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Beam-tracing prediction of room-to-room sound transmission Mahmud, Md Amin 2017

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Beam-Tracing Prediction of Room-To-Room Sound Transmission   by Md Amin Mahmud B.Sc., Islamic University of Technology (IUT), Bangladesh, 2014   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) December 2017 © Md Amin Mahmud, 2017   ii   Abstract  Modeling sound transmission is a challenging task. An existing beam-tracing model for empty, parallelepiped rooms with specularly-reflecting surfaces is extended to predict room-to-room sound transmission between a source and receiver rooms separated by a common wall. This wall is modeled as one locally-reacting homogenous partition with frequency-independent transmission loss. Besides, sound transmission is modeled in Ray-Tracing (CATT-TM) and FEM (COMSOL). A reference configuration consists of two identical reverberation rooms is chosen following the recommendations of the literature and most of the prescriptions of the reverberation room standard, ASTM 3423. The capability of various room-to-room predictions models, in particular, the phase and energy-based beam tracing models (PBTM, EBTM) in reproducing the results of the diffuse-field theory is investigated. Both EBTM and CATT-TM are found to be reasonably accurate in reproducing the diffuse sound field for a reverberation room (i.e. for diffuse sound fields). However, the predicted levels deviate considerably from the diffuse-field theory with changes in the acoustical characteristics of the room (room aspect ratio, the magnitude of the surface absorption and surface absorption distribution (i.e. for non-diffuse sound fields). EBTM has been validated in both source and receiver rooms through existing results from ODEON in the literature and by comparing the prediction results with the new CATT-TM for the reference configuration. PBTM has been compared with finite element method (COMSOL) results in the low-frequency region. Both phase-based models match well in source room with a reasonable discrepancy. However, the PBTM has not reproduced the sound field predicted by COMSOL in the receiver room. Moreover, Waterhouse effect is studied by both PBTM and EBTM model in the reverberation rooms which is ignored in the classical diffuse-field concept. However, its significant effect is exhibited near the reflecting boundaries inside the reverberation room only in the PBTM predictions. Hence, based on recommendations of the ASTM standards during measuring sound transmission between rooms, sources and receivers should be placed sufficiently far away from the reflecting surfaces, edges and corners of the rooms to avoid the errors due to the Waterhouse effect.  iii  Lay Summary                          Predicting sound transmission between two rooms and predicting the sound field contains in them are crucial for achieving an acceptable acoustical environment, e.g. speech privacy between coupled office rooms. Likewise, it is also essential to know whether the sound field is sufficiently diffuse/reverberant for achieving the optimum acoustical condition for rooms. This knowledge can be helpful in designing the optimum classroom for speech communication and auditorium or musical recording studio for the appropriate acoustical environment. This research is conducted to study the sound transmission between two adjacent rooms separated by a common wall. Three predictions models are developed for modeling sound transmission between rooms, a new beam tracing model, the ray-tracing model in CATT-Acoustics and FEM model in COMSOL. The author has assessed how different modeling approaches can reproduce the results of diffuse-field theory (DFT).                iv  Preface This thesis is an original, unpublished and independent work of the author performed under the direct supervision of Late Professor Murray Hodgson, Director of the Acoustics and Noise Research Group at UBC. A part of Chapter 4 and 5 has been published in the Acoustics Week’17 Conference Proceedings of the Canadian Acoustical Association [Md Amin Mahmud, Md Mehadi Hasan and Murray Hodgson, Beam-Tracing Prediction Of Room-To-Room Sound Transmission And The Accuracy Of Diffuse-Field Theory, Canadian Acoustics, Vol. 45 No. 3 (2017)].                                                                                                                       v  Table of Contents   Abstract……………………………………………………………………………………………………ii Lay Summary……………………………………………………………………………………………..iii Preface…………………………………………………………………………………………………......iv Table of Contents……………………………………………………………………………………….....v List of Tables  .............................................................................................................................................ix List of Figures  ...........................................................................................................................................xi Glossary………………………………………………………………………………………………….xiii Acknowledgements ..................................................................................................................................xiv Dedication  .................................................................................................................................................xv Chapter 1 Introduction………………………………………………………………………….1  1.1 Research Objectives .............................................................................................................. 4 1.2 Thesis Overview .................................................................................................................... 5 Chapter 2 Background and Literature Review ......................................................................... 6     2.1 Diffuse-Field Theory and Diffuse-Field Rooms ................................................................... 6  2.1.1 Diffuse-Field Theories: Sabine and Eyring Versions .................................................... 8  2.1.2 Classical Formula of Diffuse-field Room-to-Room Sound Transmission .................... 9  2.1.3 Waterhouse Effect ....................................................................................................... 11     2.2 Existing Prediction Models for Room-to-Room Sound Transmission ................................ 12   2.2.1 Transmission Calculation using Functional Basis ...................................................... 12   2.2.2 Transmission Model by Gaussian Distribution .......................................................... 12   2.2.3 Diffusion Model for Sound Transmission Betweeen Coupled Rooms ...................... 14   2.2.4 Airborne Sound Transmission between Coupled Rooms (ODEON) ......................... 16    2.3 Application of Other Developed Models in This Work ....................................................... 17  2.3.1 Reference Configuration (Diffuse-Room) ................................................................... 17 vi    2.3.2 Diffuse-Field Room-To-Room Sound Transmission Theoretical Model (DFT) ....... 19   2.3.3 CATT-Acoustic Room-to-Room Sound Transmission (CATT-TM) ......................... 20   2.3.4 Room_to_Room_Sound Transmission (COMSOL) ................................................... 22            2.3.5 Applicability of All Applied Models and Motivation Behind Developing the New Beam Tracing Model…………………………………………………………………………….24 Chapter 3 Development of the New Model……………………………………………………26     3.1 Existing Beam-Tracing Algorithms……………………………………………………….26     3.2 Development of the New Model…………………………………………………………..26                                                                3.2.1 Assumptions………………………………………………………………..............27             3.2.2 Methodology……………………………………………………………….............28             3.2.3 Omni-directional Source Modelling……………………………………………….29              3.2.4 Surface Boundary Conditions………………………………………………..........30                     3.2.4.1 Normal Incidence Specific Surface Impedance……………………….........31                     3.2.4.2 Normal Incidence Absorption Coefficient……………………………….....31              3.2.5 Main Components of the New Beam Tracing Algorithm………………………....32   Chapter 4 Preliminary Studies………………………………………………………………...37    4.1 Phase-Based Studies ............................................................................................................. 37       4.1.1 Convergence .................................................................................................................. 37            4.1.1.1 Number of Reflections ........................................................................................... 38            4.1.1.2 Number of Beams .................................................................................................. 39      4.1.2 Study for Reference (Diffuse) Configuration ................................................................. 41              4.1.2.1  Octave-Band Study………………………………………………………………41                      4.1.2.1.1 Source Room…………………………………………………………….41                      4.1.2.1.2 Receiver Room…………………………………………………………..42            4.1.2.2 Third-Octave-Band Study………………………………………………………..43 vii                4.1.2.2.1 Source Room………………………………………………………………….43              4.1.2.2.2 Receiver Room………………………………………………………………...43      4.1.3 Validation of the New Phase-Based Model (PBTM) ..................................................... 44             4.1.3.1 Source Room ......................................................................................................... 44             4.1.3.2 Receiver room ....................................................................................................... 45   4.2 Energy-Based Studies ........................................................................................................... 46       4.2.1 Convergence .................................................................................................................. 46              4.2.1.1 Number of Beams ................................................................................................ 46       4.2.2 Study for Reference (Diffuse) Configuration ................................................................ 48       4.2.3 Validation of the Energy-Based Beam Tracing (EBTM) model ................................... 49   4.3 Summary ............................................................................................................................... 50 Chapter 5 Objective Studies…………………………………………………………………...52   5.1 Case Studies .......................................................................................................................... 52      5.1.1 Effect of Magnitude of Surface Absorption .................................................................... 52             5.1.1.1 Source Room ......................................................................................................... 53             5.1.1.2 Receiver Room...................................................................................................... 54     5.1.2 Effect Of Room Shape (For Uniform Absorption) .......................................................... 55            5.1.2.1 Between a Small Cubic Office Room and a Large Room. .................................... 55            5.1.2.2 Between a Large Room and a Small Cubic Office Room. .................................... 56            5.1.2.3 Between Two Large Rooms (10m×5m×5m) ......................................................... 57            5.1.2.4 Between Two Large Rooms (25m×5m×5m) ......................................................... 57       5.1.3 Effect of Surface Absorption Distribution ...................................................................... 58           5.1.3.1 Most Absorptions in Ceiling ................................................................................... 58           5.1.3.2 Most Absorptions in Floor………………………………………………………...59     5.1.4 Combined Effects (Room Shape & Absorption Distribution) ......................................... 59  viii                5.1.4.1 Between a Small Cubic Office Room and Large Room with Most Absorptions in Ceiling ........................................................................................................................................... 60              5.1.4.2 Between a Large Room and Small Cubic Office Room with Most Absorptions in Ceiling ........................................................................................................................................... 60              5.1.4.3 Between Two Large Rooms with Most Absorptions in Ceiling .......................... 61    5.2 Waterhouse Effect Study ..................................................................................................... 62        5.2.1 Phase Approach ............................................................................................................ 62               5.2.1.1 Phase Approach (Source Room) ......................................................................... 62               5.2.1.2 Phase Approach (Receiver Room) ...................................................................... 63        5.2.2 Energy Approach (Source Room, Receiver Room) ...................................................... 64   5.3 Summary……………………………………………………………………………………65 Chapter 6 Conclusions………………………………………………………………………....67    6.1 Contributions……………………………………………………………………………….67   6.2 Summary of Results .............................................................................................................. 68   6.3 Limitations and Future Work……………………………………………………………….70 Bibliography…………………………………………………………………………………….73 Appendices………………………………………………………………………………………76   Appendix A: DFT for Reference Room Configuration..……………………………………….76   Appendix B: CATT-TM Project Files for Reference Room Configuration……………………77   Appendix C: New Beam Tracing Algorithm-Matlab Code ……………………………………81        ix  List of Tables   Table 2.1: DFT result for the reference room configuration…………………………………….20 Table 2.2: Material properties of the gypsum board………………………………………….....22 Table 2.3: Applicability of each applied room-to-room predictions models in this work……....25 Table 4.1: Variation of the source room octave-band levels with number of beams for a 1-Hz  frequency interval………………………………………………………………………………..41 Table 4.2: Variation of the receiver room octave-band levels with number of beams for a 1-Hz  frequency interval………………………………………………………………………………..41 Table 4.3: Comparison between DFT and Third-octave-band levels in source room.…………..43 Table 4.4: Comparison between DFT and Third-octave-band levels in receiver room…………43 Table 4.5: Variation of SPL with the number of beams for 50 reflections, for 170 Hz frequency  for the reference room configuration……………………………………………………………47 Table 4.6: Variation of the source room octave-band levels with number of beams for a 1-Hz  frequency interval………………………………………………………………………………..47 Table 4.7: Variation of the receiver room octave-band levels with number of beams for 1-Hz  frequency interval………………………………………………………………………………..48 Table 4.8: Comparison between the DFT and EBTM for the reference room configuration…...48 Table 4.9: Comparison between the DFT and other energy-based prediction methods for the  reference room configuration……………………………………………………………………49 x  Table 5.1: Comparison between DFT and EBTM levels between a small cubic office room and large room……………………………………………………………………………………….56 Table 5.2: Comparison between DFT and  EBTM levels between a large room and a small cubic office room……………………………………………………………………………………...56 Table 5.3: Comparison between the DFT and the EBTM predictions between two large rooms (10m×5m×5m)…………………………………………………………………………………..57 Table 5.4: Comparison between the DFT and EBTM results for two large rooms (25m×5m×5m).............................................................................................................................57 Table 5.5: Comparison between the DFT and EBTM for the non-uniform absorption (most absorption in ceiling)……………………………………………………………………………58 Table 5.6: Comparison between the DFT and EBTM for the non-uniform absorption (most absorptions in floor)……………………………………………………………………………..59 Table 5.7: Comparison between the DFT and EBTM levels between a small cubic office room and large room with most absorptions in the ceiling……………………………………………60 Table 5.8: Comparison between the DFT and  EBTM levels between a large room and a small cubic office room with most absorptions in the ceiling…………………………………………61 Table 5.9: Comparison between the DFT and the EBTM predictions between two large rooms (10m×5m×5m) with most absorptions in the ceiling…………………………………………….61 Table 5.10: Waterhouse Effect (phase) results for the various receiver and source positions in source room……………………………………………………………………………………....63 Table 5.11: Waterhouse Effect (phase) results for the various receiver and source positions in receiver room…………………………………………………………………………………….64 Table 5.12: Waterhouse Effect (energy) results for the various receiver and source positions in both source and receiver room…………………………………………………………………...65       xi  List of Figures  Figure 2.1: Acoustic images at a) one-wall b) two-wall and C) three-wall reflectors for one incident wave. ………………………………………………………………………………….. 11 Figure 2.2: Reference configuration showing identical source and receiver rooms, source (S), receivers (R1 and R2) and common wall………………………………………………………...19 Figure 2.3: Relations between absorbed, transmitted and reflected energy……………………..21 Figure 2.4: CATT-TM model showing source and receiver rooms, source, receivers and transmitting surface……………………………………………………………………………....21 Figure 2.5: Meshed COMSOL model of both source and receiver rooms………………………23 Figure 2.6: Solved COMSOL model of a source room showing the SPL distribution at 200 Hz  Frequency………………………………………………………………………………………...24 Figure 3.1: A flow chart of the new beam tracing room-to-room sound transmission method….29 Figure 3.2: Flow Chart of the New Beam Tracing Algorithm for Room-to-Room Sound Transmission……………………………………………………………………………………..36 Figure 4.1: Variation of source room SPL (dB) with various number of reflections for 1280 beams at 170-200 Hz frequencies with a 1-Hz frequency interval……………………………....38 Figure 4.2: Variation of receiver room SPL (dB) with various number of reflections for 1280 beams at 170-200 Hz frequencies with a 1-Hz frequency interval………………………………38 Figure 4.3: Variation of SPL (dB) in source room with a number of beams for a 1-Hz frequency interval…………………………………………………………………………………………...39Figure 4.4: Variation of SPL (dB) in receiver room with a number of beams for a 1-Hz frequency interval………………………………………………………………………………………….  40 Figure 4.5: Comparison between the DFT and PBTM octave predictions for the reference source room……………………………………………………………………………………………...42Figure 4.6: Comparison between the DFT and PBTM octave predictions for the reference receiver room…………………………………………………………………………………….42 Figure 4.7: Comparison of the PBTM and COMSOL predictions in source room for the reference room configuration within 170-200 Hz………………………………………………………….45 Figure 4.8: Comparison of the PBTM and COMSOL predictions in receiver room for the reference room configuration within 170-200 Hz……………………………………………….45 xii  Figure 5.1: Comparison between DFT, EBTM and CAT-TM levels with varying surface absorptions, α in the source room……………………………………………………………….53 Figure 5.2: Comparison between DFT, EBTM and CATT-TM levels with varying surface absorption, α in the receiver room………………………………………………………………54                          xiii  Glossary   Acronym Explanation SPL Sound Pressure Level in decibels (dB) without any weighting.   DFT Diffuse-Field Room-To-Room Sound Transmission Theoretical Model   CATT-TM CATT-Acoustic Room-to-Room Sound Transmission   PBTM Phase-Based Beam Tracing Model   EBTM  Energy-Based Beam Tracing Model ODEON Odeon Model of Airborne Sound Transmission  between Coupled Rooms                xiv  Acknowledgements  First and foremost, I would like to express my enduring gratitude to my supervisor, (Late) Dr. Murray Hodgson, Director of Acoustics and Noise Research Group at UBC. I am very grateful for his endless care, time and encouragement for the last two years until his last day at this world.  Thanks a lot, Murray!!. I would like to extend my sincere gratitude and thanks to Dr. Behrooz Yousefzadeh, a former Acoustic Lab member and postdoc at Caltech, for his precious time in the evaluation of this thesis and to help in the improving the technical contents and writing quality of this work. Without his invaluable contribution, it would have been extremely difficult to complete this work within the stipulated time. I would also like to thank my fellow Bangladeshi Acoustic Lab colleague Md Mehadi Hasan for his significant contribution throughout this thesis, in particular in Matlab coding and COMSOL modeling.  My enduring gratitude to all members of the thesis supervisory committee for their valuable contribution and guidance in completing this work. A toast to all of my acoustical lab members: Alice, Danny, Banda, Vivek, Hanna, James and all my friends/well-wishers for their support and encouragement throughout my education at UBC. In fine, all praises go to the almighty God ALLAH (SWT), whose blessing and mercy have given me the strength to continue the work in the sad situation of losing my supervisor before the completion of this work.         xv  Dedication  To my mentor and caring supervisor “ (Late) Murray R. Hodgson’’ and his wife, Bernadette Duffy who have supported and encouraged me to finish this work in good time.  To my well-wishers and family members, in particular, my loving mother, Monira Begum, and elder sister, Hosniara Shemu, who have always been there for me during my toughest moments. 1  Chapter 1 Introduction  We spend most of our lives inside the closed environments. Though we may not be mindful of it, our activities are largely affected by the acoustical conditions of the built environment. In an acoustically poor office space, occupants can feel the space too reverberant (noisy) with the significant lack of speech privacy. As a result, they can face health problems such as hearing loss, stress, and fatigue, in addition to the loss of productivity. This problem of speech privacy could be significant between a meeting room and a large open plan office separated by a common transmitting partition. For example, the talker, i.e. sound source could be in the meeting room (source room), and the listener could be in both source room and adjacent open place office (receiver room). Therefore, it is crucial to know the sound transmission between two adjacent rooms as well the sound fields of rooms on both sides of the common wall. Likewise, it is also important to predict the sound field in both source and receiver rooms.  In room acoustics, computer prediction models have been extensively used for over two decades. It involves computer simulation to predict the sound field of a room without the need for the actual physical environment or doing costly experiments. This knowledge is essential for acoustical practitioners and engineers for evaluating the pre-construction design plans for optimal acoustical conditions inside the rooms. Acoustical design of rooms is a challenging task when attempts are made to incorporate some complex room-acoustical phenomena [1]. The research presented here is a solution to modeling sound transmission between two adjacent rooms. Room prediction models try to obtain the temporal and spatial distributions of sound pressure or energy inside the room according to the source and boundary conditions of the room. Room prediction models are either wave (phase)-based or energy-based. In the wave/phase-model, interference effects are incorporated. On the other hand, energy-based models assume sound waves to be incoherent, implying that there is no interference with each other [2]. The classical diffuse-field theory is the most widely used theoretical model for predicting room sound fields and is widely used by practitioners for its simplicity [3]. It is an inherently energy-based prediction tool. The sound field of a room is diffuse if the reverberant sound field is the same 2  at every position in the room and reverberant sound waves are incident from all directions with equal intensity and random phase relations [4]. In a diffuse room, the sound energy of all points in the room is assumed to be close to the asymptotic value (a range/limiting value) of the reverberant sound field. However, diffuse-field theory ignores the effect of interference patterns near the edges and corners of room inside a reverberant field, a feature known as the Waterhouse effect in acoustics. These interference patterns are caused due to lack of phase-randomness and can result in a significant rise of energy level from the value near the reflecting boundaries in a reverberant field thus violating diffuse room assumptions [5].  