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Predicting sound propagation in fitted workrooms Li, Ke 1995

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P R E D I C T I N G S O U N D P R O P A G A T I O N I N F I T T E D W O R K R O O M S B y K e L i B . Sc. (Mechanical Engineering) Shenyang Polytechnic University, 1982 M . Sc. (Mechanical Engineering) Shenyang Polytechnic University, 1987 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E STUDIES M E C H A N I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A August 1995 © K e L i , 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of l^tcAa^C^L ^ M ^ - u j The University of British Columbia Vancouver, Canada Date #{u^UAf>8 , fflf DE-6 (2/88) In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the Universi ty of Br i t i sh Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Mechanical Engineering The University of Br i t i sh Columbia 2324 M a i n M a l l Vancouver, Canada V 6 T 1Z4 Date: Abstract When predicting sound propagation in rooms such as industrial workrooms, a major factor that must be taken into consideration is the presence of 'fittings' — obstacles such as machines and stockpiles — in the room. Besides the fitting spatial distribution, there are two important parameters used in prediction models to describe the fittings — one is the fitting density — a measure of the number of fittings and the average fitting cross-section area — and the other is the fitting absorption coefficient. Whi le ranges of typical fitting densities are known, no method exists for measuring or estimating the fitting density in a given factory. Furthermore, theoretical expressions for calculating fitting density assume small fittings and high frequency. The a im of this research project is to develop and test a method for determining the fitting density in industrial workrooms. To achieve this objective a correction formula was derived for calculating the fitting density in the case of large fitting dimensions. The variation of fitting density wi th frequency was found from sound propagation measurements in large fitted regions; a formula to express the relationship is determined by statistical methods and this model was validated experimentally in a scale-model workroom and in a machine shop wi th the help of prediction models. A correction formula for calculating fitting absorption coefficient using empty and fitted room absorption coefficients was derived and validated using measurement in a machine shop. A n image-source model — based on improving an existing model used for infinite regions — was developed to predict sound propagation in fitted rooms and validated in several workrooms. This model provided a fast, workable and accurate alternative to existing fitted-room models. 11 Table of Contents Abstract ii Table of Contents iii List of Tables vi List of Figures vii Acknowledgement xii 1 Introduction 1 2 Theoretical Background 8 2.1 Classical (Diffuse-field) Description of Sound Propagation in Rooms . . . 8 2.2 Predict ion Methods for Industrial Workrooms 10 2.2.1 Physical Scale Models 10 2.2.2 The Image-source Method 11 2.2.3 The Ray-tracing Method 12 2.2.4 Empi r ica l Models 13 2.2.5 Models Used in the Research Project 13 2.3 Model l ing of Sound Wave Propagation in Fi t ted Regions 14 3 Theoretical Developments 18 3.1 Correction of the F i t t ing Density for Large F i t t ing Dimension 18 3.1.1 Introductory Discussion 18 i i i 3.1.2 Correction for Large F i t t ing Density 20 3.2 A Correction Formula for Calculating F i t t ing Absorption Coefficient . . . 