (Hz)Pressure (Pa)0 1 2 3 4-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #2Time (s)Level (dB) T60 = 4.046 s0 1-50050Time (s)60 80-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #2Time (s)Level (dB) T60 = 2.805 s0 1-2000200Time (s)100 150-500005000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #2Time (s)Level (dB) T60 = 2.096 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 63-Hz band at receiver #1Time (s)Level (dB) T60 = 3.141 s0 1-40-2002040Time (s)60 80-4000-20000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2 2.5 3-60-40-200Predicted EDC\RT, FR, IR: 125-Hz band at receiver #1Time (s)Level (dB) T60 = 2.949 s0 1-2000200Time (s)100 150-4000-200002000400060008000Frequency (Hz)Pressure (Pa)0 0.5 1 1.5 2-60-40-200Predicted EDC\RT, FR, IR: 250-Hz band at receiver #1Time (s)Level (dB) T60 = 2.189 s0 1-5000500Time (s)200 300-101x 104Frequency (Hz)Pressure (Pa)
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivs 2.5 Canada
Graduate
Diffuse sound fields, reverberation-room methods and the effectiveness of reverberation-room designs
Text