In practice, the diffuse-field seldom exists in a real room; it can only exist in a perfect reverberation room [4]. A reverberation room is a very rare and especially-designed cubic or quasi-cubic room with diffusely reflecting surfaces and uniform absorption in all surfaces, i.e. a diffuse sound field. One such reverberation room exists in the national research council of Canada.  The classical diffuse-field theory of sound transmission between rooms is the most notable one for studying sound transmission between two adjacent rooms separated by a common wall. The steady-state sound pressure levels in both source and receiver room can be calculated by this theory. However, the applicability of this classical theory has not been studied in great detail in previous works. A few attempts have been made to investigate the application of this theory in last decade (presented in section 2.2.3 and 2.2.4). Another approach for modeling sound transmission between two rooms is to use the diffusion model. It calculates the energy density on both side of the partition wall based on the diffusion of particles in a scattering medium. This model can predict the sound-field between two coupled diffuse and non-diffuse rooms [6]. Later on, a ray-tracing based room acoustic model, Odeon is developed to accurately predict the reverberant sound field transmission between coupled rooms for both diffuse and non-diffuse configurations [7]. However, since both are energy-based models, they can't take the wave-based effects into account. Two room-acoustical prediction models have been developed in the past by the UBC Acoustics and Noise Research Group and used for predicting the steady-state responses in an empty room of various configurations. The first model is PRAY, developed by Chan and Cousins is a wave-based ray-tracing model which can account for fittings, diffuse surface reflection, and sound diffraction and predict both the steady state and transient intensity levels [8]. The second model is a beam tracing model with phase and specular reflection developed by Yousefzadeh for predicting steady 3  state and transient pressure levels for a single empty, parallelepiped room [2]. Both of these models can predict sound field inside a room by taking account of the absorption and reflection from surfaces. Since all the rays are reflected back into the source room, they can’t model the transmission into the adjacent room. In ray tracing, rays are propagated in random directions, however in beam tracing rays travel with pre-defined direction and resolution (number of beams and reflections). For an empty parallelepiped room with walls that reflect sound purely specularly, beam tracing needs fewer rays compared to ray models. Furthermore, ray tracing models the receiver as a volume (sphere) which results in the ray-divergence problem. Ray-divergence causes under-sampling errors in higher reflections even when using a large number of rays as rays don’t penetrate the receiver volume [1]. A way to solve the ray-divergence problem is to replace rays with beams since beam-tracing considers the point-receiver instead of a volume-receiver. Therefore, beam-tracing model is computationally similar to ray-tracing, but with better efficiency for similar accuracy because of its deterministic approach [1]. For the above advantages, the beam tracing model of Yousefzadeh was chosen for this work instead of the ray model of Cousins for implementing room-to-room sound transmission.   In order to predict sound transmission between rooms, a model is needed that can effectively take into account all the acoustical phenomena that occur when a sound wave strikes a transmitting surface, i.e. absorption, reflection and transmission into the adjacent receiver room. The magnitude and phase of the reflected wave are calculated from the acoustical characteristics of the surface, as well as the frequency and angle of the incident wave. The magnitude and phase of the transmitted wave depend on the propagating medium on both sides. If the medium of both sides is the same, i.e. if refraction and diffraction from the edges are ignorable, the direction of the transmitted wave can be considered same as the incident wave.   The acoustical characteristics of the surface can be defined by the impedance of a surface which is a function of both frequencies and the angle of incidence. A surface can be assumed either locally-reacting or extended reaction. A locally-reacting surface is such that if the impedance of the surface is independent of the angle of the incident sound, and depends only on frequency. Extended reaction denotes that the impedance of the surface changes with the angle of incident with frequency. Local reaction is encountered whenever the wall itself or the space behind it is 4  unable to propagate waves or vibrations in a direction parallel to its surface. This is true if transmitted wave propagates in a direction along the surface normal, for all angle of incidents. Typically, this is a reasonable assumption for simple absorptive walls, since they dissipate the energy of sound waves effectively. On the contrary, local reaction is not a realistic assumption for surfaces having significant elastic or poroelastic layers [2]. In ray tracing and beam tracing model, reflection and absorption coefficients are often used instead of surface impedances.  1.1 Research Objectives  The introduction indicates there is a lack a phase-based room prediction model including the Waterhouse effect phenomena. Therefore, the existing beam-tracing model for predicting sound pressure in a room with phase and energy has been adapted to predict room-to-room sound transmission in this work.  The new beam-tracing models (PBTM and EBTM) have been validated by comparing the COMSOL model and with the ray-tracing models (CATT-TM, ODEON) respectively. The range of applicability of each of the applied predictions tools for room-to-room sound transmission to this work is studied. The overall objective of this research work was to assess how well various modeling approaches can reproduce the results of the diffuse-field theory. The objectives of this thesis can be summarized as follows:   To extend the existing beam-tracing method to implement steady-state sound transmission between rooms separated by a homogenous partition.  To investigate whether the new beam-tracing model can reproduce the results from the diffuse field theory (DFT).   To investigate the Waterhouse effect and its impacts on the accuracy of diffuse-field theory prediction in both the source and receiver room by both the phase and energy method.   5  1.2 Thesis Overview  Chapter 2 consist of two sections. The first section presents a literature review on the theoretical background on diffuse-fields, diffuse-field rooms, and classical diffuse-field theory, Waterhouse effect and room-to-room prediction models. The modeling principle and the applicability of each sound transmission model applied to the current work are presented in the second section.   In Chapter 3, the assumptions, derivation and programming structure of the new beam-tracing model is discussed thoroughly along with the modifications made to the existing model, and its new features.   Chapter 4 contains the results, discussion, and summary of preliminary studies for both phase and energy approaches for a reference room configuration. For both approaches, the convergence study of new beam tracing model, beam tracing prediction for reference room configuration and comparison of the beam tracing results with other applied models are presented in this section.  In chapter 5, results of objective studies are presented with a discussion. The capability of the new beam tracing model to reproduce the results of diffuse field theory is investigated in particular, for rooms having non diffuse-sound fields. Next, the results of the Waterhouse effect study in the reverberation rooms for both phase and energy approaches are presented in this chapter.  Finally, a summary of all the results is presented in Chapter 6, along with conclusions, limitations, and suggestions for future work.        6  Chapter 2 Literature Review and Background  Chapter 2 begins with a literature review on the concept of diffuse-field theory, diffuse-field rooms, Schroeder frequency, the classical diffuse-field theory of room-to-room sound transmission, Waterhouse effect. We then provide an overview of the existing sound transmission prediction models are presented. Next is presented the other developed prediction tools applied to this work.   2.1 Diffuse-Field Theory and Diffuse-Field Rooms Diffuse-field theory used by practitioners to predict sound fields in all types of rooms. The basic assumption of the diffuse-field theory is that the sound field in the room is diffuse. The sound field to be diffuse if following two conditions are applicable [4]:  At  any  position  in  the  room  the  reverberant  sound  waves  are  incident  from  all  directions  with  equal  intensity  and  random  phase  relations;   The reverberant sound field is identical at every location in the room. That means the room should have equal mean energy density at all points in the room and equal mean energy flow in all directions at all points in the field [5].  However, the theory is based on assumptions that can restrict its applicability. In fact, the sound field in a room can only be diffuse which have following characteristics [4]:   Empty   Is quasi-cubic in shape  Has uniformly distributed surface absorption  Has diffusely reflecting surfaces.  Real rooms are seldom quasi-cubic, empty or with uniformly distributed surface absorption. In practice, a diffuse-field can only exist approximately in a perfect reverberation or similar room -7  an empty room with quasi-cubic dimensions, specularly reflecting surfaces, low absorption and uniformly distributed surface absorption [4]. Hence, the theory includes only a few of the important features of a room such as room geometry and source directivity, but it can’t model the distribution of surface absorption and the presence of furnishings or barriers [2]. However, it is difficult to ensure that all the surfaces reflect only diffusely without any specular reflections [9]. Hodgson [4] experimentally proved as the room aspect ratio increases, diffuse-field theory becomes less accurate. In spite of all these limitations, the diffuse-field theory is a very useful concept. It allows us to make many simplifying assumptions for various room-acoustical predictions. The concepts of the diffuse-field theory and reverberation room designs are thoroughly discussed by Hasan [9]. Ramakrishnan & Grewal [10] states that the volume of the reverberation room needs to be adequate since it fixes the low-frequency limit of the room. Above this low-frequency limit, the room responds to bands of sound uniformly thus ensuring spatial constancy of the sound levels. Methods to determine the low-frequency limit are different [11]. One limit is the Schroeder frequency [11] and expressed as                                                               𝑓𝑠 ≈ 2000 √𝑇60𝑉                                                                 (2.1) This equation only works if quantities given in SI units. Here, T60, the sound decay time or reverberation time, is related to the average surface absorption coefficient  room surface S and room volume V by the following Sabine's formula [11].                                                                𝑇60 =0.163 V S α                                                          (2.2) This frequency is defined in a way so that on average three Eigen frequencies can exist into one resonance half-width. Diffuse-field theory is based on the random interference of many simultaneously excited normal modes of a room. Generally, the random interference occurs for frequencies above this 𝑓𝑠 [10]. However, Ramakrishnan & Grewal [11] states that the widely used Schroeder test is too restrictive in practice. Hence, they investigated the most reasonable low-frequency limit of reverberation chamber in practice. This work reports that having 20 modes per 1/3 octave band frequency is a better test than Schroeder test. Predicted 1/3 octave results show 8  that even if the minimum volume requirement isn’t met, the spatial uniformity of the reverberation chamber sound levels can be satisfied for broadband sound levels [11].  2.1.1 Diffuse-Field Theories: Sabine and Eyring Versions In room acoustics, the diffuse-field theory is classically presented in two versions; Sabine and Eyring. Practitioners frequently use both versions to predict the reverberation time and steady-state sound pressure level in rooms. Sabine diffuse-field formula is based on the average room absorption coefficient that describes the spatial distribution and magnitude of the surface absorption [12]. Erying’s formula is an improvement of Sabine’s formula and more accurate for high absorptions [6]. Hodgson has performed an experimental evaluation of the accuracy of the Sabine and Eyring version of the diffuse-field theory. The experiment is done in a 1:2.5 scale reverberation room by varying the absorption coefficient of the surfaces in an approximately diffuse-field. From the experimental results. It is concluded that the Eyring  theory is  moderately more accurate than the Sabine  theory, which  overestimates  levels. This, combined  with the case that the Sabine model has limitations related to high surface-absorption coefficients, suggests that the Eyring model is the best choice for predictions in high average surface absorption. Moreover, the omission of (1- α) factor overstimates the reverberant levels by 8-20 dB, (Here α is the average diffuse-field  absorption  coefficient  of the  room  surfaces). Hence, this factor must be included in both versions to obtain an accuracy within 0.5-1 dB [13]. To study the applicability of the diffuse-field theories (Sabine and Eyring versions, Hodgson [4] considered four room-acoustical parameters– room shape, surface absorption, surface reflection and fittings for the diffuse-field theory to accurately predict sound decay/reverberation time and  steady-state sound pressure level. He concluded with these following key findings. Eyring decay-rate/reverberation-time predictions are highly accurate for an empty, quasi-cubic empty room with diffusely reflecting surfaces, a low magnitude of the absorption coefficient, uniform absorption distribution and optimum fitting density [4].  Diffuse-field (Erying version) can accurately predict both steady-state sound-pressure-levels and reverberation time for an empty, regular shaped room with specularly-reflecting surfaces, the low 9  magnitude of the absorption coefficient and any distribution of surface absorption- either uniformly or non-uniformly [4].  2.1.2 Classical Formula of Diffuse-Field Room-to-Room Sound Transmission When sound strikes a partially absorbing and partially transmitting partition between two rooms, some is reflected back into the room; some transmits into the adjacent room [14, 15]. Airborne sound transmission loss STL is defined as:                   𝑆𝑇𝐿 = 10. log101τ (dB)                                    (2.3) Where, τ is the transmission coefficient, a frequency dependent property of the material, defined by the ratio of the transmitted intensity 𝐼𝑡  to the incident intensity 𝐼𝑖 [14]:                                  τ =𝐼𝑡𝐼𝑖                                                                        (2.4) In a diffuse-field, the sound intensity at the wall in the source room can be expressed by [14]                                                                      𝐼𝑖 =𝑃124𝜌0𝑐                                                                       (2.5) Where 𝑃1 is the reverberant sound pressure in the source room; note that 𝐼𝑖 for a plane is ¼ of the intensity of a plane wave, 𝜌0 and c are the density and speed of the sound in the air [15]. Hence, the sound power incident on the common wall from the source side can be expressed as:                                                            𝑊𝑖 = 𝐼𝑖𝑆𝑤 =𝑃124𝜌0𝑐𝑆𝑤                                                     (2.6) Where 𝑆𝑤 is the common wall area.                 In the receiving room, the power transmitted through the common wall must eventually be absorbed in that room [14, 15].                                                             𝑊𝑡=𝐼𝑡𝑆2𝛼 =𝑃224𝜌0𝑐𝑆2 𝛼                                              (2.7) 10  Where, 𝑃2 is the reverberant sound pressure in the receiver room, S2 is the area of the receiving room, 𝛼 is the average room absorption coefficient.  Using the definition of τ [14, 15]:                  τ =𝑊𝑡𝑊𝑖=𝑃22𝑆2𝛼𝑃12𝑆𝑤 =𝑃22𝐴2𝑃12𝑆𝑤                                       (2.8)     Where, A2= 𝑆2𝛼,  equivalent absorption of the receiving room, S2 is the area of the receiving room, and 𝛼 is the average room absorption coefficient. Taking log and rearranging:            𝑆𝑇𝐿 = 10𝑙𝑜𝑔101τ = 10𝑙𝑜𝑔10 (𝑃12𝑃22) + 10𝑙𝑜𝑔10𝑆𝑤𝐴2                                                  = 20𝑙𝑜𝑔10𝑃1𝑃2+ 10𝑙𝑜𝑔10𝑆𝑤𝐴2                                                        (2.9) Now 𝑃1/𝑃2 is the ratio of pressures in the source and the receiving room, which can be expressed with reference to the standard pressure 20 μPa.                                       𝑆𝑇𝐿 = 20𝑙𝑜𝑔10𝑃1𝑃𝑟𝑒𝑓− 20𝑙𝑜𝑔10𝑃2𝑃𝑟𝑒𝑓 +10𝑙𝑜𝑔10𝑆𝑤𝐴2                             (2.10) Therefore,                                                  𝑆𝑇𝐿 = 𝐿1 − 𝐿2 + 10𝑙𝑜𝑔10(𝑆𝑤𝐴2)                                                    (2.11)                                         𝐿2 = 𝐿1 − 𝑆𝑇𝐿 + 10𝑙𝑜𝑔10(𝑆𝑤𝐴2)                                                      (2.12) Where, L1 and L2 are reverberant sound pressure level (dB) in the source and receiver room, respectively, Sw is the common wall area, and 𝐴2 is the equivalent absorption in the receiving room [14, 15]. This is the classical diffuse-field expression used in a standard laboratory procedure based on measurements of the reverberant sound pressure levels [16]. To investigate the capability of various modeling approaching in reproducing the SPL values of this expression is the objective of 11  the thesis to investigate, in particular for rooms that do not have characteristics associated with diffuse-fields (non-diffuse sound fields). 2.1.3 Waterhouse Effect Waterhouse showed that reverberant sound fields depart from the diffuse-field conditions significantly at rectangular edges and corners of a room [5].  The energy density is higher near one-wall reflector, Figure 2.1 (a) where it rises to 2.2 dB higher than its asymptotic value. Although equal energy is flowing in all directions in the field, the sound field is not diffuse due to the non-uniform energy density [5].       Similarly, in the case of two plane reflectors at right angles (two-wall reflector), Figure 2.1 b) the energy density is quadrupled hence the mean square pressure rises to 6 dB. The largest deviation from uniformity occurs in the intersection of three-wall reflectors at right angles at Figure 2.1 c), where the mean squared pressure is 9 dB higher than at remote points [5].  Since there is no simple way to remove the interference pattern itself, modifications are required for some measuring techniques to avoid the error due to the pattern. For the measurement of sound transmission loss (STL), it is an accepted practice to place the microphones in the source room close to the wall under test. In this case, 3 dB is subtracted from the measured STL to get the actual STL [5]. In light of above possibilities of error from placing the microphone near a wall, for avoiding errors of the order of 1 dB, pressure microphones should be positioned at least 0.7λ from the corners and edges of the room, and 0.25λ from the walls [5].  Figure 2.1: Acoustic images at a) one-wall b) two-wall and C) three-wall reflectors for one incident wave [5].   12  However, current standards [17], [18] recommends to place the microphones away from the walls and all other reflecting surfaces in the room in the measurement of sound transmission loss (TL) of partitions. ASTM E336-11 standard [17] needs the microphones to be located or scanned in an area more than 1 m from all major extended surfaces. It also requires the sound source to be placed at least 5 m from the common wall unless the room dimensions prohibit this. ASTM standard practices [18] recommends that during transmission (TL) measurement, position of the microphones should be at minimum 0.9 m from an interior surface and 1.8 m from a corner or an edge of the rooms. Moreover, the microphones in the source room requires to be minimum 1.2 m, preferably more, away from the sound source.   2.2 Existing Prediction Models for Room-to-Room Sound Transmission  2.2.1 Transmission Calculation using Functional Basis L Gagliardini developed a formula to calculate the sound transmission between rooms using functional expansions for room pressures and wall velocities [19]. The complete theory was presented for sound transmission through walls in a laboratory-like configuration. A one-third octave analysis of the transmission loss of multiple walls (including air gaps and mechanical connections) is performed with considering the actual geometry of the laboratory. This work allows to take into consideration of the parameters like source position, room dimensions, wall dimensions and room absorption which was ignored in conventional approaches to sound transmissions. Compared to experimental results, the theory gave acceptable results particularly in the low-frequency range with one-third octane frequency averaging.  For a 150 mm thick concrete test wall, the source position and the wall size produced 6 dB or more discrepancies in the low-frequency range, and of 2 dB at 800 Hz. The effect of the room's volume behind these discrepancies was minimal. The absorption area in the receiving room affects the TL below and around the critical frequency. Less absorption leads to a higher value of TL [19].  13  This study shows that in room-to-room sound transmission calculation, greater discrepancies exists in the low frequency region while compared to the high frequency region. However, this work hasn’t investigated the applicability of the diffuse-field theory of sound transmission between rooms. Moreover, it hasn’t studied the effect of change of receiver position, distribution of surface absorption and room shape [19].  2.2.2 Transmission Model by Gaussian Distribution  Hyun-Ju Kang et al. found the existence significant discrepancies in calculating the STL of multilayer panels when compared with the experimental results for conventional methods such as random or field incidence approach [20]. Hence, a new prediction method using the Gaussian function was proposed to predict the STL of various multilayered panel configurations of a window composed of the glass with an air cavity along with the single panels. In that work, correct directional distributions of incident energy to estimate the STL of multilayered panels were presented. The average sound transmission coefficient is measured from the oblique incidence transmission coefficients of the panels by weighting the directional energy density and integrating over the solid angle. It was assumed that plane waves are incident from all directions with equal probability, which is a perfect diffuse sound field condition [20]. Numerical simulations by using a ray-tracing technique were carried out to obtain a weighting function to represent the directional distribution of incident energy on the wall in a reverberation chamber. Three room models (sphere, reverberant room, and cube) were chosen for investigating the angular dependency of the incident energy on the bounding surface. The dimensions of three models were adjusted so that the reverberation time (7s) and the volume (225 m3) were almost identical. The diffusivity of the sound field depends on the number of rays. Hence, a large number of rays were generated from the omnidirectional point source to make the sound field spatially diffuse by testing sound pressures at various locations. A frequency independent absorption coefficient of 0.02 was used for all the room surfaces [20]. Results revealed that that the incident energy decreases with increasing angle of incidence and varies with four source positions, especially at near-normal incidence. The study found that the contribution to the sound transmission through the common wall between two cubic rooms is mainly due to the sound energy of near-normal incident angles. It was shown that the directional 14  distribution of incident energy varies with change of angle of incidence, room shape, frequency and source positions, especially at near-normal incidence. However, the effect of varying the receiver positions in the incident energy hasn’t been studied in this work. Compared with the ray-tracing simulation result, the Gaussian curve indicated that the Gaussian curve with a proper value of β within the range of 1–2 agrees satisfactorily to the simulation curves of a reverberation chamber as well as sphere and cubic room model.  