24 4 Measurement Methods 26 4.1 Principles of Scale Model l ing and Choice of Scale Factor 26 4.2 Measurement System 28 4.3 M L S S A System 30 4.4 Scale-model Sound Source 31 4.5 Correction for Loudspeaker Distortion 34 4.5.1 Correlation and Spectral Analysis Method 35 4.5.2 Equalization Method 41 4.5.3 Bias Error of Loundspeaker Distortion 41 4.6 Scale-Model Fitt ings 42 4.7 Test Environments 43 4.7.1 Anechoic Chamber 43 4.7.2 Scale-model Workroom 46 5 Experiments in the Anechoic Chamber with Fittings 49 5.1 Sound Propagation in the Presence of a Single Obstacle 49 5.2 Measurement and Analysis 51 5.3 Calculat ion of the F i t t ing Density from the Measured Data 54 5.3.1 Sound Energy 54 5.3.2 A Model for Q(f) ' 59 5.3.3 Correlation Test of Q(f) 61 5.3.4 F ina l Mode l for Q(f) 61 5.4 Comparison of Experiment with Predictions by Exist ing Models 63 iv 6 Experiments in a Scale Model Workroom 70 6.1 Randomly Distributed Fittings 70 6.2 Fit t ings on the Floor 77 6.3 Fit t ings on the Floor and Absorbent Ceil ing 82 7 Validation of Q(f) by a Measurement in a Machine Shop 90 8 Summary and Conclusion 98 Nomenclature 102 Bibliography 104 Appendices 106 A An Improved Prediction Model for Fitted Rooms 106 A . l Mode l Development 106 A . 1.1 Infinite Region 106 A.1.2 Fini te Bounded Region 108 A . 2 Experimental Validat ion of the Model I l l A.2.1 Hypothetical Workroom I l l A.2.2 Real Workroom 113 A.2.3 Scale Mode l Workroom 116 A . 3 The Program of the New Model 118 v List of Tables 3.1 Some typical corrected fitting densities ( m _ 1 ) 23 4.1 Background noise in the empty anechoic chamber 44 5.1 Changes in SPL measured in the presence of a single obstacle (in dB) . . . 51 5.2 Q{f) determined from the measured data using E q . 5.8 57 5.3 The fitting density by three methods (in m - 1 ) 59 5.4 The parameters used in prediction (full scale values) 64 6.1 The parameters in full scale values used in prediction for the scale-model workroom with randomly distributed fittings 71 6.2 The parameters used in prediction for fitting on the floor 78 6.3 The parameters in full scale values used in prediction for the scale-model workroom with uniformly distributed fittings on the floor 86 7.1 Values of parameters used for ray-tracing prediction of SP in the machine shop 91 7.2 The parameters used for prediction using Q(f) 92 vi List of Figures 1.1 Sound-ray paths in a fitted region 3 1.2 Illustration of impulse response 5 2.1 The total sound level in a diffuse field, showing the direct and diffuse contribution 9 2.2 The image-source method: image source positions in the case of a single plane and in an empty 2D rectangular room 11 2.3 Tracing a sound ray from sound source S to receiver R 12 2.4 A single sound-ray path in a fitted region 15 3.1 Two overlapping spheres of radius R 20 3.2 Three spheres of radius R used to estimate the mean free path reduction. 21 3.3 Dimension of spheres and distance between them 24 4.1 Measurement system showing the test environment and the measuring instruments 29 4.2 The cone used to reduce loudspeaker directivity 31 4.3 Measuring loudspeaker directivity 32 4.4 Directivities of speaker 1 with the cone in 4 octave bands 32 4.5 Directivities of speaker 2 in 4 octave bands 33 4.6 Ideal single input/single output linear system 35 4.7 Schematic diagram of the acoustic system 37 4.8 (a) input and (b) output signals of the acoustic system 38 v i i 4.9 The calculated ordinary coherence function between the input and output signals for the system shown in F ig . 4.7 38 4.10 The cross-correlation and auto-correlation functions, (a) gxy, (b) gyx, (c) 9xx, (d) gyy, of the input and output signals of the system shown in F i g . 4.7 . 39 4.11 Comparison of true transfer functions for 3 cases (see text) 40 4.12 Demonstration of the error in the system 41 4.13 The dimensions of the bottles used as 1:8 scale fittings 42 4.14 Octave-band sound pressure level variation with distance in the empty anechoic chamber 45 4.15 Comparison SP curves, (a) Q = 0.1 m - 1 , (b) Q — 0.5 m _ 1 , (—) in an fitted very large region; ( ) in the fitted anechoic chamber 45 4.16 Photograph of the scale model 47 4.17 Dimensions of the scale model space 47 5.1 The positions of the obstacle 50 5.2 Impulse responses at r = l m , (a) <5=0m _ 1; (b) <2=0.2m - 1; (c) <5=0.6m _ 1. 