Here, β is a constant to reflect the effects of the parameters mentioned above [20]. The theoretical STLs were calculated for a single steel panel, double and triple- glazed window composed of glass-air-glass and showed good agreement with the measured data. For the single panel, the predicted STL curves agree reasonably well, except for about 3 dB differences in the coincidence region, explained by the finite size effect near the critical frequency. The predicted STL of a double panel with an air cavity showed much lower values than the measured one at the critical frequency, e.g. 250 Hz. In practice, this discrepancy can be explained by the friction between panel edges and the frame of test opening while the panels and air cavity acts as a mass and spring [20]. The weighting function is derived by ignoring the Waterhouse effect at the boundaries and considering the sound field at the middle positions in the room is diffuse. Hence, the previous overall framework of the infinite panel approach wasn’t altered. Besides, the complex mass of panels was introduced to consider boundary damping as a function of frequency allowing more accurate prediction at the lowest mass–spring–mass resonance frequency [20].  Similar to the functional model, this model hasn’t studied the classical diffuse theory of sound transmission between rooms for the reverberation or other rooms. Moreover, since this is an energy based model, so it couldn’t predict the wave phenomena like Waterhouse effect. 2.2.3 Diffusion Model for Sound Transmission Between Coupled Rooms Billon et al. proposed a diffusion model to account for the sound transmission between a source room and an adjacent room, coupled through a common wall. The model was validated numerically, comparing with the diffuse-field theory for several coupled-room configurations by 15  varying the coupling area surface, the absorption coefficient of each room, and the volume of the adjacent room, and by an experiment for two coupled classrooms [6].  The diffusion model can be used as an alternative to the diffuse-field theory, particularly when the diffuse-field theory was not applicable. Besides, the diffusion model allows the prediction of the spatial distribution of sound energy within each coupled room. It can be extended to arbitrary numbers of coupled rooms, whereas the diffuse-field theory provides only one reverberant sound level for each room. The diffusion model for a single room was extended to two enclosures coupled through a partition wall by considering two diffusion equations, one for each enclosure using the coupled energy balance equation and applying the boundary conditions [6]. The results of this work showed that compared to the diffuse-field theory, diffusion model sound level in the room doesnot vary with the change of the transmission loss values. The diffusion model was used to predict the sound pressure levels in both source and receiver room; each cubic room (5m×5m×5m), all the surfaces of both the rooms having 0.1 absorption coefficient and a frequency-independent transmission loss value between 10 and 50 dB. It was not expected of the diffusion model to predict the modal behavior of the room. Good agreement with a maximum discrepancies of SPL less than 0.5 dB was obtained between the diffusion model and the diffuse-field theory for the cubic room configuration. Moreover, the study was done to investigate the effect of the absorption coefficients in both the source and adjacent room. The results revealed that the discrepancy of the diffusion model from the diffuse-field theory increases by 1.5 dB for changing the absorption value from 0.05 to 0.5 dB [6].   Since the diffusion model is an energy-based approach, it can’t predict the frequency-dependent SPL with changes of phase and interference effect like a wave model. Hence, diffusion model predicted values should deviate significantly from the wave-based predictions particularly at the low frequency region [6].      16  2.2.4: Airborne Sound Transmission between Coupled Rooms (ODEON) Rindel & Christensen integrated the sound transmission between rooms by including auralization in the receiving room in the room-acoustic software Odeon. Odeon is built on ray tracing combined with a secondary source radiation method for typically 0 – 3 orders of reflection. This method was considered to be particularly useful for the prediction of sound isolation between spaces with non-diffuse sound fields, like rooms with very uneven distributions of sound absorption and unusual room shapes [7]. In the ray tracing, sound particles (a large number of rays), starting from a point in the source room are propagated, and when they hit a normal surface, the particles are reflected back into the room according to a reflection and scattering model. However, when the particles hit the transmission surface, a certain fraction of the particles are transmitted, i.e. they propagate through the surface into the adjacent room where they continue propagation. As a result, the transmitting surface turns into a sound source that radiates into the receiver room. The transmitting surface was modeled either a single surface or two parallel surfaces to allow different absorption materials at the two sides of the surface [7].   The optimum fraction of the transmitting particles wasn’t well defined.  50% transmission particles causes a high percentage of the particles to transmit back into the source room from the receiver room. On the other hand, if 1% of the transmission particle is used, a large number of rays would be required (as a minimum 100 times more rays would be needed to achieve a sufficient reflection density in the receiver room. To overcome these difficulties, the Odeon set the fraction of the transmitted particles to 10%. This implies that transmitted particles have the energy increased by a factor of 10 (i.e. + 10 dB) before the values of the transmission loss are subtracted, and the reflected particles have the energy increased by a factor of 10/9 = 1.1111 (i.e., + 0.5 dB) [7].   The test room configurations are identical to the reference configuration presented in section 2.3.1, page 16. The sound pressure levels in each room were calculated by averaging over the three receivers positions in a line across the middle of the room. The number of rays used for the calculations is 10,000 [7]. 17   2.3 Application of Other Models in This Work Section 2.3 presents the reference (diffuse) room configuration, application of other sound transmission prediction models (DFT, CATT-TM, and COMSOL) to this work. It is concluded with a discussion of the applicability of each of these prediction models in this work and motivation behind developing the beam tracing models.   2.3.1 Reference Configuration (Diffuse-Room) Section 2.1 recommends a reverberation room with diffusely-reflecting surfaces for achieving a diffuse-sound field. However, the diffuse-field (Eyring) theory is found reasonably accurate with rooms with specularly-reflecting surfaces. Moreover, since the existing beam tracing version of diffusely reflecting surfaces had limited success, the new model has been adapted from the version of specularly-reflecting surfaces. Hence, the reference configuration (diffuse-room) has been chosen based on the recommendations for a diffuse-room presented in section 2.1 with specularly-reflecting surfaces. Same reference configuration is also used by both the Diffusion and Odeon models (see sections 2.3.3 and 2.3.4). Preliminary studies of chapter 4 and Waterhouse effect study of chapter 5 are done for same reverberant room configuration in both source and receiver rooms. The reverberation rooms have been designed following most of the prescriptions of existing standards. ASTM C423 standard [21] specify a minimum room volume of 125 m3, heavy construction materials (e.g. concrete) to obtain least absorption at surfaces, average absorption coefficient 𝛼 = 0.05 for 250 < f < 2500 Hz and 𝛼 = 0.1 for f ≥ 2500 Hz, 1 or 2 sources near room trihedral corners, 3 or 5 receivers; 1.5 m apart, 2 m from sound source, 1 m from reflective surface [9].    Frequency range: 88 Hz to 5656 Hz with a 1 Hz frequency interval (Schroder Frequency 193 Hz): Predictions are done to the frequency range below and above the Schroeder limit to find the effect of this cut-off frequency in the accuracy of the diffuse-field theory.   Average Absorption coefficient: 0.1 (kept constant for all frequency range to maintain consistency with background studies: Diffusion model and Odeon) 18    Wall impedance : 15759 Pa.m.s-1 (Normal), Specific Impedance: 37.97   Source power: Beam tracing, CAT-TM: 100 dB, FEM : 0:01 W   Material: concrete    Room size: 5m ×5m× 5m, volume 125 m3   Source: 1, S (0.35m ×0.35m × 0.35m) (Source is placed at tetrahedral corners, 0.25λ distances far apart from the surfaces as suggested by Waterhouse to ignore the Waterhouse effect [5].   Receivers: 1 in each room near the middle (1.5 m apart, 2 m from sound source, 1 m from reflective surface). R1 (2.5m× 2.5m×2.5m) and R2 (7.5m× 2.5m×2.5m). In this study, we have chosen one receiver in each room because it is expected that this reference room configuration should have the same reverberant sound field except at the corners and edges although ASTM C423 suggests minimum 3 receivers in a room. The receivers are placed near the middle and at least 0.25λ distances away from the surfaces to ignore the Waterhouse effect [5]. Moreover, receivers are located sufficiently far away from the source to ignore the direct sound.   Transmission loss of the common wall: 20 dB (frequency independent)   Since SPL value predicted by the diffuse-field theory doesn’t vary with frequency. Hence, it can’t agree with the phase-based SPL value at a single frequency particularly in the low-frequency region. However, it can agree to an average band SPL value over a certain range of frequency above the Schroeder frequency. Hence, octave band SPL is calculated by averaging the frequency-dependent levels over the range of frequencies within the 1/1 octave band.   The 1/3-octave filter provides a finer spectral analysis but at an increased cost and computational time. It requires 30 filters versus 10 for the full octave filter to cover the audio range (20-20000 Hz) [22]. Hence, most studies of chapter 4 and 5 are shown in the 1/1 octave band to save the computational time. The reference configuration is illustrated in Figure 2.2. 19            2.3.2 Diffuse-Field Room-To-Room Sound Transmission Theoretical Model (DFT)  Within the limitations of the diffuse-field theory discussed previously in section 2.1, DFT Model is developed to calculate the theoretical values from the classical diffuse-field theory as illustrated in section 2.2. This model can predict the steady state SPL in both source and receiver room for any given room configuration with known room-acoustical parameters (see the attached model on page 76, appendix A). It is applied for comparing with the predicted sound pressure values by the other prediction models. Section 2.1 shows that diffuse-sound theory is not accurate at the low frequency region below the Schroder frequency. Source room reverberant sound pressure level (L1) is calculated using the following Sabine formula of the diffuse-field in the steady state [23].                                        L1 = Lw+10log (4𝑅1)                                                                         (2.13) Ignoring the direct field contribution (𝑄4𝜋𝑟2), the room constant R1 is                                          𝑅1 = 𝐴1(1 − α)                                                                           (2.14)  Figure 2.2: Reference configuration showing identical source and receiver rooms, source (S), receivers (R1 and R2) and common wall. 20  Where, A1 (equivalent absorption area) is calculated by the Sabine formula including air absorption, 𝐴1 = 𝑆α,  S is the total surface area, and 𝛼 is the average absorption coefficient of the identical source and receiver rooms. Next, the receiver room SPL is obtained from the classical diffuse-field formula as follows [15]: L2=L1-TL+10 log [Sw/A2]                                                                (2.15) Where L1= Reverberant sound pressure level at the receiver R1 of source room (dB)          L2= Reverberant sound pressure level at the receiver R2 of receiver room (dB)          TL= Transmission loss of the sound transmitting partition (dB)          Sw= area of sound transmitting partition (m2)           A2= Equivalent absorption area of the receiver room (m2) It should be noted that, in equation 2.15, both TL and A2 can vary with frequency. However, DFT considers the same average absorption coefficient 0.1 and transmission loss value (TL=20) for all frequency region even below the Schroder frequency to be consistent with background studies such as Odeon [7].  Room Configuration L1 (dB) TL Correction Term L2 (dB) Reference 93.8 20 2.21  76.02  2.3.3 CATT-Acoustic Room-to-Room Sound Transmission (CATT-TM) A new room-to-room sound transmission model has been developed in the commercial distribution of CATT-Acoustic (energy approach) named as CATT-TM. The purpose of developing CATT-TM model is to validate the energy beam tracing model (EBTM) and compare it with the DFT levels for the reference room configuration. CATT-Acoustics also works on the ray-tracing principle similar to the Odeon model [7] presented in section 2.2.3. Moreover, it is quite similar to the EBTM since it also calculates the pressure of a receiver through tracing of a large number of rays. CATT-Table 2.1: DFT result for the reference room configuration 21  TM can also model sound transmission between two rooms by taking account of the principle of transparency as follows [24].  Direct sound passing through a maximum of one semi-transparent surface is attenuated by (1 - α) τ. Higher order transmission is handled by a random process. 1st order reflections are correspondingly attenuated by (1 - α) (1 - τ) and are not transmitted. Since the transmitted sound would be extremely weak, the transmitted part is assumed absorbed. 2nd order and higher transmission are random depending on the transmission coefficient τ. The sum of the three components is equal to 1 so that the energy is conserved as shown in Figure 2.3 [24]:   Besides, few conditions and assumptions are fulfilled for planes with transparency. The transmitting surface is modeled as a doubled-sided wall to enable transmission from the source room to the receiver room. Moreover, both sides of the room have the same material, i.e. air.  For simulating sound transmission between two cubic rooms (5m×5m×5m), i.e. reference configuration of section 2.3.1. Ray-convergence was ensured; 31390 rays and 0.88 sec ray truncation time are needed in CATT-TM to get the converged results. The project files of the CATT-TM for the reference configuration are attached in appendix B, page 77-80 as illustrated in Figure 2.4 below.     Figure 2.3: Relations between absorbed, transmitted and reflected energy [24].     Figure 2.4: CATT-TM model showing source and receiver rooms, source, receivers and transmitting surface.  22  2.3.4 Room-to-Room-Sound Transmission (COMSOL) The purpose of using the COMSOL model is to validate the PBTM prediction results for the diffuse room configuration in the low-frequency region. A numerical, FEM-based, modal approach is adopted in this study. It solves the classical Helmholtz equation (frequency domain) with the impedance boundary condition [9].  At first, the reference room configuration is modeled in COMSOL; two cubic rooms (5m×5m×5m) with the defined source and receiver positions (see section 2.3.1, page 17-18). Similar to that configuration, the transmitting wall has a pre-calculated [25] frequency-dependent transmission loss value of around 20 dB. The material properties of the gypsum board are given in Table 2.2. The basic characteristics properties of the air; density of 1.2 kg/m3 and speed of 343 m/s are used here. Table 2.2: Material properties of the gypsum board Property Value  Density 760 kg/m3 Speed of sound 1670 m/s Young’s modulus 2.1 GPa Poisson’s ratio 0.24 Isotropic structural loss factor 0.001 In COMSOL model, two physics modes are used in the frequency domain; pressure acoustics and acoustic-solid interaction. Pressure acoustics mode; sound hard boundary (wall) is used for all room surfaces except the common wall with a specific impedance value of 37.97 rayl. This is corresponding to the 0.1 average absorptions of the reference configuration. This modeling of sound transmission between rooms involves the acoustic-solid interaction of fluid domain with the solid domain. For this study, air is the acoustic domain inside the rooms, and a particular gypsum board is a solid domain, used for room surfaces. This boundary condition consists of the fluid load and structural acceleration, used as default boundary condition for acoustic-structure interaction and is not applicable to other boundaries. Besides, damping–isotropic loss factor is used to account for the structural loss during the interaction of acoustic and solid medium (see table 3.2). The sound power of a point source is inputted as reference power (RMS), Pref  equal to 0.01 W [26]. 23  Solving acoustical problems numerically, while maintaining the desired accuracy level, is very challenging. This is because 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. COMSOL recommends tetrahedral elements for diffuse sound fields, due to their anisotropic form [27]. Bibby studied the mesh convergence in COMSOL for different numbers of elements per wavelength, varying from 0.5 to 15, and found that 3.5 elements per wavelength, all parameters converged within 10%, or 0.4 dB. Hence, five elements per wavelength were used in all the predictions [27]. Therefore, in this study, we have selected a 3D, quadratic, tetrahedral elements to discretize our domains; 5 elements per wavelength are considered to ensure sufficient accuracy. The meshing of the geometry considering five elements per wavelength is shown in Figure 2.5. The FEM mesh-convergence procedure is described in details in [9, 27].        Figure 2.5: Meshed COMSOL model for both source and receiver rooms.   Because of the limitation on computational capacity, this study is restricted to the low-frequency region (170-200 Hz) with 1 Hz interval. To perform the numerical simulation, a FE-based commercial software (COMSOL 4.3b) is used on a desktop computer with a 2.4 GHz processor and 24 GB of memory. Usually run times are highly dependent on geometry, mesh and frequency resolution in a FEM model. Hence, run times were typically under 24 hours to predict steady-state sound pressures for the mentioned range of frequencies considered in this study. COMSOL solves the Helmholtz equation, gives the acoustic-pressure solution for the entire acoustic domain and acoustic-solid solution for the acoustic-fluid domain at each frequency of interest. In this study,  Figure 2.5: Meshed COMSOL model of both source and receiver rooms.   24  we have used the steady-state, frequency-domain interface of the COMSOL acoustic module to simulate the pressure field at different receiver locations. The solved COMSOL model for a source room is illustrated in Figure 2.6.            2.3.5 Applicability of All Applied Models and Motivation Behind Developing the New Beam Tracing Model  Section 2.3.5 summarizes the range of applicability of each applied models to this work. Based on this discussion, the reasons behind developing and using the new beam tracing model as the main model for achieving the research objectives of this work are also discussed. An overview of all the other applied room-to-room sound transmission models (DFT, ODEON, CATT-TM, and COMSOL) presented in section 2.2 along with beam tracing model is shown in Table 2.3. This overview covers the principle, range of applicability, advantages/ limitations, and applications of all applied models in this work.  It is clear from the table 2.3 that the beam tracing has the best range of applicability and computational efficiency among all the predictions models applied to this work. Hence, it has been adopted as the main model for all subsequent predictions in the work to achieve the research   Figure 2.6: Solved COMSOL model of a source room showing the SPL distribution at 200 Hz frequency.  25  objectives of this work. The development of the new beam tracing model is presented next in chapter 3. Table 2.3: Applicability of each applied room-to-room predictions models in this work. Model Name Principle Range of Applicability Limitations/Advantages Applications to this work DFT Diffuse-field theory of room-to-room sound transmission Inherently energy-based method Can’t predict the frequency-varying sound pressure levels including wave effect (interference effect)  To predict the diffuse-field theory sound pressure level and to investigate its accuracy by using other models.  ODEON,CATT-TM Ray-tracing Energy-based method Same as DFT To validate the energy-beam tracing model and investigate the accuracy of the DFT. COMSOL  Finite Element Method (FEM) Only phase-based model Advantage: Can predict the frequency-varying sound pressure levels including wave (interference effect) Limitations: Can’t do energy-based prediction. Computationally very expensive, so its applications are usually restricted to predictions for low frequency and simple room shapes.  To validate the phase-based beam tracing model in the low-frequency region. Beam Tracing Geometrical room acoustics Both phase and energy model Advantage: can predict the frequency-varying sound pressure levels including wave effect (interference effect) in all frequency range. Achieve the research objectives of this work by comparing with DFT values for diffuse and non-diffuse rooms.  26  Chapter 3 Presentation of the New Model Chapter 3 starts with an overview of existing beam tracing algorithms followed by the development of the new model. The assumptions, methodology, source modeling, surface boundary conditions, main components and programming structure of the new beam tracing model are discussed.  3.1 Existing Beam-Tracing Algorithms The triangular-beam tracing approach was described by Lewers [28]. Later, Wareing developed a program in MATLAB, upgraded to include phase and extended reaction boundary conditions [1]. Yousefzadeh extended it to include the impulse response and energy based predictions (sound-pressure level and echogram) [2].  The new model for room-to-room sound transmission would be a beam-tracing model as mentioned in chapter 1. It is developed based on the existing beam-tracing model for empty, parallelepiped rooms with specular reflection with phase by Yousefzadeh [2]. This is the first work of room-to-room sound transmission using beam tracing method. The Phase and energy beam tracing model are used as PBTM and EBTM respectively in all subsequent sections of this thesis. The general structure of the new algorithm is identical to the previous one. But modifications have been made to incorporate the new section for the transmission of the beams from the source room into the receiver room through the transmitting surface and multiple reflections of the transmitted beams in the receiver room. The previous model is only restricted to one source and one receiver position. Hence, the feature for adding multiple receivers has been added to the new model. In beam-tracing, rays are generated with variable resolution and pre-determined directions. Beams with triangular cross sections are employed because a point source can be modeled without any overlapping beams and uniform energy distribution across the beam face. In beam-tracing, the source generates rays through the area centers of a set of spherical triangles covering a whole sphere. These spherical triangles become the face of each beam in the beam tracing model allowing 27  variable resolution. The receivers now become points instead of spheres. A single triangular beam can be identified by the center axis c with the addition of three boundary planes (B1, B2, B3) and cross sectional plane X. The beam originates from the vertex which is a single point S. A beam’s vertex is the real source in the room for the direct source-receiver path. Otherwise, new image source S’ is the vertex for any beam having a reflected source-receiver path. The beam is reflected by determining the surface of the impact of the central ray and then reflecting the normal to the three boundary planes. With this geometric information, beams are traceable, but further acoustical information is required. The complete details of the beam tracing algorithm are discussed in the work of Wareing [1].  3.2 Development of the New Model  The new model is developed by adapting the existing beam tracing method, similar to the image source method used by Guo & Hodgson [29]. Beam transmission is implemented from the source room to the adjacent receiver to predict the steady-state sound pressure level in both source and receiver room. The methodology and structure of the PBTM and EBTM models are the main focus of this chapter. The details about the new model are discussed in this section.   3.2.1 Assumptions Following assumptions are made in the new model for implementing room-to-room sound transmission. According to ODEON [7], incorporating sound transmission in the model should not change the sound field in the source room. It was reported that the transmission model works satisfactorily for a transmission loss of 10 dB or above. To ensure that in the beam tracing, no sound transmission from receiver to source room is considered in this model. This is a realistic assumption for the new model since its input transmission loss of the common wall is 20 dB, higher than the suggested 10 dB. The partition is defined as a single homogenous partition with a frequency independent single number transmission loss. 28  The source and receiver positions are located at least 0.2 wavelengths away from walls and reflecting surfaces to avoid the Waterhouse effect [5]. Moreover, the source and receivers are kept far apart from each other to ignore the direct sound.  The room surfaces are modeling as specularly-reflected and locally-reacting surfaces. The new model aims to simulate the same physical environment as the diffuse-field theory of room-to-room sound transmission. Diffuse-field theory only accounts for direct airborne sound transmission through the common wall. It ignores the structure-borne and flanking sound transmission. Hence, the structure-borne sound propagation from the room surfaces and flanking transmission haven’t been considered in the new model.    3.2.2 Methodology  The source was modeled as an omnidirectional sphere with a sound power level of 100 dB. Both receivers were placed near the middle of each reverberation room. At each frequency, the source, receiver room, source and receivers, beam resolution (number of beam and reflections) and surface properties are defined in the model. Then, each beam was generated and propagated in source room. If a beam hits the transmitting surface in source room, part of the beam is reflected back and continued to bounce in the source room. Within the source room, a test was done if the beam path contains the receiver R1. For a positive receiver-detection test, the complex-pressure contribution was calculated at the receiver R1 for the current beam. Another part of the beam was transmitted into the receiver room through the common wall with the transmitted energy. The direction of the transmitted beam was considered same as the incident beam. The transmitted beam was traced for pre-defined reflections, and the receiver-detection test was performed for receiver R2 in the receiver room just as the source room. The complex-pressure contribution was recorded for the current beam at the receiver R1 of the receiver room. Upon the completion of tracing all the beams until pre-defined reflections, the summation of the complex pressure contributions of all beams was converted to decibels to obtain the steady-state sound pressure level at the receiver R1 and R2 in source and receiver room. These results could be expressed in octave or third-octave from the phase results if required. Figure 3.1 briefly illustrates the above-described methodology of the new beam tracing model. 29                         3.2.3 Omni-directional Source Modelling The sound intensity I (W/m2) in a spherical wave can be determined by the following equation, the same expression for that of a plane wave [23]:                                                           𝐼 =𝑝22𝜌𝑐                                                                               (3.1)              Eq. (3.1) [1] can be expressed with regards to amplitude constant A as                                                           𝐼 =𝐴22𝑟2𝜌𝑐                                                                            (3.2)   Figure 3.1: A flow chart of the new beam tracing room-to-room sound transmission method.  30  Hence, the sound intensity in a spherical wave is inversely proportional to the square of the distance. The acoustical output power of the point source, W, can be calculated by multiplying the area of a sphere 4𝜋𝑟2 by the acoustical intensity I:    = 4𝜋𝑟2𝐴22𝑟2𝜌𝑐                                                                     =2𝜋𝐴22𝜌𝑐                                                                           (3.3) Therefore, the amplitude constant A can be expressed in terms of sound power W [1]:                                                                 𝐴 = √𝜌𝑐𝑊2𝜋                                                                                     (3.4) To start the beam tracing program, it was necessary to set the initial phase angle to zero. The beam-tracing technique thus models a spherical wave propagating from a defined point source with pressure amplitude A to calculate the initial acoustical pressure of a beam [1].  The sound pressure of the spherical wave decays inversely with distance L from the source, with phase angle −kL, where k is the wave number [1].  3.2.4 Surface Boundary Conditions Accurate definition of acoustical boundary conditions is quite significant for room acoustic predictions. Phased geometrical acoustics methods such as beam tracing method use both absorption coefficient and surface impedance boundary condition to predict the sound field in a room. However, surface impedance boundary conditions are likely to be superior to absorption boundary conditions because they fully describe the physical characteristics of the boundary, i.e. the magnitude and phase changes on reflections [30]. Suh & Nelson found that the phased geometrical acoustics simulations using surface impedances as boundary conditions agree well with the measurements and reference calculations [31].    𝑊 = 4𝜋𝑟2𝐼  31  3.2.4.1 Normal Incidence Specific Surface Impedance  The normal incidence specific surface impedance (ζ𝑛𝑜𝑟) is typically used as a boundary condition in the phased beam tracing as well as the boundary element simulation. This boundary condition assumes that the surface impedance is thought to be same over the angle of incidence. The normal incidence surface impedance is complex-valued and frequency-dependent but often approximated to real-valued; therefore the reflection coefficient is also real-valued. The specific surface impedance is normalized by the characteristic impedance of air as follows [30].                                                                             ζ𝑛𝑜𝑟 =𝑍𝑛𝑜𝑟𝜌𝑐                                                                       (3.5)  For a locally-reacting surface, the pressure reflection coefficient is calculated for the angle of incidence 𝜃 as follows [30]:                                             𝑅(𝜃𝑖) =ζ𝑛𝑜𝑟 cos 𝜃𝑖−1ζ𝑛𝑜𝑟 cos 𝜃𝑖+1                                                                    (3.6)  The details derivation of this impedance boundary condition is given by Yousefzadeh [2]. The reflection coefficient is a function of the angle of incident although the impedance is not angle dependent [30]. This reflection coefficient is used as the input to the beam-tracing algorithm to model the reflection of sound waves from locally-reacting surfaces.   3.2.4.2 Normal Incidence Absorption Coefficient   For absorption coefficient boundary conditions, the angle dependence and phase shift on reflection are ignored. The absorption coefficient was calculated for a defined input value of impedance in beam tracing. The obtained absorption value was inputted into the DFT model to predict the reverberant sound pressure levels in both rooms. The normal incidence absorption coefficient was calculated using an impedance tube [30].                                                         α𝑛𝑜𝑟 = 1 − |ζ𝑛𝑜𝑟−1ζ𝑛𝑜𝑟+1|2                                                         (3.7)  32  Which can be simplified from the relation of the reflection coefficient in eqn. 3.7 after neglecting the angle dependence as [30]                                                                          α𝑛𝑜𝑟 = 1 − 𝑅2                                                        (3.8)  However, this equation only applies if we consider the amount of energy transmitted through the wall and the energy lost inside it same. For instance, at the transmitting surface between a source and receiver room, the amount of energy transmitted and lost inside are separated to calculate the energy of a transmitted beam in the receiver room, Eq. (3.11) becomes [24]   𝑅𝑡 = √(1 + α𝑛𝑜𝑟)𝜏                                                                                                               (3.10)  Where 𝜏 is the transmission coefficient of the transmitting surface pre-defined in the beam tracing algorithm. This reflection coefficient Rt was used only for the transmitting wall in the new model.  3.2.5 Main Components of the New Beam Tracing Algorithm  The main components of the new beam tracing algorithm consist of a. Frequency loop b. Source beam generation loop: b1) Receiver detection loop for source room b2) Primary Reflection order loop for both reflection and transmission of a beam. c. Secondary reflection order loop for receiver room: receiver detection loops for receiver room. d. Calculation of the steady-state sound pressure level at a single frequency and octave band prediction.  a) Frequency Loop The primary component of the beam tracing algorithm is the frequency loop, executed for each frequency over the range of frequencies. Common input parameters for propagating medium, i.e., air are defined for each frequency. Moreover, surface geometry, source and receiver positions, 33  beam resolution, source power, surface boundary conditions; properties with transmitting surface are pre-defined in the input file. Surface boundary conditions are described in details next.  b) Source Beam Generation Loop The fundamental component of the beam tracing algorithm is the beam loop. Beams are generated from an omnidirectional source S with initial complex pressure amplitude A and direction vectors. Executed for each beam, it couples the process of finding the nearest permissible surface that is in the path of the centre ray, the new image source, the point of impact, the director vectors and angle of incidence of the reflected beam. The new image source now becomes the new start point for the next reflected beam trajectory.  b1) Receiver Detection Loop for Source Room Inside the reflection loop, a test is done to detect the receiver point within the trajectory from the previous source to the new image source. If the receiver point R1 is detected within that beam’s path, the complex pressure contribution is calculated for the beam at that receiver position for the prediction frequency. For new beam tracing (PBTM) model, the beam’s contribution to the acoustical pressure at the receiver point R1 is calculated considering the air absorption [2] by                                                   𝑃𝑟 =𝐴𝐿. 𝑅𝑒𝑓𝑓.𝑒−𝑗𝑘𝐿 . 𝑒−𝑚𝐿/2                                                      (3.11)                             Where A is the amplitude constant of the source, L is the path length of the beam, Reff is the reflection coefficient for all reflected surfaces encountered by the beam from S to R1, a wave number k contains a half of the attenuation exponent m, which is a function of temperature, relative humidity, and frequency. The complex pressure at the point R1 is obtained for a single frequency, and for a single beam, between reflections r and r+1. However, for the EBTM, the sound energy of a beam is calculated instead of sound pressure by Eqn. (3.11) after ignoring the phase term 𝑒−𝑗𝑘𝐿 . When a beam trajectory is detected at a receiver point R1, its sound energy is calculated by squaring the absolute value of its complex pressure amplitude. Then, the total energy-based sound-pressure level at the receiver is calculated by taking the sum of the total energy contributions of all beams detected at the receiver point and converting the sum to decibels.  34  b2) Primary Reflection Order Loop for Reflection and Transmission in Source Room Within a trajectory loop, a new test is done if the beam trajectory strikes the transmitting surface, i.e. the common wall in the source room, if it hits the transmitting surface, part of the beam is reflected back and continues to propagate in the source room. Another part of the beam is transmitted through the transmitting surface into the receiver room with transmitted energy, √𝑅(1- ) [24] similar to the mechanism of CATT-TM presented in the section 2.3.3. Here,  is the transmission coefficient, pressure reflection coefficient 𝑅 discussed in the section 3.4.2.1 and 𝛼 is the absorption coefficient of the transmitting surface.  c) Secondary Reflection Order Loop For Receiver Room  Each transmitted beam then continues to propagate until the pre-defined maximum reflection order in the secondary reflection order loop. Within the new trajectory loop for receiver room, another check is done whether it encounters the receiver point R2 within that beam’s path; the complex-pressure contribution is calculated for that beam at that receiver position for the prediction frequency similar to the Eqn. (3.11). Both reflection order loops in the source and receiver rooms are executed for each beam and repeated until the pre-defined maximum reflection order is reached. Upon completion of the primary reflection loop, the beam number is incremented, and the next beam is traced by resetting the source, power and direction vectors.  d) Calculation of the Steady State Sound Pressure Level at a Single Frequency and Octave Band Prediction After completion of tracing all the beams, the complex pressure contributions of all beams are added, and results are converted to decibels to obtain the steady-state sound pressure level at the receiver point. The octave band value is calculated from the obtained frequency-dependent SPL from the PBTM and EBTM models. To calculate the octave band level, pressure spectrum of the room must be accurately filtered for that frequency bandwidth. In this research, filtering is performed only in the frequency domain; i.e., the transfer function is multiplied by an octave-band filter in the frequency domain with a certain frequency step. The rectangular filter is the simplest band-pass filter commonly used in the 35  frequency domain. This filter only transmits energy within that frequency band, i.e. have zero values outside the range of the corresponding bandwidth [2]. For further details, mathematics about the filtering and Fourier transfer, see page 49-50, [2].  Each octave and 1/3 octave bands are represented by a middle frequency, a lower frequency bound and an upper one. Hence, in this study 1/1 octave band is used. The reason for choosing the 1/1 octave is mentioned on page 19, section 2.3.1. The reference middle frequency in acoustics is 1000 Hz. An octave implies to the interval between one frequency, and it's double or its half. There is a one-octave band between frequencies 1000 Hz and 2000 Hz. In the available standard, the computed octave middle frequencies vary from 16 Hz to 16 kHz as shown in Table 1 [32]. In this study, a 1/1 octave band prediction was performed from only 125 Hz to 4 kHz band by averaging the SPL from 88 Hz to 5656 Hz frequency with each 1 Hz interval.  Programming flow chart of the new beam tracing algorithm is illustrated in Figure 3.2. The MATLAB code for the new algorithm is attached on page 81-100, Appendix C.                  36                                                                                                                      Figure 3.2: Flow Chart of the New Beam Tracing Algorithm for Room-to-Room Sound Transmission. For beam number b=1: Number of beams, generate beam at defined source S with direction vectors Obtain image source, reflection coefficient, surface struck and director vectors of the reflected beam  Receiver R1 detected?    Increase the beam number and reset source, power and direction vectors No Yes Define surface geometry, acoustical properties, source and receiver positions, Initialize internal variables, transmitting surface, source power, and beam resolution  Calculate steady-state sound pressure level L1 and L2 for at the receiver point R1 and R2 for each frequency, f  Calculate complex pressure contribution of current beam: add to counter Primary Reflection order loop complete? Beam loop complete? Yes No No Yes Frequency loop: For each frequency f Frequency loop complete? Check for transmission: If incident beam strikes the transmitting partition, part of the ray is transmitted to receiver room with its transmitted energy. No No Receiver R2 detected?    Calculate complex pressure contribution of current beam: add to counter Yes No Check complete for transmission? Yes Secondary Reflection order loop complete in receiver room? Yes Primary Reflection Loop: For reflection order r, r=1: Number of reflections  Obtain image source, reflection coefficient, surface struck and director vectors of the reflected beam  Secondary Reflection Loop for only receiver room: For reflection order r1, r1=1: Number of reflections  Yes No No Calculate Octave level from SPL value at single frequency f 37  Chapter 4 Preliminary Studies In the previous chapter, the theory and programming structure are presented for the new beam tracing model. But, it is also necessary for the prediction results of the new model to achieve a desired level of accuracy. Wareing mentioned that the beam-tracing predictions require the number of beams, reflections, and frequencies within a frequency band must be adequately high to achieve a high accuracy [1]. Therefore, it is important in this study to investigate how the predicted SPL levels vary with change in these parameters in both PBTM and EBTM models. Because the new model predicts the sound fields in both the source and receiver rooms, it needs a larger number of beams, i.e. computational time than the existing model. However, the PBTM and EBTM models have an acceptable run time of around 8 and 5 hours respectively in MATLAB 2013 in a computer with a 2.6 GHz processor and 16 GB of memory. Chapter 4, preliminary studies is divided into two sections: phase-based and energy-based predictions of the sound field. Each section begins with the convergence study of the new model. Next, a case study of the reference room configuration is performed by comparing with either the PBTM or EBTM prediction results with DFT levels. Afterwards, the validation of the either PBTM or EBTM is shown by comparing with other applied models for the same room configuration.   4.1 Phase-Based Studies  4.1.1 Convergence To give meaningful results, the prediction model must achieve all types of convergence: number of beams, reflections number and frequency sampling. For saving computational time, only the predictions of beam and reflection order have been shown in the low-frequency region where the interference and modal effects are usually dominant. For all predictions in this chapter, a frequency interval of 1 Hz is used since it ensured the convergence of levels within 0.1 dB in the Wareing beam-tracing predictions [1]. 38  4.1.1.1 Number of Reflections The appropriate number of reflections needed to acquire high accuracy in each room is investigated next. Predictions are done for 20, 30, 40 and 50 reflections in each room for 1280 beams. The variation of steady-state SPL levels with the number of reflections are presented in Figure 4.1 and 4.2 for the source and receiver rooms respectively for 170-200 Hz with a 1-Hz frequency resolution.                     Figure 4.1: Variation of source room SPL (dB) with various number of reflections for 1280 beams at 170-200 Hz frequencies with a 1-Hz frequency interval.   Figure 4.2: Variation of receiver room SPL (dB) with various number of reflections for 1280 beams at 170-200 Hz frequencies with a 1-Hz frequency interval.  39  SPL levels in the 170-200 Hz shows fluctuation with the number of reflections. This is what would be anticipated for PBTM model, showing that the phase has a significant effect on the levels in the low-frequency region. Predictions showed that all levels has converged within about 0.3 dB in the case of 1280 beams, 50 reflections, and a 1-Hz frequency interval. Therefore, 50 reflections have been chosen as the appropriate for achieving an acceptable accuracy and used in all further studies of this work. 4.1.1.2 Number of Beams The appropriate number of reflections was determined using 1280 beams. It is to be noted that increasing the number of beams would not require a higher number of reflections. Therefore, the beam number is varied from 1000 to 15000 with a constant interval of 1000 for 50 reflections; for 170-200 Hz frequency with a 1-Hz interval.  50 reflections    the energy of the transmitted . This is expected as the transmitted beams enter the receiver room with reduced energy due to 50 reflections in the source room. Hence, fewer beams and reflections are needed for getting a converged value in the receiver room while compared to the source room.                  For brevity, the predicted levels for only two beam resolutions; 14000 and 15000 beams are shown here in Figure 4.1 and 4.2 for the source and receiver room respectively. Figure 4.3 reveals that source room levels converge within 1 dB except for around 1.5 dB in 180-185 Hz. On the other hand, in the receiver room shown in Figure 4.4, SPL levels converge within 0.52 dB in most frequencies. However, 2.3 dB difference can be observed in 185 Hz which can be influenced by constructive or destructive interference due to the presence of phase. It is noticeable that the  Figure 4.3: Variation of SPL (dB) in source room with a number of beams for a 1-Hz frequency interval.  40  receiver room has a better accuracy while compared to that of the source room. The energy contribution of a beam decreases monotonically with each reflection. As a result, the magnitude of the total energy contribution of a beam reaches to a converged value which shows negligible change after addition of any contribution energy loss. Each transmitted beams has already lost a significant amount of energy due to reflecting 50 times in source room during transmitting into the receiver room. Hence, fewer beams and reflections are required to converge in the receiver room. Therefore, for same number of beams, the receiver rooms levels converges faster compared to source room.      Table 4.2: Variation of SPL (dB) in receiver room with a number of beams for a fixed reflection and constant 5-Hz frequency interval.      Octave band levels are calculated from phase results by averaging the frequency-dependent SPL from 125 to 4 kHz band with a 1 Hz frequency interval within the 1/1 octave band. PBTM octave bands levels are shown in Table 4.1 and 4.2 respectively for the source and receiver room respectively. Predicted octave-band SPL levels have converged within 0.65 and 0.4 dB in all bands for source and receiver room respectively for 10580 beams and 50 reflections. Lowest accuracy is obtained at 125 Hz band for both rooms as expected since modal effects are dominant in the low-frequency region.  