52 5.3 The filtered cumulative energy curves 53 5.4 Variat ion of fitting density wi th frequency, (a) Q0 = 0.2 m " 1 ; (b) Q0 = 0.4 m _ 1 ; (c) Qo = 0.6 m _ 1 , calculated from the experimental data using E q . 5.8 . Also shown are the fitting densities Q 0 , QL and Q' 58 5.5 Comparison of two models for Q(f) 62 5.6 Comparison between Q(f), as (+ +) measured, and as predicted (—) by E q . 5.15 and (---) by E q . 5.14 63 5.7 Octave-band sound propagation in the anechoic chamber used as a 1:8 scale model room with randomly distributed 81 bottles (Q0 = 0.025 m _ 1 ) , as measured and predicted by the ray-tracing model 66 v m 5.8 As for F i g . 5.7 but the bottles covered with absorbent (Q0 — 0.025 m x ) . 66 5.9 As for F i g . 5.7 but wi th 162 bottles (Q = 0.05 m " 1 ) 67 5.10 As for F i g . 5.7 but with 243 bottles {Q = 0.075 m " 1 ) 67 5.11 Comparison of 500 Hz octave-band SP curves for 243 bottles, Q(f) — 0.088 m - 1 , as measured and as predicted by R A Y C U B , Lindqvist and new models. 68 5.12 As for F i g . 5.11 but for 1000 Hz , Q(f) = 0.101 m " 1 68 5.13 As for F i g . 5.11 but for 2000 Hz , Q(f) = 0.109 m " 1 69 5.14 A s for F i g . 5.11 but for 4000 Hz , Q(f) = 0.113 m " 1 69 6.1 Octave-band sound propagation in the empty 1:8 scale model workroom, as measured and predicted by the ray-tracing model 72 6.2 Octave-band sound propagation in the empty 1:8 scale model workroom, as measured and predicted the new image-source model 72 6.3 Octave-band sound propagation in the 1:8 scale model workroom with 23 randomly distributed bottles (Q0 = 0.025 m _ 1 ) , as measured and predicted by the ray-tracing model 73 6.4 As for F i g . 6.3 but wi th 46 bottles (Q0 = 0.05 m " 1 ) 74 6.5 As for F i g . 6.3 but with 69 bottles (Q0 = 0.075 m _ 1 ) 74 6.6 S P ( d B ) wi th 69 randomly distributed bottles, as measured and predicted by the ray-tracing model using Q{f) 75 6.7 As for F i g . 6.6 but predicted by the new model 75 6.8 Comparing the differences predicted by ray-tracing model: using Q0 (0); using Q(f) (+) 76 6.9 Uniform distribution of 32 bottles on the floor of the scale-model workroom. 77 6.10 £ P ( d B ) measured in the empty room compared with that predicted by the ray-tracing model 79 ix 6.11 Octave-band sound propagation in the 1:8 scale model workroom wi th 16 randomly distributed bottles on the floor (Qo = 0.0625 m _ 1 ) , as measured and predicted by the ray-tracing model 80 6.12 A s for F i g . 6.11 but wi th 32 bottles.{Q = 0.125 m " 1 ) 80 6.13 Comparison of 250 Hz octave-band SP curves for 32 bottles on the floor, as measured and as predicted by ray-tracing for four configurations (see text) 82 6.14 As for F i g . 6.13 but for 500 Hz 83 6.15 As for F i g . 6.13 but for 1000 Hz 83 6.16 As for F i g . 6.13 but for 2000 Hz 84 6.17 As for F i g . 6.13 but for 4000 Hz 84 6.18 Comparing the difference of SP wi th 32 bottles on the floor predicted by ray-tracing using Q0 (o) and using Q(f) (+) 85 6.19 S P ( d B ) — empty 5 m high scale-model workroom with absorbent ceiling, compared wi th R A Y C U B 87 6.20 Comparison of 250 Hz octave-band SP curves for 31 uniform distributed bottles on the floor, as measured and as predicted by ray-tracing for two configurations (see text) and by the Lindqvist and the new models. . . . 87 6.21 As for F i g . 6.20 but for 500 Hz 88 6.22 As for F i g . 6.20 but for 1000 Hz 88 6.23 As for F i g . 6.20 but for 2000 Hz 89 6.24 As for F i g . 6.20 but for 4000 Hz 89 7.1 250 Hz S P ( d B ) curves measured and predicted by R A Y C U B using Q(f) = 0.1 m " 1 and af = 0.2 93 x 7.2 500 Hz £ P ( d B ) curves measured and predicted by R A Y C U B using Q(f) = 0.142 m " 1 and af = 0.15. 