Figure 4.4: Variation of SPL (dB) in receiver room with a number of beams for a 1-Hz frequency interval.  41  Table 4.1: Variation of the source room octave-band levels with the number of beams for a 1-Hz frequency interval  SPL levels (dB) for octave-band center frequency (Hz) Beam Number 125 250 500 1000 2000 4000 9000 89.52 92.99 95.30 96.27 96.42 93.10 10580 90.16 93.01 95.86 96.72 96.97 93.70 Difference (dB) 0.64 0.02 0.56 0.45 0.55 0.60  Table 4.2: Variation of the receiver room octave-band levels with the number of beams for a 1-Hz frequency interval.  SPL levels (dB) for octave-band center frequency (Hz) Beam Number 125 250 500 1000 2000 4000 9000 66.02 73.15 77.77 77.53 78.90 78.32 10580 66.40 73.09 77.57 77.73 78.93 78.60 Difference (dB) 0.39 -0.06 0.20 0.20 0.03 0.27 4.1.2 Study for Reference (Diffuse) Room Configuration The accuracy of the PBTM model in reproducing the diffuse-sound field (DFT) values is investigated for the reference room configuration for both 1/1 octave and 1/3-octave bands. 4.1.2.1 Octave-Band Study for Reference Configuration The comparison between DFT and 1/1 octave-band PBTM prediction results are shown in both source and receiver rooms in Figure 4.5 and 4.6 respectively. 4.1.2.1.1 Source room  PBTM octave levels are 3 dB lower than DFT at the 125 Hz band; around 0.5 dB at the 250 Hz band and 2-4 dB higher from the 500-4KHz bands. These fluctuations show the dominant effect of the phase throughout the octave frequency region. 125 Hz band (88-176 Hz) is below the Schroeder frequency (193 Hz) for the reference configuration. As discussed in section 2.1, sound field a room is not sufficiently diffuse below the Schroeder frequency due to lack of field 42  diffuseness. Hence, PBTM 125 band levels are lower than the DFT since it is below the Schroder frequency.         4.1.2.1.2 Receiver room  PBTM octave levels are 9.5 dB lower then DFT levels in 125 bands; 3 dB lower in 250 Hz and 2-3 dB higher from the 500-4KHz band. These variations are also due to the constructive and destructive interferences. The PBTM band average results are within 3 dB of the DFT levels beyond the 125 Hz band i.e. above Schroder frequency. PBTM 125 band levels are significantly lower than the DFT levels below the Schroeder frequency. The reason for this underestimation is described in the previous discussion of the source room. This implies that the PBTM model can’t be used to reproduce the diffuse sound field below the Schroder frequency. However, the DFT values are in the vicinity of the PBTM prediction results in most bands above the Schroeder frequency. Significant phase effects are evident particularly in the low-frequency region, in both the source and receiver room causing in a marked deviation of the SPL levels from the DFT values.                Figure 4.5: Comparison between the DFT and PBTM octave predictions for the reference source room.   Figure 4.6: Comparison between the DFT and PBTM octave predictions for the reference receiver room.  43  4.1.2.2 Third-octave-Band Study for Reference Configuration The comparison between DFT and 1/3-octave-band PBTM converged results for 10580 beams are shown in next both source and receiver rooms. 4.1.2.2.1 Source room  Table 4.3: Comparison between DFT and third-octave-band levels in source room. DFT (dB) 1/3–octave-center frequency (Hz) for PBTM  93.80   100 125 160 200 250 315 400 500 630 800  92.50  84.95  92.88  87.92  91.86  96.91  94.38  96.31  96.41  97.07 Third –octave-center frequency (Hz) for PBTM 1000 1250 1600 2000 2500 3150 4000    95.73 97.28 95.91 96.52 98.00 97.49 97.41     The comparison between DFT and third-octave-band PBTM levels for source room is shown in Table 4.3. It is noticeable that PBTM levels are within the vicinity of DFT with 2-4 dB discrepancy in the higher frequency bands above 200 octave i.e. Schroeder frequency (198 Hz). Higher discrepancy between PBTM and DFT can be observed below the Schroeder frequency: 8.8 dB at 125 and 5.9 dB at 200 Hz. Above results indicate the dominant effects of modal effect at the low frequency region particularly below the Schroeder frequency, at 125 and 200 Hz since the wavelength at those bands are almost equal to the length of the room. Hence, PBTM cannot reproduce the sufficient diffuse-field at the low-frequency region below the Schroeder frequency.  4.1.2.2.2 Receiver room Table 4.4: Comparison between DFT and Third-octave-band levels in receiver room. DFT (dB) Third–octave-center frequency (Hz) 76.02    100 125 160 200 250 315 400 500 630 800 67.09 62.46 69.37 68.08 71.51 77.47 76.02 78.58 78.08 79.61 Third –octave-center frequency (Hz) 1000 1250 1600 2000 2500 3150 4000    79.70 77.89 77.29 77.95 79.92 79.25 78.96    44   The comparison between DFT and third-octave-band PBTM levels for receiver room is shown in Table 4.3. It can be observed that PBTM levels are within the vicinity of DFT with 2-4 dB discrepancy in the higher frequency bands above 315 octave i.e. Schroeder frequency (198 Hz). Higher discrepancy between PBTM and DFT can be observed below the Schroeder frequency bands: 9-14 dB at 100-200 octave bands. However, the discrepancy between PBTM and DFT in 400-4000 third-octave are comparatively less in receiver room. Receiver room results indicate that PBTM cannot reproduce the sufficient diffuse-field at the low-frequency region 100-250 third octave bands, below and close to the Schroeder frequency. Based on the above third-octave study, it can be concluded that although PBTM levels are in the vicinity of the diffuse-field in the mid and higher frequency bands, significantly deviation can be observed at the low-frequency particularly below the Schroeder frequency.  4.1.3 Validation of the New Phase-Based Model (PBTM) The applicability of the PBTM model has been further tested by comparing it to the COMSOL model. Details about the COMSOL model are available on page 22-23, section 2.3.4. Predictions are shown for the source room first and receiver room next.  Both the PBTM and COMSOL models are applied for the reference room configuration, cubic rooms (5m×5m×5m), (for details, see page 17-18, section 2.3.1). Both models calculate the RMS sound pressure levels at the receivers for 170-200 Hz with a frequency interval of 1 Hz. COMSOL simulations were preformed following recommended mesh convergence criteria (details are given in section 2.3.4).  4.1.3.1 Source Room PBTM and COMSOL predictions are shown in Figure 4.7 for the reference source room in the 170-200 Hz. Since these predictions are done by phase-based models, it is anticipated that the peaks and dips in the curves are the representation of the excited modes in the room. These local minima and maxima are a close cluster of excited modes. The maxima relate to a grouping of modes that are in similar phase while the minima are a group of modes that are nearly opposite in phase. The SPL curve between two models matches at most frequencies within 1.5 dB. However, SPL at 185-190 Hz frequency and 197-200 Hz frequency region deviate within 5 dB possibly due to the presence of any excited mode. 45             4.1.3.2 Receiver room PBTM and COMSOL predictions are presented in Figure 4.8 for the reference receiver room in the 170-200 Hz. The curves of both models fluctuate with frequencies as expected for phase-based models which resemble the source room. However, there are higher discrepancies in the predictions by the two models at more frequencies in the receiver room compared to the source room predictions. The SPL curve between PBTM and COMSOL matches at most frequencies within 4 dB. However, SPL at 185-190 Hz frequency and 197-200 Hz frequency region exhibits significant discrepancies.         Figure 4.7: Comparison of the PBTM and COMSOL predictions in the source room for the reference room configuration within 170-200 Hz.   Figure 4.8: Comparison of the PBTM and COMSOL predictions in the receiver room for the reference room configuration within 170-200 Hz.  46  Following points can explain the reasons behind the underestimation of the PBTM levels from the COMSOL particularly in the receiver room: COMSOL models the common wall boundary as infinite impedance boundary, while the PBTM applies the same, specific impedance boundary condition at all boundaries. An attempt has been made to fix this mismatch by using an infinite impedance condition for the common wall in the PBTM but has not reduced the discrepancies significantly. Despite the differences in predictions between the two models, the agreement trend is similar at the majority of the peaks and dips for the two models at the same frequency, particularly in the source room. From the above discussion, it is evident that the PBTM model has failed to reproduce the sound field predicted by COMSOL particularly in the receiver room. However, PBTM is used for achieving the research objectives because it somewhat reproduces the results that the diffuse-field theory predicts. 4.2 Energy-Based Studies Section 4.2 covers the preliminary studies of the energy-based studies; convergence, study for reference room configuration and the validation of the EBTM model. EBTM takes the absolute value of complex pressure amplitude ignoring the phase information [2]. All predictions shown in this work are performed including the air-absorption to be consistent with the CATT-TM and Odeon predictions. 4.2.1 Convergence      Similar to phase approaches, the convergence of the EBTM model is studied based on the number of the beam and reflections. However, the variation with the number of reflections is not shown for EBTM since the results of convergence for the number of reflections of EBTM are quite similar to PBTM given in the previous section. 4.2.1.1 Number of Beams  The number of beams is varied for 50 reflections, chosen from the reflection order convergence study in section 4.1.1. For brevity, the predictions results for 170 to 200 Hz with a frequency resolution of 10 Hz interval are shown in this section. The predicted SPL levels converge within 47  0.2 and 0.1 dB for the source and receiver room respectively as shown in Table 4.5. This simulation needs 12000 beams and 50 reflections. It is noticeable that EBTM model requires less number of beams and gives better accuracy (within 0.2 dB of converged value) while compared to around 1 dB for PBTM model. Since the EBTM levels do not vary significantly with frequencies, the predicted values are identical within 170-200 Hz frequencies. Hence, results are only shown for 170 Hz frequency region for both rooms in Table 4.5. However, the fluctuation of SPL can be exhibited at higher frequency octave bands. Table 4.5: Variation of SPL with the number of beams for 50 reflections, for 170 Hz frequency for the reference room configuration. Source Room Receiver Room  Beam Number  Beam Number  Frequency(Hz) 10000 12000 Discrepancy (dB) 10000 12000 Discrepancy (dB) 170 94.03 94.21 0.19 74.15 74.24 0.09  Octave band levels are calculated from by averaging the frequency-dependent EBTM SPL levels for 125 to 4 kHz band. The steady-state SPL from 89 to 5656 Hz with a frequency interval of 1 Hz are predicted by the EBTM model. This prediction needs 15680 beams with 50 reflections. The source and receiver octave band predictions are shown in Table 4.6 and 4.7 respectively. Predicted octave-band SPL levels converge within 0.69 and 0.11 dB in all bands for the source and receiver room respectively. Table 4.6: Variation of the source room octave-band levels with number of beams for a 1-Hz frequency interval  SPL levels (dB) for octave-band center frequency (Hz) Beam Resolution 125 250 500 1000 2000 4000 10580 Beams & 50 Reflections 94.03 94.00 93.95 93.84 93.47 93.05 12000 Beams & 50 Reflections 94.21 94.20 94.17 94.12 93.94 93.74 Discrepancy (dB) 0.19 0.20 0.23 0.28 0.47 0.69  48  Table 4.7: Variation of the receiver room octave-band levels with number of beams for 1-Hz frequency interval  SPL levels (dB) for octave-band center frequency (Hz) Beam Resolution 125 250 500 1000 2000 4000 10580 Beams & 50 Reflections 74.14 74.11 74.07 73.97 73.62 73.22 12000 Beams & 50 Reflections 74.24 74.21 74.16 74.07 73.72 73.33 Discrepancy (dB) 0.10 0.10 0.10 0.10 0.11 0.11  4.2.2 Study for Reference (Diffuse) Room Configuration         Section 4.2 shows a comparison between the DFT levels with that of the EBTM for the reference (diffuse) room configuration in Table 4.8 for both the source and receiver room.  Table 4.8: Comparison between the DFT and EBTM levels for the reference room configuration. Room DFT (dB) EBTM 1 kHz (dB) Discrepancy (dB) Source Room 93.80 94.12 0.32 Receiver room 76.02 74.07 1.95         It should be mentioned that the prediction uncertainty is high at low frequency. Therefore, the accuracy of DFT is tested by comparing with the EBTM levels at middle frequency band-1 kHz octave band above the Schroeder frequency, 189 Hz.  It is noticeable from the prediction results shown in table 4.5 that in source room, EBTM 1 kHz octave band levels are within 0.32 dB of the DFT levels for the reference source room (i.e. diffuse sound field). In the receiver room, EBTM 1 kHz octave levels are 1.95 dB of the DFT levels for the reference receiver room.  49  4.2.3 Validation of the Energy-Based Beam Tracing (EBTM) Model  The suitability of the EBTM model is tested further by comparing with the other applied energy-based prediction models; CATT-TM and ODEON [7]. Both beam and ray tracing models are applied here to reproduce the result from the diffuse-field theory for the reference (diffuse) room configuration. Table 4.9 demonstrates the comparison of all four energy methods for both reference source and receiver room.  Table 4.9: Comparison between the DFT and other energy-based predicted levels at 1 kHz octave for the reference room configuration. Room DFT EBTM (1 kHz) CATT-TM (1 kHz) ODEON (1 kHz) Source 93.80 94.12 93.30 93.25 Receiver 76.02 74.07 76.00 75.11  In the source room, the departure of the DFT levels from the 1 kHz levels of other three models are within 0.5 dB. Therefore, all the energy-based based methods are highly accurate in predicting the diffuse-field theory in the reference source room of cubic shape and uniform absorption of 0.1 (diffuse sound field).  In the receiver room, the deviation of the DFT levels from the EBTM 1 kHz levels is 1.95 dB; 0.02 from CATT-TM and 0.91 dB from ODEON. However, EBTM 1 kHz values are within 1 dB of ODEON and 2 dB of CATT-TM. Therefore, the energy-based methods are comparatively less accurate in predicting the diffuse-field theory in receiver room. Although all these energy models are different, their results are somewhat similar to the diffuse room configuration.    50  4.3 Summary  In chapter 4, convergence study is performed for both the PBTM and EBTM models with respect to the number of beam and reflections. Next, the second research objective is studied by investigating the comparing the 1/1 and 1/3 octave results of beam tracing model with the diffuse-field theory for a room configuration with diffuse sound field. Moreover, validation for both PBTM and EBTM models against other prediction models are shown. Here is the summary of the results obtained in this chapter:  Number of Reflections: 50 reflections are sufficient to ensure convergence in both the source and receiver room for both phase and energy predictions.  Number of Beams: Phase-based predictions needs 15000 beams for achieving an accuracy within 1 dB at most frequencies for both the source and receiver room while compared to 10580 beams are required to converge within 0.5 dB for the EBTM model. Therefore, the run time of the PBTM is significantly higher, 8 hours while compared to 5 hours for the predicting SPL values from 170-200 Hz in both the source and receiver room. For both phase and energy approaches, octave band values converges within 0.69 dB for a 1 Hz frequency interval for the source room. While, receiver room values converge within 0.4 dB for the same frequency interval.   Comparison between phase and energy predictions: Phase-based predictions exhibits greater fluctuation in sound pressure level with frequencies, particularly in the low-frequency region. On the contrary, energy-based predictions show much smoother and slighter variations with frequency for the same parameter.   Study for Reference (Diffuse Room Configuration):  Phase-based Study: PBTM predictions show 1-4 dB deviation in SPL levels from the diffuse-field theory in the source room throughout the octave bands. On the other hand, the departure of receiver room SPL levels from the diffuse-field levels are within 3 dB in the octave bands above the Schroeder frequency. PBTM octave results indicate that the PBTM model cannot reproduce the diffuse sound field below the Schroder frequency as expected.   Energy-based Study: Energy-based predictions show better accuracy in predicting diffuse-field theory in both source and receiver room while comparing with phase-based predictions. 51  The discrepancy of the EBTM levels from the diffuse-field levels are within 0.5 dB and 2 dB throughout the octave bands for the source and receiver room respectively.  Validation of the New Beam Tracing Model  Phase-based Approach: PBTM predictions show somewhat acceptable agreement with COMSOL predictions at the low-frequency region in source room. However, the PBTM model has failed to reproduce the sound field predicted by COMSOL in the receiver room.     Energy-based Approach: EBTM predictions show better agreement with both CATT-TM and ODEON models in the source room. The discrepancies between the EBTM and ray models are within 1 and 2 dB for the reference (diffuse) source and the receiver room respectively.  Overall, EBTM predictions showed faster convergence compared to PBTM and better agreement with the DFT for the reference room configuration. Hence, only the EBTM model is applied to study the non-diffuse room configurations in the next chapter.                       52  Chapter 5 Objective Studies   Chapter 5 begins with the application of the EBTM model for the non-diffuse room configurations. The aim is to investigate the second objective by studying the effect of varying different room characteristics (e.g. magnitude of surface absorption, room aspect ratio, surface absorption distribution, combined effects of room shape and absorption distribution) on predicting the diffuse sound field. This is done by calculating the steady-state SPL for each test configurations and comparing with the DFT levels. In addition to that, the third objective, Waterhouse effect is investigated by both the PBTM and EBTM model for the reference room configuration in section 5.2. Waterhouse effect should only exist in the PBTM model due to the presence of phase effect. To scientifically confirm this fact, the capability of the EBTM model to predict the Waterhouse effect is tested.  5.1 Case Studies  All case studies were performed in section 5.1 using the common parameter values provided in page 16-17, section 2.3.1 reference configuration.   5.1.1 Effect of Magnitude of Surface Absorption  The lower is the average absorption coefficient, the more diffuse is the sound field [4]. The effect of varying the surface absorption on the accuracy of the DFT is investigated by applying the EBTM and CATT-TM 125 octave band levels. For both the source and receiver room, the average surface absorptions are varied from low to high absorption while keeping the shape of room unchanged from the reference configuration. Seven different absorption cases are studied for both the rooms for the value of absorption coefficient α ranging from 0.03 to 0.4.   53  5.1.1.1 Source Room   Figure 5.1 shows the variation of the energy-based predicted levels with the magnitude of surface absorption for the source room.                             Figure 5.1 shows a trend of the discrepancy between the DFT, EBTM and CATT-TM levels for varying surface absorptions in the source room. It is noticeable that the discrepancy of both the energy models; EBTM and CATT-TM levels from DFT levels increases gradually with the increase of the average surface absorption value. Following findings can be drawn from the comparison between the three prediction models levels in the source room. EBTM levels depart from the DFT levels within 0.15 dB from 0.05-0.1, low-medium absorption cases but this departure increases to 0.73 dB for the 0.4 absorption case. This is expected as the accuracy of the diffuse-field reduces with increasing of average surface absorption.  The difference between the DFT and CATT-TM levels doesnot show much variation with varying the surface absorptions; discrepancies are within 0.56 dB for all the absorption cases.   Figure 5.1: Comparison between the DFT, EBTM and CAT-TM levels with varying surface absorptions, α in the source room. 54  Moreover, the difference between the CATT-TM and EBTM levels are around 0.5 dB up-to absorption 0.1 case however it increases to 0.7 and 1.15 dB for the 0.2 and 0.4 absorption case respectively. It can be concluded that the difference between the energy-based predicted levels increases with increasing of the average surface absorption in the source room. 5.1.1.2 Receiver Room   The variation of energy-based predicted levels with the magnitude of surface absorption for the receiver room is demonstrated in Figure 5.2.                        Receiver room EBTM predictions show some contrasting results compared to the source room. It can be observed that the departure of EBTM levels from the DFT decreases with an increase of its average absorption. The agreement between these two methods is not as good as the source room particularly in the low absorption cases as expected. EBTM levels underestimate the DFT levels by 3-2 dB for 0.03-0.07: low absorption; 1-2 dB for medium absorptions: 0.1-0.2 and 0.08 dB for 0.4 absorption. For the reference case: absorption of 0.1, 1.78 dB discrepancy is observed between the EBTM and DFT predicted levels.     Figure 5.2: Comparison between the DFT, EBTM and CATT-TM levels with varying surface absorption, α in the receiver room. 55  The difference between the DFT and CATT-TM is within 1 dB from 0.03-0.2 absorption cases but 1.92 dB for the 0.4 absorption case. These results show that lower is the average surface absorption, more accurate is the diffuse-field theory. CATT-TM levels closely match with the DFT values within 0.5 dB in the low absorption values. The discrepancies from DFT level rises with the increase of the value of surface absorption as expected. Besides, the comparison between the CATT-TM and EBTM exhibits that both methods match well at around the medium absorption 0.15 but deviates more; 2-2.5 dB in absorption 0.03-0.1 and 0.4 cases. These discrepancies are comparatively higher at both the low and higher absorption cases. Overall, best agreement between them is obtained in the medium absorption for the receiver room. In fine, the receiver room EBTM predictions haven’t produced the same conclusion as the source room that the less is the absorption value, more diffuse is the sound field. EBTM is close to predicting the diffuse-field theory for the medium absorptions while compared to the low and high absorption cases. However, CATT-TM model has shown better agreement in predicting the diffuse-field theory particularly in the low absorption of the receiver room.  5.1.2 Effect Of Room Shape (For Uniform Absorption) In section 5.1.2, the shapes of both source and receiver rooms are varied from the reference room while keeping the absorption of the surfaces uniform. For all four room configurations, the room aspect ratio is increased by increasing the length of the cubic receiver room by two and five times from the reference length respectively by always keeping the constant width and height of 5 m.   5.1.2.1 Between a Small Cubic Office Room and a Large Room.      The new model is applied to study the effect of room shape on the accuracy of DFT between a  small empty cubic office room (source room) and a large room (receiver room). The dimensions of the source and receiver rooms are 5m×5m×5m and 10m×5m×5m respectively; all surfaces have a uniform average absorption of 0.1, same as the one used by the reference configuration. EBTM predictions for both source and receiver room are presented in Table 5.1. 56  Table 5.1: Comparison between the DFT and EBTM levels between a small cubic office room and large room.   Steady-State SPL Levels (dB)  Room Dimensions DFT EBTM Discrepancy (dB) Source 5m×5m×5m  93.80 94.03 0.23 Receiver 10m×5m×5m 73.80 71.59 2.10  SPL levels remained same for both models in the small office room from 5.1.1. This is expected since the shape of the source room is kept unchanged.  The discrepancy of the EBTM levels from the DFT rises to 2.21 dB from 1.87 dB in the large room. So, 0.5 dB more discrepancy is observed in the large receiver room for increasing the room aspect ratio of the receiver room from the predicted value of the diffuse configuration study of section 4.2.2.  5.1.2.2 Between a Large Room and a Small Cubic Office Room.      The second test configuration is a small empty cubic office room (source room) and a large room (receiver room). The shape of the receiver room is varied while keeping the source room same as the 5.1.1. The dimensions of the source and receiver rooms are 5m×5m×5m and 10m×10m×10m respectively; all surfaces have a uniform average absorption of 0.1, identical to the one used in the reference configuration. EBTM prediction results are given in Table 5.2 for both source and receiver room. Table 5.2: Comparison between the DFT and EBTM levels between a large room and a small cubic office room.   Steady-State SPL Levels (dB)  Room Dimensions DFT EBTM Discrepancy (dB) Source 10m×5m×5m  91.58 91.15 0.43 Receiver 5m×5m×5m 73.80 71.30 2.50  The discrepancy of the EBTM levels from DFT increases from 0.23 to 0.43 dB in the large source room; from 2.21 to 2.5 dB in the small office room. This is only for doubling the length of source room (more non-diffuse sound field) from reference (diffuse) room configuration levels of section 57  4.2.2. So, increasing the aspect ratio of the source room results in an increase of 0.4 dB discrepancy between the receiver room EBTM and DFT levels. 5.1.2.3 Between Two Large Rooms (10m×5m×5m) The third test configuration is two large rooms, identical source and receiver room of 10m×5m×5m. Both the source and receiver room prediction results are shown in Table 5.3. Table 5.3: Comparison between the DFT and the EBTM predictions between two large rooms (10m×5m×5m).   Steady-State SPL Levels (dB)  Room Dimensions DFT EBTM Discrepancy (dB) Source 10m×5m×5m 91.58 91.15 0.43 Receiver 10m×5m×5m 68.93 71.3 2.65  The discrepancy of the EBTM levels from DFT levels are 0.43 and 2.5 dB for two large rooms. However, the discrepancy between EBTM and DFT levels increases by 0.23 and 0.78 dB for only doubling the length of the reference source and receiver room respectively. Therefore, it can be concluded that the discrepancy of the EBTM levels from DFT levels decrease with increasing the aspect ratio of the room from cubic shape (non-diffuse sound field).  5.1.2.4 Between Two Large Rooms (25m×5m×5m)   The fourth test configuration is more large rooms; identical source and receiver room of 25m×25m×25m. Both the source and receiver room results are shown in Table 5.4. Table 5.4: Comparison between the DFT and EBTM results for two large rooms (25m×5m×5m).   Steady-State SPL Levels (dB)  Room Dimensions DFT EBTM Discrepancy (dB) Source 25m×5m×5m 88.16 86.85 1.31 Receiver 25m×5m×5m 64.74 60.58 4.16  58  EBTM levels are lower than the DFT levels by 1.31 and 4.16 dB in the large source and the receiver room respectively. DFT is 1.08 and 2.29 dB higher than EBTM levels for just increasing the length of the reference source and receiver room by five times respectively. Therefore, the discrepancies of the EBTM levels from the diffuse-field rises gradually with increasing the aspect ratio of both source and receiver rooms for long and narrow rooms (non-diffuse sound field) from the reference room configuration (diffuse sound field) as expected.   5.1.3 Effect of Surface Absorption Distribution  The absorption distribution is made non-uniform from the reference room configuration of 4.2.2 by placing most absorptions in one of the room surfaces while always keeping the same equivalent absorption area in both the source and receiver room. 5.1.3.1 Most Absorptions in the Ceiling  In section 5.1.3.1, the absorption distribution is made non-uniform by placing most absorptions in the ceiling. Equivalent absorption area for reference configurations, 0.1×150 m2 i.e. 15m2 is distributed among the room surfaces keeping same equivalent absorption area as follows; 0.55×25 m2 (ceiling)+ 0.01×125 m2 (remaining 5 surfaces except ceiling) = 13.75+1.25 = 15 m2. The results for the most absorptions in the ceiling case are shown for both source and receiver room in Table 5.5.  Table 5.5: Comparison between the DFT and EBTM levels for the non-uniform absorption (most absorption in the ceiling).   Steady-State SPL Levels (dB)  Room Dimensions DFT EBTM Discrepancy (dB) Source 5m×5m×5m  93.8 95.45 1.6 Receiver 5m×5m×5m 76.02 79.42 3.4  For the same cubic room with most absorptions in the ceiling (non-diffuse sound field), the discrepancy of the EBTM levels from the DFT levels increases to 1.6 dB and 3.4 dB. These discrepancies are 1.42 dB and 1.53 dB higher in the source and receiver room respectively 59  compared to reference configuration with uniform absorption (diffuse sound field). This shows that the surface absorption distribution has a substantial effect on the accuracy of the diffuse-field theory.  5.1.3.2 Most Absorptions in the Floor  The results for the non-uniform surface absorption distribution with most absorptions in the floor are shown in Table 5.6 for both the source and receiver room.  Table 5.6: Comparison between DFT and EBTM for the non-uniform absorption (most absorptions in the floor)   Steady-State SPL Levels (dB)  Room Dimensions DFT                      EBTM Discrepancy (dB) Source 5m×5m×5m  95.32 93.80 1.52 Receiver 5m×5m×5m 80.01 76.02 3.99  For the same cubic room with most absorptions in the floor (non-diffuse sound field), the discrepancy of the EBTM levels with the DFT increases to 1.52 dB and 3.99 dB in the source and receiver room respectively. These discrepancies are 1.29 dB and 2.12 dB higher than the reference configuration with uniform absorption (diffuse sound field).  Moreover, the discrepancy between the EBTM and the DFT levels decreases by 0.1 dB in the source room but rises by 0.59 dB in the receiver room compared to the 5.1.3.1; ceiling case. This implies that changing the surface index: (floor or ceiling) doesn’t affect the accuracy of DFT significantly.  Overall, both these predictions show that the surface absorption distribution (magnitude of absorption in each surface has a major effect on the accuracy of the diffuse-field theory.  5.1.4 Combined Effects (Room Shape & Absorption Distribution) The combined effects of room shape (Section 5.1.2) and absorption distribution (Section 5.1.3) in the applicability of the DFT are investigated by using the EBTM model. This analysis is performed for the combination of three previous room configurations, applied in section 5.1.2.1, 5.1.2.2 and 60  5.1.2.3 with most absorptions in the ceiling (5.1.3.1). The predictions results of each configuration are described as follows.  5.1.4.1 Between a Small Cubic Office Room and Large Room with Most Absorptions in Ceiling  The DFT and EBTM predictions of the first case between a small empty cubic office room (source room) and large room (receiver room) are presented in Table 5.7.  Table 5.7: Comparison between the DFT and EBTM levels between a small cubic office room and large room with most absorptions in the ceiling.   Steady-State SPL Levels (dB)  Room Dimensions DFT EBTM Discrepancy (dB) Source 5m×5m×5m  93.80 94.45 0.65 Receiver 10m×5m×5m 73.8 76.69 2.89  For changing both the room shape and absorption distribution, the discrepancy of EBTM levels from the DFT values are 0.65 dB and 2.89 dB. These discrepancies are 0.42 dB and 0.78 higher from the uniform absorption case 5.1.2.1 for the source and receiver room respectively. This shows that the departure of the EBTM values from DFT values decreases further for combined effects of room shape and non-uniform absorption distribution while compared to the uniform absorption case (same room shape). Receiver room levels show greater discrepancy from the DFT in comparison with the source room, since both the shape and absorption distribution of only the receiver room are varied from the reference room configuration of 4.2.2. 5.1.4.2 Between a Large Room and Small Cubic Office Room with Most Absorptions in the Ceiling The DFT and EBTM predictions of the second case for both source and receiver room are shown in Table 5.8.  61  Table 5.8: Comparison between the DFT and  EBTM levels between a large room and a small cubic office room with most absorptions in the ceiling.   Steady-State SPL Levels (dB)  Room Dimensions DFT EBTM Discrepancy (dB) Source 10m×5m×5m  91.58 93.66 2.08 Receiver 5m×5m×10m 73.8 77.01 3.21  Combined effects of both room shape and absorption distribution resulted in a discrepancy of 2.08 and 3.21 dB between the EBTM and DFT levels. These discrepancies are 1.3 dB and 0.71 dB more from the uniform absorption case 5.1.2.2 for the source and receiver room respectively. This shows that the departure of the EBTM levels from DFT increases considerably for changing both room shape and non-uniform absorption distribution from uniform absorption case 5.1.2.2. The departure is, in particular, greater in the source room compared to the receiver room as both the shape and absorption distribution of only the source room are varied from the reference cubic room configuration of 4.2.2. 5.1.4.3 Between Two Large Rooms with Most Absorptions in Ceiling  The DFT and EBTM results of the third case between two large rooms, identical source and receiver room are demonstrated in Table 5.9.  Table 5.9: Comparison between the DFT and the EBTM predictions between two large rooms (10m×5m×5m) with most absorptions in the ceiling.   Steady-State SPL Levels (dB)  Room Dimensions DFT EBTM Discrepancy (dB) Source 10m×5m×5m 91.58 92.58 1.22 Receiver 10m×5m×5m 71.58 74.37 2.78  For changing both room shape and absorption distribution between two large rooms, the discrepancy between the EBTM and DFT levels becomes 1.22 dB and 2.78 dB. These discrepancies are 0.82 dB and 0.55 dB more from the uniform absorption case 5.1.2.3 for the source and receiver room respectively. This shows that the discrepancy of the EBTM levels from 62  DFT rises further for combined effects of room shape and non-uniform absorption distribution while compared to the uniform absorption case 5.1.2.3. Receiver room levels show a greater discrepancy from the diffuse-field theory in comparison with the source room of identical shape which resembles with the prediction for reference configuration 4.2.2.  5.2 Waterhouse Effect Study   Section 5.2 investigates the Waterhouse effect using the PBTM and EBTM models and to compare with the prediction results reported in the literature [5]. For both the PBTM and EBTM model, the steady-state pressure levels are calculated at four source and receivers positions; middle, single-corner, double corner and triple corner in both the reference source and receiver room. For both the studies, simulations are done for 15000 beams and 50 reflections. 5.2.1 Phase Approach   PBTM model is used to calculate the SPL levels by varying the receiver positions for same source position and vice versa. The receiver SPL are expressed in 125-octave band at four receiver cases by averaging the levels at each 1 Hz interval over total ten receiver positions; (three in the middle) and two receivers in each one, double and triple corners in both the source and receiver room.  5.2.1.1 Phase Approach (Source Room)  Following findings can be obtained from the phase predictions in the source room shown in Table 5.10. For the source sufficiently far away, the SPL level at one-wall, two-wall, and three-wall reflectors are found to be 2.32 dB, 3.12 dB, and 4.63 dB higher than at the middle where the interference patterns are negligible. These obtained rise of levels at the corners of the room are somewhat consistent with that of Waterhouse study that the energy density rises by 2.2 dB, 6 dB, and 9 dB in the one-wall, two-wall, and three-wall reflectors of a rectangular room [5]. The SPL level at the middle increases by 6.71 dB, 7.75 dB, and 13.8 dB when the source is placed at one-wall, two-wall, and three-wall reflectors respectively for the source room.  63  For placing both the source and receiver at one-wall and two-wall, the SPL level at the middle rises to around 9-10 dB than at remote points near the middle of the room. The largest departure from uniformity occurs when both source and receiver are at three-wall reflectors. Recorded source room values are 17 dB, and 14 dB more than the recommended case in the source and receiver are kept far apart from the surfaces, i.e. middle.  Table 5.10: Waterhouse effect (phase) results for the various receiver and source positions in the source room.   Receiver Positions  Source Positions Middle (R1) One-wall (R2) Two-wall (R3) Three-wall (R4) 1  Away from walls (S1) 85.56 87.88 88.78 90.19  Departure(dB) Reference 2.32 3.22 4.63 2 One-wall (S2) 92.27 97.20 97.29 99.19  Departure(dB) 6.71 4.93 5.02 6.92 3 Two-wall (S3) 93.31 98.01 98.45 101.25  Departure(dB) 7.75 4.70 5.14 7.94 4 Three-wall (S4) 99.32 100.48 102.31 107.32  Departure(dB) 13.80 1.160 2.99 8.00  5.2.1.2 Phase Approach (Receiver Room)  Following outcomes are achieved from the phase predictions in the receiver room shown in Table 5.11.  For the source sufficiently far away, the SPL level at one-wall, two-wall, and three-wall reflectors are 2.73 dB, 3.71 dB, and 6.43 dB higher than at the middle where the interference patterns are negligible. These significant rise of levels at the corners of the room are consistent with the observation made by Waterhouse that the energy density rises by 2.2 dB, 6 dB and 9 dB in the one-wall, two-wall, and three-wall reflectors of a rectangular room. In the receiver room, the SPL level at the middle increases by 6.86 dB, 9.09 dB, and 14.36 dB when the source is placed at one-wall, two-wall and three-wall reflectors respectively.  64  For placing both the source and receiver at one-wall and two-wall reflectors, the SPL level at the middle rises to around 9-10 dB than at remote points near the middle in both source and receiver room. The largest departure from uniformity occurs when both source and receiver are at three-wall reflectors. Recorded source and receiver room values are 14 dB more than the recommended case in the source and receiver are kept far apart from the surfaces, i.e. middle.  Table 5.11: Waterhouse effect (phase) results for the various receiver and source positions in the receiver room.   Receiver Positions Receiver Position  Source Positions Middle (R1) One-wall (R2) Two-wall (R3) Three-wall (R4) 1  Away from wall (S1) 66.11 68.84 69.83 72.54  Departure(dB) Reference  2.73 3.71 6.43 2 One-wall (S2) 72.97 73.37 76.73 77.11  Departure(dB) 6.86 0.4 3.76 4.14 3 Two-wall (S3) 75.21 77.44 78.63 79.93  Departure(dB) 9.09 2.23 3.42 4.72 4 Three-wall (S4) 80.47 82.88 82.91 85.10  Departure(dB) 14.36 2.41 2.44 4.63  5.2.2 Energy Approach (Source Room, Receiver Room)  EBTM results are shown by neglecting the effect of air-absorption at various receiver positions for each source position (see Table 5.12). The SPL at one-wall, two-wall and three-wall reflectors decrease by 0.53 dB, 1.89 dB and 3.8 dB respectively from the recorded value at the middle in the source room. In the receiver room, levels at one-wall, two-wall, and three-wall reflectors drop by 0.25 dB, 1.49 dB and 3.36 dB respectively from the value at the middle. The largest decrease from uniformity occurs when both source and receiver are at triple corner. These energy results are quite contrasting to the obtained phase results at the reflectors for both source and receiver position. This could be explained by the spatial decay of SPL only decreases inversely with increasing the 65  source-receiver distance as there is no effect of phase in the EBTM model. Therefore, it can be concluded that the Waterhouse effects can’t be obtained by this model.   Table 5.12: Waterhouse effect (energy) results for the various receiver and source positions in both the source and receiver room. Beam/Reflection-15000/50   Receiver Positions Receiver Position Room Source positions   Middle (R1) One-wall (R2) Two-wall  (R3) Three-wall (R4) Source Away from wall-S1  SPL (dB)  94.21 93.68 92.32 90.41 Departure (dB)   Reference -0.53 -1.89 -3.8 Receiver SPL (dB)  74.24 73.99 72.74 70.88 Departure   Reference -0.25 -1.49 -3.36   5.3 Summary   In chapter 5, the second research objective is further studied by investigating the capability of EBTM in reproducing the results of the diffuse-field theory for non-diffuse rooms. In particular, the effects of varying the room aspect ratio, magnitude of surface absorption, surface absorption distribution, combined effects of room shape and absorption distribution) were studied. The third objective to investigate the Waterhouse effect is studied by applying both the PBTM and EBTM models. This is carried out by investigating the effects of varying both source and receiver positions for the reference room configuration. The results obtained in this chapter can be summarized as follows:  Effect of magnitude of surface absorption: The discrepancies of the EBTM and CATT-TM models increases with an increase of its average absorption in the source room. Both models agree well with the DFT levels mostly within 0.5 dB departure for the low-medium absorptions cases in the source room. On the other hand, in the receiver room, EBTM shows better agreement with the DFT values for the medium absorptions while compared to the low and high absorption cases. Compared to the EBTM, CATT-TM has shown better agreement in predicting the DFT particularly in the low absorption of the receiver room. 66   Effect of room shape (for uniform absorption): The agreement between the EBTM and the DFT value decreases significantly with the increase of aspect ratio of the room; for long rooms (non-diffuse sound field).   Effect of surface absorption distribution (non-uniform absorption): The discrepancy from the diffuse-field for the non-uniform absorption distribution are around 1.5 and 2.2 dB higher compared to the uniform distribution for the rooms of the same shape.   Combined effect of room shape and surface absorption distribution: The discrepancy of the EBTM levels from the DFT increases further by around 1.3 and 0.8 dB in the source and receiver rooms respectively for combined effects of room shape and absorption distribution while compared to that of the uniform absorption case.   Comparison between the effect of room shape and absorption distribution: The discrepancies between the EBTM and DFT are comparatively less for changes in the shape of the room compared to the non-uniform absorption distribution case.   Waterhouse effect study: These phase studies validate the findings of Waterhouse [5].  Following findings can be drawn from the PBTM Waterhouse effect study:  For the source sufficiently far away from the surface, in both source and receiver room, the SPL level at one-wall, two-wall, and three-wall reflectors are 2.32-2.73 dB, 3.12-3.71 dB, and 4.63-6.43 dB higher than at the middle of the cubic room where the interference patterns are negligible.  In both source room and receiver room, the SPL level at the middle of the room increases considerably by around 6 to 14 dB when the source is placed at one-wall, two-wall, and three-wall reflectors of the reverberation room.  The largest departure from uniformity occurs when both or either of source and receiver is at the three-wall reflector. The SPL value are found to be around 17 dB, and 14 dB higher than the recommended case suggested by Waterhouse paper [5] when both source and receiver are kept far away from the surfaces.  67  Chapter 6 Conclusions The focus of this thesis was to study steady-state sound transmission between rooms with an aim to achieve three main objectives. The first objective was to extend the existing beam-tracing model to implement steady-state sound transmission between rooms separated by a homogenous partition. The second objective was to investigate whether the new beam-tracing models can reproduce the results of the diffuse-field theory. And, the third one was to investigate the Waterhouse effect phenomena using the beam tracing models. Based on these objectives, main contributions of this work can be summarized as:  6.1 Contributions We started by presenting the importance of the sound transmission between rooms in room acoustics and available state of works on it from the literature review. It was revealed that there was a lack of a phase-based room prediction model for predicting sound transmission between rooms. Therefore, steady-state sound transmission was modeled in both phase and energy-based beam tracing models (PBTM and EBTM). This is the first implementation of sound transmission in a beam tracing model. Feature for adding any number of receivers in both the source and receiver room was incorporated into the new model. Convergence of both the models was studied with respect to the number of beams and reflections. A room configuration with a diffuse sound field (reverberation room) was chosen based on the requirements of the reverberation room standard (ASTM 3423) and recommendations from the literature. Besides, other predictions tools were developed to predict room-to-room transmissions such as CATT-TM, COMSOL, and DFT. The range of applicability of each of these prediction models were discussed. DFT model was built to calculate the theoretical value of the diffuse-field theory for both the source and receiver room. For the chosen reverberation room, the capability of the PBTM, EBTM, and CATT-TM in predicting the diffuse-field theory were investigated.  68  The EBTM was then used to predict the sound fields when the following attributes of the reference room configuration were changed: the room aspect ratio, the magnitude of the surface absorption, and the surface absorption distribution. These changes are known to result in a non-diffuse sound field. As expected, the EBTM predictions deviated from DFT as the room characteristics deviated from the reference configuration.  The work is concluded by studying the Waterhouse effect using both EBTM and PBTM in both reverberation source and receiver rooms.  6.2 Summary of Results The following points summarize the key findings of this work. Phase and Energy-based approach: Results of varying number of reflections revealed that and 12000 beams and 50 reflections are sufficient to obtain convergence within 0.5 dB. For a given set of parameters, PBTM results exhibit greater fluctuations with frequency than EBTM. These fluctuations are caused by the interference patterns that cannot be captured using the EBTM. Hence, the energy-based model needs fewer number of beams than the phase-based model.  Phase-based model shows an acceptable agreement with results from finite element analysis (COMSOL) at the low-frequency region in source room. However, the beam model has failed to reproduce the sound field predicted by the COMSOL in the receiver room. On the other hand, the EBTM results match well with the diffuse-field and ray models (ODEON and CATT-TM) for the diffuse room configuration. The predicted levels agree among all four models are within a reasonable discrepancy of 0.5 dB in the source room and 2 dB in the receiver room.  Studies for Reverberation Room (Phase, Energy): Phase-based beam tracing shows 0.5-4 dB discrepancy from the diffuse-field theory in source room throughout the octave band levels. On the other hand, the receiver room octave levels show similar range of discrepancy with the diffuse-field levels above the 125 Hz octave band. PBTM cannot reproduce the results of the diffuse field theory near and below the Schroeder frequency in the receiver. At higher frequencies, the PBTM results have an acceptable agreement with the diffuse-field theory. 69  Energy-based predictions show better accuracy in predicting the diffuse-field theory in both the source and receiver room compared to phase-based predictions. The discrepancy of the energy-predictions from the diffuse-field levels are within 0.5 dB and 2 dB throughout the octave bands for the source and receiver room respectively. EBTM predictions for non-diffuse rooms:  Effect of Surface Absorption: Diffuse-field levels show excellent agreement with EBTM results in the source room for low-medium absorption absorptions cases. However, the beam tracing results departs considerably from the diffuse-field values in the receiver room, particularly in the low and medium absorption cases.  Effect of Room Aspect Ratio: The accuracy of the EBTM in reproducing the diffuse sound field decreases considerably with the increase of aspect ratio of the room. This discrepancy increases more significantly as the room shape (aspect ratio) deviates more from a cube.  Effect of Surface Absorption Distribution: Non-uniform surface absorption results in a considerable deviation from the diffuse-field theory compared to the room with a uniform distribution of its absorption. Waterhouse Effect Study: The results of the PBTM model in both reverberation rooms validate the findings of Waterhouse in a rectangular room [5]. The EBTM cannot capture the Waterhouse effect because it is caused by the interference effects near a reflecting surface. The Waterhouse effect results in increase of sound pressure levels near reflecting surfaces (compared to the diffuse field value). These rises of SPL are higher for the receiver room than the source room. Moreover, placing sources at the reflectors results in a greater rise of levels compared to receivers. The largest rise of SPL occurs when either source or receiver is at three-wall reflectors.       70  6.2 Limitations and Future Work Following points discuss the limitations of the work:  Uses a constant frequency independent transmission loss and average absorption value for all frequency regions in the new models: DFT, PBTM and EBTM and CATT-TM. ASTM 3423 recommends different absorption values for the low and high frequency region.  Greater discrepancy between the EBTM with DFT and CATT-TM results of the receiver room was exhibited than the source room.  Validation of PBTM with Finite element method (COMSOL) was presented only in the in low-frequency region (170-200 Hz).   SPL was measured in 1 receiver in each reverberation room compared to recommended 3 or 5 receivers in the reverberation room standard ASTM 3423 standard.  Study of the modal characteristics of the reverberation room (in particular, modal density and 1/3 octave test for low-frequency limit were not performed).   Experimental validation of the predictions models, in particular beam-tracing and FEM were not presented.   Computational time of the new beam tracing model has increased significantly from the previous model.  The sound pressure levels (dB) were not expressed in weighted decibels (dBA) to quantify the sensitivity of human ears at different intensities of the sound.  The detailed recommendations are presented below to improve the limitations of the current work in future.  Reverberation-Room Study:  Because of both time and computational restrictions, we were not able to predict other parameters except for sound pressure levels. Also, all the features of a reverberation room as prescribed by standard ASTM 3423 weren’t maintained (e.g. placing 3 or 5 receivers in each room and choosing different average absorption coefficient value for low and high-frequency region). Moreover, the modal characteristics of the tested reverberation room configuration (e.g. modal density, spacing, 71  etc. should be studied particularly at the low-frequency-region. Moreover, 1/3 octave low-frequency test needs to be studied instead of Schroeder-test since it provides better spatial uniformity in a reverberation room. Experimental Validation: The new beam tracing model (both PBTM and EBTM) and COMSOL predictions needs to be validated experimentally particularly in the receiver room. It was not possible due to lack of required experimental facility (e.g. a transmission suite having two reverberation rooms separated by a homogenous common wall) and also lack of matching experimental data in the available literature.  Improving the Computational Efficiency: The computation time of the new beam tracing model has increased considerably from the previous model. This is largely caused by the addition of a separate reflection loop for the receiver room. Moreover, a higher number of beams are needed for simulating two rooms in the new model compared to prediction within a single room in the previous model. However, the coding efficiency could be improved further by enabling the reflection and transmission in both the source and receiver by using only one reflection loop. Incorporating Weighted Decibel: The decibel (dB) quantifies the physical amplitude of a sound, independent of the frequency. It clearly does not quantify the sensitivity of humans to sound since this varies with frequency. For instance, SPL (dB) can be filtered through A filtering network to obtain A-weighted decibel (dBA), quantifying the human hearing response at low intensities. Future work could involve expressing the predicted SPL in rooms in dBA. Predicting More Arbitrary Shape Rooms: Present work has studied the effect of the shape of the room in predicting the diffuse sound field. This is done by only by increasing the aspect ratio with changing only the length of the reverberation room. Further works could involve examining the accuracy of the new model in producing a diffuse sound field for more arbitrary shape rooms. This could be performed by changing all three dimensions of the room.  Modelling Composite Partitions: In the case of a source and a receiver room separated by a homogenous partition of a single value of Sound Transmission Class (STC), SPL between rooms is not expected to vary much with source and receiver position since all parts of the partition has the same material, i.e. transmission loss. In the case of a composite partition made of different 72  materials (e.g. a door, a window, drywall), however, the transmission loss of different parts of the partition may be significantly different. The resulting speech pressure level may vary remarkably with source and receiver position which can cause significant reduction of the speech privacy condition between two rooms. Therefore, future work could model the room-to-room sound transmission between the composite transmitting partitions.  Modelling Multilayer Surfaces: Only single-layered room surfaces are applied in the current work. However, predictions could be done for multilayer room surfaces, i.e. solid, fluid or poroelastic materials in both the source and receiver room. This could be accomplished by integrating the transfer matrix algorithm into the new beam tracing model.                73  Bibliography 1. Wareing, A. (2000). Acoustical modeling of rooms with extended-reaction surfaces (M.A.Sc. Thesis, University of British Columbia).  2. Yousefzadeh, B. (2010). Acoustical modeling of the transient response of rooms using a beam-tracing model (M.A.Sc. Thesis, University of British Columbia).  3. Hodgson, M., & Bradley, J. (2006). Sound in rooms. Noise and Vibration Control Engineering: Principles and Applications, Second Edition, 181-214.  4. Hodgson, M. (1996). When is diffuse-field theory applicable?. Applied Acoustics, 49(3), 197-207.  5. Waterhouse, R. V. (1955). Interference patterns in reverberant sound fields. The Journal of the Acoustical Society of America, 27(2), 247-258. 6. Billon, A., Foy, C., Picaut, J., Valeau, V., & Sakout, A. (2008). Modeling the sound transmission between rooms coupled through partition walls by using a diffusion model. The Journal of the Acoustical Society of America, 123(6), 4261-4271.  7. Rindel, J. H., & Christensen, C. L. (2008). Modelling airborne sound transmission between coupled rooms. Proceedings of Baltic-Nordic Acoustic Meeting (BNAM), Iceland.  8. Cousins, O. M. (2008). Prediction of sound pressure and intensity fields in rooms and near surfaces by ray tracing (M.A.Sc. Thesis, University of British Columbia).  9. Hasan, M. M. (2015). Diffuse sound fields, reverberation-room methods and the effectiveness of reverberation-room designs (M.A.Sc. Thesis, University of British Columbia).  10. Ramakrishnan, R., & Grewal, A. (2008). Reverberation rooms and spatial uniformity. Canadian Acoustics, 36(3), 28-29.  11. Schroeder, M. R. (1962). Frequency‐Correlation Functions of Frequency Responses in Rooms. The Journal of the Acoustical Society of America, 34(12), 1819-1823. 12. Sabine W.C. (1932). Collected Papers on Acoustics, New York, Dover reprint. 13. Hodgson, M. (1993). Experimental evaluation of the accuracy of the Sabine and Eyring theories in the case of non‐low surface absorption. The Journal of the Acoustical Society of America, 94(2), 835-840.  14. Vigran, T. E. (2008). Building acoustics. CRC Press.  15. Lamancusa J. S. (2000) Noise control 9.1: Transmission, Penn State.  74   16. ISO, B. 140-4: 1998 Acoustics–Measurement of sound insulation in buildings and of building elements. Part 4. Field measurements of airborne sound insulation between rooms.  17. ASTM, E336-11: 2014. Standard Test Method for Measurement of Airborne Sound Attenuation between Rooms in Buildings. 18. Ingard, K. U., Huntley, R., Jack, W. A., Summerfield, M. B., Tyzzer, F., & Waterhouse, R. V. (1959). American Standard Recommended Practice for Laboratory Measurement of Air‐Borne Sound Transmission Loss of Building Floors and Walls. The Journal of the Acoustical Society of America, 31(1), 76-78. 19. Gagliardini, L., Roland, J., & Guyader, J. L. (1991). The use of a functional basis to calculate acoustic transmission between rooms. Journal of Sound and Vibration, 145(3), 457-478.  20. Kang, H. J., Ih, J. G., Kim, J. S., & Kim, H. S. (2000). Prediction of sound transmission loss through multilayered panels by using Gaussian distribution of directional incident energy. The Journal of the Acoustical Society of America, 107(3), 1413-1420.  21. ASTM C423, (2009). Standard test method for sound absorption and sound absorption coefficients by the reverberation room method, American Society for Testing and Materials International, West Conshohocken, PA, USA  22. MATLAB (2017) R2017b documentation, Octave-Band and Fractional Octave-Band Filters.  23. Kuttruff, H. (2009). Room acoustics. Crc Press.  24. CATT-Acoustic (2007) v8.0 user manual © CATT 2007.  25. Logawa, B. (2017). Improving the sound absorption of cross-laminated timber panels using resonant absorbent layer (M.A.Sc. Thesis, University of British Columbia).  26. COMSOL (2013) 4.3b. Acoustics Module User Guide, Chapter 4, The Acoustic-Solid Interaction, Frequency Domain User Interface.   27. Bibby, C. (2011). Investigation of the acoustical and airflow performance of interior natural ventilation openings (M.A.Sc. Thesis, University of British Columbia).  28. Lewers, T. (1993). A combined beam tracing and radiatn exchange computer model of room acoustics. Applied Acoustics, 38(2-4), 161-178.  29. Guo, J., & Hodgson, M. (1999, December). Investigation of active noise control in non-diffuse sound fields. In INTER-NOISE and NOISE-CON Congress and Conference Proceedings (Vol. 1999, No. 5, pp. 621-632). Institute of Noise Control Engineering. 75   30. Jeong, C. H. (2012). Absorption and impedance boundary conditions for phased geometrical-acoustics methods. The Journal of the Acoustical Society of America, 132(4), 2347-2358.  31. Suh, J. S., & Nelson, P. A. (1999). Measurement of transient response of rooms and comparison with geometrical acoustic models. The Journal of the Acoustical Society of America, 105(4), 2304-2317.  32. Jaouen luc (2007-2011) http://apmr.matelys.com/Standards/OctaveBands.html, MATLAB’s documentation on octave standards, © APMR site.                          76  Appendices  Appendix A: DFT for Reference (Diffuse) Room Configuration  Sound Power Lw (dB) 100 Source Power W 0.01 Average Absorption Coefficient α 0.1 Room Coefficient R 16.6 Sum Of Surface Areas S 150 Eq. Absorption (Room1) A1=αS 15 Eq. Absorption (Room2) A2 (considering same as room 1) 15 Density rho 1.2 Air Speed c 343 Common Wall Area Sw 25 Transmission Loss TL 20 Tao 10𝑇𝐿10−1 0.001 L1 (dB) L1 = Lw+10log (4𝑅1)                                                                                                               93.80 Critical Radius r 1.81 Correction Factor 10log(Sw/A2) 2.22 L2 (From L1) (dB) L1-TL+10log(Sw/A2) 76.02                        77  Appendix B: CATT-TM Project Files for Reference Room Configuration  ;MASTER.GEO ;PROJECT=Room1  ;INCLUDE  GLOBAL W = 5 GLOBAL L = 5 GLOBAL H = 5 GlOBAL P= 10 GLOBAL ABU = 40 GLOBAL t=1.11  ;OFFSETCO ;OFFSETPL  ;MIRROR co_add pl_add  ;ABS absname <10 10 10 10 10 10> ;L <10 10 10 10 10 10>  ABS FLOOR <ABU ABU ABU ABU ABU ABU> L <0 0 0 0 0 0>{150 128 105} ABS FWALL <ABU ABU ABU ABU ABU ABU> L <0 0 0 0 0 0> {253 251 200} ABS SWALL <ABU ABU ABU ABU ABU ABU> L <0 0 0 0 0 0> {255 255 0} ABS SWALL2 <ABU/t ABU/t ABU/t ABU/t ABU/t ABU/t> L <0 0 0 0 0 0> {253 251 200} ABS CEIL <ABU ABU ABU ABU ABU ABU> L <0 0 0 0 0 0> {111 197 225} ABS SWALL3 <ABU ABU ABU ABU ABU ABU> L <0 0 0 0 0 0> {200 121 34}  CORNERS  ;id x y z  1 0 0 0 2 0 0 H 3 W 0 H 4 W 0 0 5 0 L 0 6 0 L H 7 W L H 8 W L 0 9 P 0 0 10 P 0 H 11 P L H 12 P L 0   78  PLANES  [1 floor /1 4 8 5/ floor] [2 fwall /1 2 3 4/ fwall] [3 swall1 /1 5 6 2/ swall] [4 fwall /5 8 7 6/ fwall] [5 Ceil  /3 2 6 7/ ceil] [D 6 swall2 /4 3 7 8/ swall2] [8 floor /4 9 12 8/ floor] [9 fwall /4 3 10 9/ fwall] [10 swll3 /9 10 11 12/ swall3] [11 fwall /8 12 11 7/ fwall] [12 Ceil  /10 3 7 11/ ceil]  ;[id name  /   / absname  ] ;[id name  /   / (a /  / a_abs) (b /  / b_abs) ]  ;PROJECT=Amin-01  REC.LOC   RECEIVERS    1  2.5 2.5 2.5  2  7.5 2.5 2.5  SRC.LOC  SOURCEDEFS  ; a natural source ;id   position    directivity  aim-position [roll]  A0  0.35 0.35 0.35     OMNI     0 0 0  Lp1m_a  = <89.1 89.1 89.1 89.1 89.1 89.1>  ; at 1m on source axis   CATT-Acoustic v8.0i : Acoustic parameters        Copyright © CATT 1988-2009 =========================================================================== Original file name: C:\USERS\NOISELAB\DESKTO...OM -SIDEWALL\OUT\E_A0_01.TXT Original save time: 2017-05-16 14:26:54 ----------------------------------------------------------------------------  Project                : -   GEOMETRICAL INFORMATION 79   Src id and loc     [m] :   A0        0.350     0.350     0.350  Rec id and loc     [m] :   01        2.500     2.500     2.500   VARIABLE SOURCE DATA  Directivity    Type (library)       :  OMNI.SD0    Dir. Index (DI) [dB] :    0.0   0.0   0.0   0.0   0.0   0.0  Level    On axis 1m      [dB] :   89.1  89.1  89.1  89.1  89.1  89.1    Total power     [dB] :  100.0 100.0 100.0 100.0 100.0 100.0  Auxiliary delay   [ms] :    0.0   VARIABLE AIR DATA  Temperature       [°C] :   20  Relative humidity  [%] :   50  Density        [kg/m³] :    1.2  Sound speed      [m/s] :  343  Impedance      [Ns/m³] :  412  Diss. coeff. [0.001/m] :   0.10  0.30  0.63  1.07  2.28  6.83   (estimated)   CALCULATION RESULTS  Head direction     [m] :  01     0.350     0.350     0.350 (source)   GLOBAL RESULTS FROM TRACING - Late part ray-tracing ----------------------------------------------------------------------------   Trunc[s]   Rays[-]   Lost[-]   Absorbed[-]   Angle[°] ............................................................................       0.88     31390         1             0       1.1 ----------------------------------------------------------------------------             125   250   500   1k    2k    4k  .......... ..... ..... ..... ..... ..... ..... .............................  MFP  [m]   3.34  3.34  3.34  3.34  3.34  3.34  Diffs[%]   0.00  0.00  0.00  0.00  0.00  0.00 ............................................................................  T-15 [s]   1.54  1.53  1.52  1.49  1.43  1.23   (LS-fit -5 to -20 dB)  T-30 [s]   1.43  1.42  1.41  1.40  1.36  1.21   (LS-fit -5 to -35 dB)  EyrT [s]   1.27  1.26  1.25  1.23  1.19  1.05   (MFP, AbsC)  EyrTg[s]   1.27  1.26  1.25  1.23  1.19  1.05   (MFP, AbsCg)  SabT [s]   1.34  1.33  1.31  1.30  1.25  1.09   (Vact, Sact, AbsCg) 80   AbsC [%]   10.0  10.0  10.0  10.0  10.0  10.0   (based on tracing)  AbsCg[%]   10.0  10.0  10.0  10.0  10.0  10.0   (area-proportional)  Back[dB]   45.0  38.0  32.0  28.0  25.0  23.0   (non-individual bkg noise level)  Resi[dB]    0.0   0.0   0.0   0.0   0.0   0.0   (indiv. resid. bkg noise level) ----------------------------------------------------------------------------   AVERAGE/ACTUAL VOLUME/SURFACE INFORMATION ----------------------------------------------------------------------------  V[m³]  Lx[m]  Ly[m]  Lz[m]  S[m2] Sact[m²] Vact[m³] ............................................................................  250.2   7.63   5.73   5.73  300.0    300.0      250 ----------------------------------------------------------------------------   Initial delay     [ms] :    10.85 ----------------------------------------------------------------------------  RASTI   [%]   55.5 (FAIR)         55.5 (FAIR)      with background noise  STI orig[%]   57.3 (FAIR)         57.3 (FAIR)      with background noise ----------------------------------------------------------------------------                 125      250      500      1k       2k       4k  ...........   ......   ......   ......   ......   ......   ......  TI   [%]      54.5     54.7     54.9     55.2     56.0     58.7     66.5  TI(n)[%]      54.5     54.7     54.9     55.2     56.0     58.7     66.5  Weights        130      140      110      120      190      170      140    x 0.001 (original)  ----------------------------------------------------------------------------  Parameter      125      250      500      1k       2k       4k      sum  ...........   ......   ......   ......   ......   ......   ......   ......  Ts   [ms]     85.2     84.6     83.8     82.6     79.8     70.4     76.7  D-50  [%]     44.1     44.3     44.6     45.0     46.0     50.0     48.3  C-80 [dB]      1.9      2.0      2.0      2.1      2.4      3.2      2.7  LFC   [%]     41.7     41.6     41.6     41.6     41.6     41.2     41.0  LF    [%]     29.3     29.3     29.2     29.2     29.3     29.0     28.9  G    [dB]     24.4     24.3     24.3     24.3     24.2     23.8     23.9  SPL  [dB]     93.5     93.4     93.4     93.3     93.2     92.6    101.6     99.3    (A)  EDT   [s]     1.14     1.14     1.13     1.11     1.07     0.96      ---  T-15  [s]     1.37     1.35     1.34     1.31     1.25     1.04      ---  T-30  [s]     1.38     1.37     1.36     1.34     1.29     1.13      ---     81  Appendix C: New Beam Tracing Algorithm-Matlab Code  This is the m-file called to execute the beam tracing program and gui. %This file should always be present in the beam_tracing_model directory.    %Clear workspace and command window: clear all clc home   %prepend the matlab path with directory structure containing beam tracing files: p = fileparts(which('bt_model_old.m')); p = genpath(p); path(p, path);    %initialize variables: globals;   %inputs from file: inputvar;   #plot the rooms % plot3d;   %execute the beam tracing: bmtrace;   disp(sprintf('%d beams were traced for %d reflections in %d hours, %d minutes and %f seconds.'...      ,20*icosa_freq^2,Nr,t(1),t(2),t(3)))   %----------------------------------------------------------------------------  %inputvar.m %enter all input variables here globals;   %1)     Common Variables %       ---------------- freq =89:1:5656;%linspace(0,2000,1024*2);               %frequency (Hz) freq = freq'; Zc = 415;                %Characteristic Impedance, default value for air c = 343;                 %Sound speed, default value for air air_density = 1.2;       %standard air density T  = 20; %%%temperature in centigrades RH = 20; %%%relative humidity (percent) m  = aabsorb(freq,T,RH,1);    %%% only works for 20C and 20% right now omega = 2*pi*freq; k = omega/c;   %% Source input Source = [0.35, 0.35, 0.35]; %Source Coordinates 82  Nb = 10000;         %Minimum number of beams (20*f^2) Nr = 50;             %Reflection Order   Rec = [2.5,2.5,2.5]; % source room Rec2 = [7.5,2.5,2.5]; % receiverroom %%% receiver coordinates: %% 2)     Room Inputs %       ----------- Lx=10; % Total length of both rooms: 5m for each source and receiver room Ly=5; % width Lz=5; % height room_dim = [Lx, Ly, Lz]; Coordinates{1, 1} = [0, 0, 0; 0, Ly, 0; 0, Ly, Lz; 0, 0, Lz];     % x = 0 Coordinates{2, 1} = [0, Ly, 0; Lx, Ly, 0; Lx, Ly, Lz; 0, Ly, Lz]; % y = Ly Coordinates{3, 1} = [Lx, Ly, 0; Lx, 0, 0; Lx, 0, Lz; Lx, Ly, Lz]; % x = Lx Coordinates{4, 1} = [Lx, 0, 0; 0, 0, 0; 0, 0, Lz; Lx, 0, Lz];     % y = 0 Coordinates{5, 1} = [0, 0, 0; 0, Ly, 0; Lx, Ly, 0; Lx, 0, 0];     % z = 0 Coordinates{6, 1} = [0, 0, Lz; 0, Ly, Lz; Lx, Ly, Lz; Lx, 0, Lz]; % z = Lz Coordinates{7, 1} = [Lx/2, Ly, 0; Lx/2, 0, 0; Lx/2, 0, Lz; Lx/2, Ly, Lz]; % x = Lx/2 #Partition  %%% these are points on each plane. from the first three of these points, %%% the (outward) normal of the plane, and the coefficients of the plane %%% equation are calculated. [Coefficients, Normals] = planegeo(Source);  % [Coefficients, Normals] = planegeocorrect(Source);   %%% this is where the coefficients of each plane equation and its normal %%% are calculated.   %% Source power ----------------------------------------------------------------------------- Lw = 100;                          %Sound Power Lw = repmat(Lw, length(freq), 1);  %sound power of source in dB, frequency dependent W = (10.^(Lw./10 - 12));           %sound power for source in Watts, for each beam Po = sqrt((W*air_density*c)/(2*pi)); %%% Po is the amplitude constant for spherical waves. see page 32     %% Beam vertices and centre vectors %%% loading precalculated matrices instead of calculating them. icosa_freq = ceil(sqrt(Nb/20)); icosa1=floor(icosa_freq/100);  icosa2=floor((icosa_freq-100*icosa1)/10); icosa3=floor(icosa_freq-100*icosa1-10*icosa2); eval(['load sourcebeams',num2str(icosa1),num2str(icosa2),num2str(icosa3),' V F']) CV = zeros(size(F, 1), 3); %%% the center vector - see p. 44 for i = 1:size(F, 1)    CV(i, :) = center(V(F(i, 1), :), V(F(i, 2), :), V(F(i, 3), :)); end     83  %% 3)     Surface Input Parameters %       ------------------------ extended_reaction = [0, 0, 0, 0, 0, 0, 0];  % LOCAL REACTION SURFACES (extended_reaction = 0), LD = ['A','A','A','A','A', 'A', 'A']; rb = [1, 1, 1, 1, 1, 1, 1]; %%% rb = rigid backing. 1=yes, 0=no. %alpha=0.1; % normalized average absorption coefficient of all surfaces TL=20; %tramsmission loss (dB) of the transmitting partition tao=1/(10^(TL/10)); %tramsmission coefficient of the transmitting partition  %%% constant reflection coefficient:  SurfaceA{1,1}   = [];% sqrt(1+alpha) % absorption coefficient boundary condition  SurfaceA{1,2}   = 37.97*415; % impedance boundary condition SurfaceA{1,3}   = [];  %---------------------------------------------------------------------------   %bmtrace.m %beam tracing main program for room-to-room-sound transmission with %specularly-reflection_arbitary geometry_withphase %%% all vector inner products changed from dot() to sum(.*)   t1 = clock; %start timer   %initialize the pressure vector: pPH = zeros(length(freq), size(Rec,1)); %%% pEN = zeros(length(freq), size(Rec,1)); %%% this one is for energy-only  ppPH = zeros(length(freq), size(Rec2,1)); %%% ppEN = zeros(length(freq), size(Rec2,1)); %%% this one is for energy-only     for b = 1:size(F, 1) %%% loop over the number of beams %                      %%% the way it works is that it takes a beam coming                      %%% from the source and tracks it to the end.     u = [V(F(b, 1), :); V(F(b, 2), :); V(F(b, 3), :); CV(b, :)]; S = Source; R_eff = ones(length(freq), 1); R_eff2 = ones(length(freq), 1);            for ro = 1:Nr %%% loop over reflections (order of reflection)          clc   msg = sprintf('Beam: %d/%d\t Reflection: %d/%d', b, size(F, 1), ro, Nr);          disp(msg)            %% Receiver Detection      for reception=1:size(Rec,1) %%% loop over receiver positions             Rr=Rec(reception,:);             receiver_detected = detectr(S, Rr, u(1, :), u(2, :), u(3, :));             %%% to check whether the receiver is contained in the beam             %%% THIS NEEDS TO BE UPDATED TO ACCOUNT FOR OBSTACLES AS WELL             if receiver_detected 84                  RecPlane = u(4, :);                 RecPlane(1, 4) = sum(RecPlane.*Rr);%%%          %RP contains plane coefficients that contain the receiver point R                 r = dist(RecPlane, S, u(4, :));          %total path length from last source to R-plane along centre vector          %%% is this (r) correct? Yes!                 phase_factor = exp(-1j*k*r);                 air_factor   = exp(-m*r/2/1000);                 p_inst = 1./r.*R_eff.*(phase_factor).*air_factor;                 p_ener = 1./r.*abs(R_eff).*air_factor;                   pPH(:, reception) = pPH(:, reception) + p_inst; %%%                 pEN(:, reception) = pEN(:, reception) + p_ener.^2; %%%               end  end % end of receiver loop           %% Beam Propagation          [S_image, v, rc, surf,doi,poi] = project(S, u);        %%% this is the heart of reflection! lots of things come out of this       %%% function; e.g. diffuse reflection and diffraction                        %% Transmission into Room #2 (Receiver room)             if surf==7 % Check if beam strikes the transmitting surface                   R_eff2 = R_eff.*sqrt(tau);                   Normals(7,1:3)=Normals(7,1:3).*-1;                    %% Reflection loop for room #2 (Receiver room)          for r1=1:Nr               [I2, v2, rc2, surf2,doi2,poi2] = project(S, u); % New beam propagation loop for executing reflections in receiver rooms                                                      % Receiver Detection                  for reception=1:size(Rec2,1) %%% loop over receiver positions              Rr=Rec2(reception,:);              receiver_detected = detectr(S, Rr, u(1, :), u(2, :), u(3, :));                    %%% to check whether the receiver is contained in the beam                    %%% THIS NEEDS TO BE UPDATED TO ACCOUNT FOR OBSTACLES AS WELL                   if receiver_detected                       RecPlane = u(4, :);                       RecPlane(1, 4) = sum(RecPlane.*Rr);%%%                      %RP contains plane coefficients that contain the receiver point R                       r = dist(RecPlane, S, u(4, :));                       %total path length from last source to R-plane along centre vector                       %%% is this (r) correct? Yes!                       phase_factor = exp(-1j*k*r);                       air_factor   = exp(-m*r/2/1000); 85                        pp_inst = 1./r.*R_eff2.*(phase_factor).*air_factor;                       pp_ener = 1./r.*abs(R_eff2).*air_factor;                         ppPH(:, reception) = ppPH(:, reception) + pp_inst; %%%                       ppEN(:, reception) = ppEN(:, reception) + pp_ener.^2; %%%                     end              end % end of receiver loop in #Room2                            S=I2;              u=v2;               R_eff2 = R_eff2.*rc2;                       end % end of secondary reflection order loop in # Room2               Normals(7,1:3)=Normals(7,1:3).*-1; % switching the direction of normal of the transmitting surface to enable reflection for next trajectory in source room       end            %reset source, direction vectors, and calculate effective reflection coefficient       S = S_image;       u = v;       R_eff = R_eff.*rc;           end %end of primary reflection order loop in # Room1      end      %calculate pressure at receiver for beam in both source and receiver room (phase, energy)   pPH=pPH.*Po;  pEN=pEN.*Po.^2; ppPH=ppPH.*Po; ppEN=ppEN.*Po.^2;  %% Calculation time Calculation_Time = etime(clock, t1);  %calculation time in seconds [hrs, mins, secs] = time(Calculation_Time); t = [hrs, mins, secs];  % Phase-result (PBTM)  SPL_PBTM=10*log10(pPH(:,:).*conj(pPH(:,:))./(2e-5)^2); SPL2_PBTM=10*log10(ppPH(:,:).*conj(ppPH(:,:))./(2e-5)^2);   L1=[freq SPL_PBTM] L2=[freq SPL2_PBTM]   % Energy-result (EBTM)   l1=10*log10((pEN(:,:)./4e-10)); l2=10*log10((ppEN(:,:)./4e-10));   EBTM results=[freq l1 l2] 86  %----------------------------------------------------------------------------  % Octave-band %% Converting spl Result to Octave Bands and 1/3rd octave bands %code adapted from narrow_to_one_third_octave.m by Luc Jaouen %http://apmr.matelys.com/Standards/OctaveBands.html [28]  % temps=xlsread('myExample.xlsx'); % f=temps(:,1); % load from excel file spl=l2; % spl=temps(:,5); f=freq; % Setting up the frequency bands   oneThirdOctaveDisp=[16 20 25  31.5 40 50 63 80 100 125 160 200 250 ...                     315 400 500 630 800 1000 1250 1600 2000 2500 ...                     3150 4000 5000 6300 8000 10000 12500 16000]; octaveDisp=[16 31.5 63 125 250 500 1000 2000 4000 8000 16000];   oneThirdOctaveAct=zeros(1,length(oneThirdOctaveDisp)); oneThirdBand=zeros(2,length(oneThirdOctaveDisp)); octaveAct=zeros(1,length(octaveDisp)); octaveBand=zeros(2,length(octaveDisp));   for i=1:length(oneThirdOctaveDisp)     oneThirdOctaveAct(i)=(1000*((2^(1/3)))^(i-19));     oneThirdBand(1,i) = oneThirdOctaveAct(i)/2^(1/6);      oneThirdBand(2,i) = oneThirdOctaveAct(i)*2^(1/6); end    for i=1:length(octaveDisp)     octaveAct(i)=1000*2^(i-7);     octaveBand(1,i)=octaveAct(i)/2^(1/2);     octaveBand(2,i)=octaveAct(i)*2^(1/2); end   % Calculate spl for each third octave for a = 1:size(oneThirdBand,2),     splOneThird(a) = 0;     idx = find( f >= oneThirdBand(1,a) ...         & f < oneThirdBand(2,a) );     % If we have no 'measurement' point in this band:     if ( isempty(idx) )       fprintf('Warning: no point found in band centered at %4.0f\n',oneThirdOctaveDisp(a));     % If we have only 1 'measurement' point in this band:     elseif ( length(idx) == 1 )       fprintf('Warning: only one point found in band centered at %4.0f\n',oneThirdOctaveDisp(a));       splOneThird(a) = spl(idx); 87      % If we have more than 1 'measurement' point in this band:     elseif ( length(idx) > 1 )       for b = 1:length(idx)-1,     splOneThird(a) = splOneThird(a) + ...           ( f(idx(1)+b)-f(idx(1)+b-1) ) * ....           (spl(idx(1)+b)+spl(idx(1)+b-1) )/ 2;       end       splOneThird(a) = splOneThird(a) / ( f(idx(length(idx)))-f(idx(1)) );     end end % Calculate spl for octave band for a = 1:size(octaveBand,2),     splOctave(a) = 0;     idx = find( f >= octaveBand(1,a) ...         & f < octaveBand(2,a) );     % If we have no 'measurement' point in this band:     if ( isempty(idx) )       fprintf('Warning: no point found in band centered at %4.0f\n',octaveDisp(a));     % If we have only 1 'measurement' point in this band:     elseif ( length(idx) == 1 )       fprintf('Warning: only one point found in band centered at %4.0f\n',octaveDisp(a));       splOctave(a) = spl(idx);     % If we have more than 1 'measurement' point in this band:     elseif ( length(idx) > 1 )       for b = 1:length(idx)-1,       splOctave(a) = splOctave(a) + ...           ( f(idx(1)+b)-f(idx(1)+b-1) ) * ....           (spl(idx(1)+b)+spl(idx(1)+b-1) )/ 2;       end       splOctave(a) = splOctave(a) / ( f(idx(length(idx)))-f(idx(1)) );     end end %----------------------------------------------------------------------------   %globals.m   %declare global variables   global freq Po omega Lw global c air_density Zc global Source Lx Ly Lz xr yr zr Rec Rec2 global Coordinates Coefficients Normals    global extended_reaction diffusion_coeff global LD rb global SurfaceA  global R Z tau   88  % plot3d  % 3D GEOMETRIC PLOT OF THE ROOM, THE BARRIERS, THE NORMAL VECTORS, THE SOURCE AND THE RECEIVERS %---------------------------------------------------------------------------- w=size(Coordinates,1); Pts=Coordinates; for i=1:w    Pts{i,1}(5,:)=Pts{i,1}(1,:);     end  %Pts{7,1}(:,:)=Pts{3,1}(:,:); %Pts{7,1}(:,1)=Pts{7,1}(:,1)./2;   figure                                                               for l=1:w                                                       plot3(Pts{l,1}(:,1),Pts{l,1}(:,2),Pts{l,1}(:,3),'k','linewidth',2);               % PLOT WALLS AND BARRIERS WITH ITS NORMALS     %quiver3(Ptscntr{l,1}(1,1),Ptscntr{l,1}(1,2),Ptscntr{l,1}(1,3),Normals(l,1),Normals(l,2),Normals(l,3))     hold on                                                         % STAY ON THE SAME PLOT end   for l=1:size(Rec,1)                                                 % PLOT MATRIX RECEIVERS LIKE GREEN SPHERES EVEN IF THEY ARE PHYSICAL POINTS                                                       scatter3(Rec(l,1),Rec(l,2),Rec(l,3),45,'g','fill')                       hold on                                                         % STAY ON THE SAME PLOT end   for l=1:size(Rec2,1)                                                 % PLOT MATRIX RECEIVERS LIKE GREEN SPHERES EVEN IF THEY ARE PHYSICAL POINTS                                                       scatter3(Rec2(l,1),Rec2(l,2),Rec2(l,3),45,'g','fill')                       hold on                                                         % STAY ON THE SAME PLOT end scatter3(Source(1),Source(2),Source(3),90,'r','fill')               % PLOT SOURCE LIKE A RED SPHERE (ALMOST A REAL SPHERE FOR A BIG NUMBER OF BEAMS) axis equal grid off %----------------------------------------------------------------------------  %planegeo.m %Calculates normal vectors for planes with coefficients   %%% all vector inner products changed from dot() to sum(.*)   function [C, n] = planegeo(S)   globals;   no_of_surfaces = size(Coordinates, 1); C = zeros(no_of_surfaces, 4); n = zeros(no_of_surfaces, 3);   89  for i = 1:no_of_surfaces    P = Coordinates{i, 1}(1, :);    Q = Coordinates{i, 1}(2, :);    R = Coordinates{i, 1}(3, :);        PR = R - P;    PQ = Q - P;        N = cross(PR, PQ);      N_length = N(1)^2 + N(2)^2 + N(3)^2;    N_length = sqrt(N_length);        C(i, 1:3) = N;    C(i, 4) = sum(N.*P);%%%      n(i, 1:3) = N/N_length;        chkpt = sum(N.*S);%%%        if chkpt>C(i, 4)       n(i, 1:3) = -n(i, 1:3);    end end %----------------------------------------------------------------------------      %planegeocorrect.m %Corrects normal vectors for planes with coefficients %correct normals of walls with exterior facing source %JH   function [C, n] = planegeocorrect(S)   globals;   no_of_surfaces = size(Coordinates, 1); C = zeros(no_of_surfaces, 4); n = zeros(no_of_surfaces, 3);   for i = 1:no_of_surfaces    P = Coordinates{i, 1}(1, :);    Q = Coordinates{i, 1}(2, :);    R = Coordinates{i, 1}(3, :);        PR = R - P;    PQ = Q - P;        N = cross(PR, PQ);      %N_length = N(1)^2 + N(2)^2 + N(3)^2;    N_length = norm(N,2);        C(i, 1:3) = N;    C(i, 4) = sum(N.*P);%%%      n(i, 1:3) = N/N_length;     90     %turn all normals away from source    chkpt = sum(N.*S);%%%    if chkpt>C(i, 4)       n(i, 1:3) = -n(i, 1:3);    end         %draw line from source to middle of surface     %use validsrf to find surfaces along that line     %count surfaces with dist less than the dist to the surface of interest     %even->normal away from source, odd->normal toward source     middle = mean([P;Q;R],1);     line = middle - S;     line_norm = line/norm(line,2);     surfacesintheway = validsrf(line_norm,S);     counter = 0;     for j=1:length(surfacesintheway(:,2))         if (abs(surfacesintheway(:,2))-0.00000000001)<=norm(line)             counter = counter + 1;         end     end     counter = counter -1;     if mod(counter,2)         n(i, 1:3) = -n(i, 1:3);     end    end   for i = 2:no_of_surfaces    if Coordinates{i, 1}(1:3,1:3) == Coordinates{i-1, 1}(1:3,1:3)        n(i, 1:3) = -n(i, 1:3);        %reverse internal wall's second normal    end end end %---------------------------------------------------------------------------- %project.m %projects the beam vertices on the intersecting surface, and calculates the %reflecting direction vectors, image source, and reflection coefficient.  function [I, v, rc, rsurf,doi,poi] = project(S, u) %%% the input u contains the center ray and vertices of the incident beam. %%% the center ray is its last row; i.e. u(4,:)   % global Coordinates%%% % global Coefficients%%% % global Normals%%% % global R_Coefficients%%% globals; v = zeros(4, 3);   [rsurf,doi,poi] = intsrf(u(4,:), S);  %determines the surface intersected by a ray %%% it has been very slightly changed to allow for multi-ray (beam) input %%% rsurf: plane of intersection (surface index) %%% poi  : point of intersection %%% doi  : distance from the (image) source to the point of intersection %%% Note that these are vectors containing info for the constituent rays. 91    %%%-------------- REFLECTION --- case of similar walls %%% this is where reflection occurs. diffuse reflection is added here. if (rand > diffusion_coeff(rsurf)) %%% specular reflection case     for ii = 1:4         v(ii, :) = spec_ref(u(ii, :), rsurf);     end     I = isource(S, rsurf); %%% image source (specular reflection) else %%% diffuse reflection case (BY)     [v(4,:), nd] = diff_ref(u(4,:),rsurf);     for ii=1:3         v(ii,:)=specref2(u(ii,:),nd);     end     I = iisource(S,nd,poi);%%% image source (diffuse reflection) end   surface_class = ['Surface', LD(rsurf)];   if extended_reaction(rsurf)     aofi=anginc(u(4,:),rsurf); %%% angle of incidence (aofi) % %    eval(['layer_data = ', surface_class, ';']); % %    rc = asprop(freq, aofi, Zc, c, layer_data, rb(rsurf));      index=floor(aofi/1/pi*180)+1; %    eval(['R = ',surface_class,'{1,3};'])    rc=R(:,index); %             eval(['Z = ', surface_class, '{1, 2};']) %             rc = locreact(Z(:,index), Zc, aofi);   %    rc=abs(rc);   else   %use pre-allocated table of reflection coefficients    if eval(['isempty(', surface_class, '{1, 2})'])        eval(['rc = ', surface_class, '{1, 3};'])    elseif eval(['isempty(', surface_class, '{1, 3})'])       aofi=anginc(u(4,:),rsurf); %%% angle of incidence (aofi)       eval(['Za = ', surface_class, '{1, 2};'])       [rc] = locreact(Za, Zc, aofi); %%%local-reaction reflection coefficient %       rc=abs(rc);    end end  %----------------------------------------------------------------------------  %locreact.m   function [R] = locreact(Z, Zc, angl)   %%%Zc = repmat(Zc, length(Z), 1);   A = Z.*cos(angl) - Zc; B = Z.*cos(angl) + Zc;   R = A./B; %----------------------------------------------------------------------------  92  %intsrf.m %determines the surface that is intersected by a ray   function [srf, intsd, intsp] = intsrf(u, S) %%%   %%% global Coefficients %%% global Normals globals; srf   = zeros(size(u,1),1); intsp = zeros(size(u,1),3); intsd = zeros(size(u,1),1);   for scount=1:length(srf)       P = validsrf(u(scount,:), S);     %Calculates possible surfaces that are intersected by a ray.     %%% output: plane number, distance, intersection point            surfaces = P(:, 1);     dst = P(:, 2);       [mindst, jj] = min(dst);       srf(scount)     = surfaces(jj);     intsd(scount)   = dst(jj);     intsp(scount,:) = P(jj,3:5);   end end %----------------------------------------------------------------------------  %validsrf.m %Calculates possible surfaces that are intersected by a ray. %%% all vector inner products changed from dot() to sum(.*)    function P = validsrf(u, S)   % global Coordinates % global Coefficients globals;   jj = 1;   P = zeros(size(Normals,1),5); %%% preallocation   for ii = 1:size(Normals, 1)           possible_surface = sum(u.* Normals(ii, :));%%%        if possible_surface > 0             d = dist(ii, S, u);   %calculate distance from S to plane, along u       Q = S + d*u;          %point of intersection of ray and plane 93        hit = contain(ii, Q); %check if intersection point lies within coordinates that define surface boundary       if hit          P(jj, 1) = ii;      %record plane number          P(jj, 2) = d;       %record distance          P(jj,3:5)= Q;     %%%record point of intersection          jj = jj + 1;        %increment possible surface index       end    end        P(jj:end,:)=[]; %%% deleting extra, empty rows                    %%% added because of line 14 end    end %----------------------------------------------------------------------------  %Dist.m %Function to calculate the distance from start point S along direction vector u,  %to the intersection of the ray and a plane with coefficients in the coefficient matrix.   function s = dist(plane, S, u)   globals;   intplanecoeffs = zeros(length(Coefficients(:,1)),4); intplanecoeffs(1:length(Coefficients(:,1)),:) = Coefficients;   if length(plane) == 1    P1 = intplanecoeffs(plane, 1:3);    P2 = intplanecoeffs(plane, 4); else    P1 = plane(1, 1:3);    P2 = plane(1, 4); end   %%% s1 = dot(P1, S); %%% s2 = dot(P1, u); %%% changed for computational efficiency s1=sum(P1.*S); s2=sum(P1.*u);  s1 = P2 - s1;   if s2 == 0    s = 0;       %A "0" is returned if the direction vector is perpendicular to the normal vector for the plane else    s = s1/s2; end  %---------------------------------------------------------------------------    94   %contain.m %function to determine if the point P, the intersection of a ray and a  %plane, lies within the surface defined by the coordinates for that plane   function H = contain(plane, P)   globals; %global Normals   %extract coordinates and normal vector for surface # 'plane' coords = Coordinates{plane, 1}; n = Normals(plane, :);   %determine euler angles to rotate new coordinate axis [azimuth, elev, r] = cart2sph(n(1), n(2), n(3));   psi = azimuth - pi/2; theta = elev - pi/2; phi = 0;   %calculate transformation matrix and new coordinates E = eulerang(psi, theta, phi);   coords = coords*E'; P = P*E';   %determine if the point P lies inside the surface defined by the coordinates H = inpolygon(P(1), P(2), coords(:, 1), coords(:, 2)); %----------------------------------------------------------------------------   %function to calculate the image source of a point reflected about a plane %%% modified to include diffuse reflection   function Si = isource(S, plane)  global Normals n = Normals(plane, :); s = dist(plane, S, n); Si = S + 2*s*n; %----------------------------------------------------------------------------   %icosa.m   %function to construct an icosahedron of frequency, f.  There are 20*f^2 vertices calculated.   function [V, F] = icosa(f)   %V = vertex matrix, a 20*f^2x3 matrix of vertices that make up the icosahedron %F = face matrix, optional, used to construct patches.   deg = pi/180; tao = 0.5*(1+sqrt(5));     %golden ratio 95  v = 1;   %initialize vertex index k = 1; %CONSTRUCT THE FIRST FACE IN THE POSITIVE X, Y, Z QUADRANT. %THE OTHER 19 FACES WILL BE CONSTRUCTED FROM THIS ONE USING THE ROTATION FORMULAS   for i = 0:f    for j = 0:i       x1(v) = j;       y1(v) = i - j;       z1(v) = f - i;       v = v + 1;     %increment vertex index by 1    end end   x = x1*sin(72*deg); y = y1 + x1*cos(72*deg); z = repmat(f/2, 1, length(z1)) + z1/tao;   %rotate coordinates to establish global coordinate system: psi1 = 18*deg; th1 = 0; ph1 = 0;   D = eulerang(psi1, th1, ph1); for i = 1:length(x)    X(1:3, i) = D*[x(i); y(i); z(i)]; end   %sort the coordinates to be consistent with the F matrix m = 1; h = 1; k = 1; d1 = 1; d2 = 1;   for i = 1:f    m = m + i;    n = m + i;    X(1:3, m:n) = fliplr(X(1:3, m:n));    k = k + i;    d1 = [d1, k];    d2(i + 1) = d2(i) + (i + 1); end   %convert to spherical coordinates [theta1, phi1, r] = cart2sph(X(1, :), X(2, :), X(3, :)); r1 = ones(1, length(theta1));   %Now with r = 1 (unit vectors) for all vertices, convert back to cartesian coordinates [p1, p2, p3] = sph2cart(theta1, phi1, r1);   %Calculate F  F1 = face_mat(f); 96   %Now construct the other faces v = v - 1; %remaining top 4 faces: [theta_s1, phi_s1, F_s1] = calcface(theta1, phi1, F1, v);   %Faces 6 - 10 %The 5 triangles around the equator that point downward %Rotation (Euler) angles: psi = 126*deg; th = acos(-tao/(2 + tao)); ph = 54*deg; %calculate rotation (coordinate transformation) matrix: D = eulerang(psi, th, ph);   for i = 1:length(p1)    x(1:3, i) = inv(D)*[p1(i); p2(i); p3(i)]; end   %find spherical coordinates of new base face [theta2, phi2, r2] = cart2sph(x(1, :), x(2, :), x(3, :)); F2 = F1 + 5*repmat(v, size(F1));   [theta_s2, phi_s2, F_s2] = calcface(theta2, phi2, F2, v);   %Faces 11 - 15 %The 5 triangles around the equator that point downward %Rotation (Euler) angles: psi = pi/2; th = acos(tao/(2 + tao)); ph = -126*deg; %calculate rotation (coordinate transformation) matrix: D = eulerang(psi, th, ph); Dinv = inv(D);   for i = 1:length(p1)    x(1:3, i) = Dinv*[p1(i); p2(i); p3(i)]; end   %find spherical coordinates of new base face [theta3, phi3, r3] = cart2sph(x(1, :), x(2, :), x(3, :)); F3 = F1 + 10*repmat(v, size(F1));   [theta_s3, phi_s3, F_s3] = calcface(theta3, phi3, F3, v);   %Faces 16 - 20 %calculate coordinates for base triangle for the bottom 5: theta4 = theta1 + repmat(36*deg, size(theta1)); phi4 = -phi1; F4 = F1 + 15*repmat(v, size(F1)); %calculate other 4 bottom triangles: [theta_s4, phi_s4, F_s4] = calcface(theta4, phi4, F4, v);   %Now assemble all coordinates: theta = [theta_s1, theta_s2, theta_s3, theta_s4]; 97  phi = [phi_s1, phi_s2, phi_s3, phi_s4]; r = ones(1, length(theta));   [x, y, z] = sph2cart(theta, phi, r); V = [x', y', z'];  F = [F_s1; F_s2; F_s3; F_s4]; %----------------------------------------------------------------------------   eulerang.m %rotate euler angles psi, theta, and phi   function D = eulerang(psi, theta, phi)   c1 = cos(psi); c2 = cos(theta); c3 = cos(phi); s1 = sin(psi); s2 = sin(theta); s3 = sin(phi); D =[c1*c3-s1*c2*s3, s1*c3+c1*c2*s3, s2*s3;...       -c1*s3-s1*c2*c3, -s1*s3+c1*c2*c3, s2*c3;...       s1*s2, -c1*s2, c2]; %----------------------------------------------------------------------------   %calcface.m   %This a specific function used in icosa.m to calculate 4 additional similar faces to a base face. %In an icosahedron, there are 4 groups of 5 similar faces.  Given the coordinates for one of those faces, %the coordinates for the other four are calculated.   function [thetas, phis, Fs] = calcface(theta1, phi1, F1, v)   deg = pi/180;   phis = phi1; thetas = theta1; Fs = F1;   for k = 1:4    phi = phi1;    theta = theta1 + repmat(72*deg*k, size(theta1));    F = F1 + k*repmat(v, size(F1));        phis = [phis, phi];    thetas = [thetas, theta];    Fs = [Fs; F]; end  %----------------------------------------------------------------------------      98  elimvert.m %function to eliminate redundant vertices calculated for the source.  function [v2, f2] = elimvert(v, f)   %set the tolerance for the difference between magnitudes of a pair of vertices to distinguish if they are %equal or unique tol = 1e-6; v2 = v; f2 = f; vrows = size(v, 1); i = 1; c = 1; redundant_vertices = []; freq = sqrt(size(f, 1)/20); fv = vrows/20; vertices = 12 + 30*(freq - 1) + 20*(fv - 3*freq);   while c <= vertices    %find the magnitude of the difference between each vertex and all others    v1 = repmat(v(i, 1:3), vrows, 1);    vdiff = v1 - v;    vdiff = sqrt(vdiff(:, 1).^2 + vdiff(:, 2).^2 + vdiff(:, 3).^2);        %set the magnitude of the difference between the vertex being tested and itself to be 1,    %this is done to avoid this vertex being deleted.        %find indeces of redundant vertices    j = find(vdiff < tol);              if ~isempty(j)       %eliminate redundant vertices       redundant_vertices = [redundant_vertices; j(2:length(j))];       redundant_vertices = unique(redundant_vertices);       %replace redundant vertices with index of unique vertex in faces matrix       for k = 1:length(j)          [m, n] = find(f == j(k));          for p = 1:length(m)             f2(m(p), n(p)) = c;          end       end    end           %reset number of rows to new size of the vertices matrix, increment index into this matrix    i = i + 1;    while 1       if ~isempty(find(redundant_vertices == i))          i = i + 1;       else          c = c + 1;          break;       end    end end 99    v2(redundant_vertices, :) = []; [i, j] = find(abs(v2) < tol); for k = 1:length(i)    v2(i(k), j(k)) = 0; end %----------------------------------------------------------------------------   %center.m   %find the center vector (in cartesian coordinates) for a triangular face. %the coordinates of the three vertices for the triangle are given. %The three vertices for the triangular beam face are unit vectors that al lie on the unit sphere %that defines the source. %These points lie on a plane that is parallel to one that is tangent to the unit sphere at a point  %along the center vector for the triangular face of the beam.  This vector is a unit vector that is %the unit normal to the plane defined by the three vertices.   %%% all vector inner products changed from dot() to sum(.*)   function cv = center(v1, v2, v3)   %cv = center vector %v1, v2, v3 = vertices of triangular face   PR = v2 - v1; PQ = v3 - v1;     N = cross(PR, PQ);   N_length = N(1)^2 + N(2)^2 + N(3)^2; N_length = sqrt(N_length);     n = N/N_length;     if sum(n.*v1) < 0 %%%    n = -n; end   cv = n;  %----------------------------------------------------------------------------  %face_mat.m   %construct a matrix that contains indeces for a vertex matrix to define patches.   function F = face_mat(nu)   %F = face matrix, size nu^2x3, where nu is the frequency.   v0 = 1; 100    for i = 1:nu        v1 = v0;        for j = (i-1)^2+1:2:i^2       F(j, 1:3) = [v1, (v1+i), (v1+i+1)];       F(j+1, 1:3) = [v1+i+1, v1+1, v1];              v1 = v1 + 1;    end        v0 = v0 + i; end   F(size(F, 1), :) = [];     %delete the last row               

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