94 7.3 1000 Hz £ P ( d B ) curves measured and predicted by R A Y C U B using Q(f) = 0.182 rn"1 and af = 0.12 94 7.4 2000 Hz S P ( d B ) curves measured and predicted by R A Y C U B using Q(f) = 0.213 m " 1 and af = 0.1. 95 7.5 4000 Hz 5 P ( d B ) curves measured and predicted by R A Y C U B using Q(f) = 0.232 m _ 1 and a} = 0.1 95 A . l Comparison of SP predicted by the new model and other models for an infinite fitted space, af — 0 109 A . 2 Hypothetical workroom I l l A . 3 Comparison of the SP predicted by three models in the hypothetical work-room shown in F ig . A . 2 , for three fitting densities 112 A . 4 Floor plan of the warehouse showing dimensions and source and receiver positions 113 A . 5 Comparison of SP curves predicted by three models with that measured in the warehouse, for Q = 0.135 m _ 1 114 A . 6 Comparison of SP curves predicted by three models with measurement in the warehouse, for Q = 0.068 m _ 1 114 A . 7 Comparison of the energy impulse responses predicted by the new model and by R A Y C U B in the warehouse, with r = 20 m , Q = 0.135 m _ 1 . . . . 115 A . 8 Comparison of SP curves predicted by three models with measurement in the scale model workroom, for Q = 0.025 m - 1 116 A . 9 Comparison of SP curves predicted by three models with measurement in the scale model workroom, for Q = 0.05 m - 1 117 X I Acknowledgement First and foremost I thank my advisor, Dr . Murray Hodgson, for his valuable help and patience throughout the research. I would like to express a special word of gratitude to Dr . Jingfang L i , post-doctoral fellow in the acoustics group, for her help in the analysis of sound signals. Thanks are also due to the graduate students in the acoustics group of Mechanical Engineering — especially to Todd Busch and Nelson Heerema — for their help and availability in setting up and adjusting the measuring equipment. Thanks to M r . B r a d Beracus of Canadian Springs Water Co. L t d . for giving us the bottles. xu Chapter 1 Introduction The operation of machinery in workrooms can cause exposure to undesirable noise levels, which can cause annoyance to the workers, reduce their efficiency and, more impor-tantly, cause permanent hearing damage. To prevent this from occurring, it is necessary to control the noise to an acceptable level at the work position. Noise can be controlled in a new factory at the design stage by the appropriate choice of the workroom geometry, construction materials, machine lay out and the selection of quiet machinery. To this end, it would be useful at the design stage if the architect or acoustical consultant could evaluate design options to determine which would l imi t the noise exposure of the workers in the most cost-effective way. For existing workrooms, noise can be reduced by methods such as erecting acoustical barriers, rearranging the plant layout or adding sound absorptive materials to the room surfaces. These methods may be costly, time consuming and disruptive to production, so it would be expedient if the effects on the noise exposure of such changes were known sufficiently accurately before actual implementation. Noise levels in new and existing workrooms can be predicted using computer-based prediction models. They can be used to predict noise levels before and after measures are introduced in order to evaluate their effectiveness. The objective of this thesis is to make a contribution to industrial acoustics wi th the a im of reducing noise levels by improving the accuracy of prediction models. 1 Chapter 1. Introduction 2 Workrooms are enclosed volumes of complex geometry. They are of complex con-struction, this denning the acoustical properties of the bounding surfaces. These prop-erties are usually described by the surface absorption coefficient — the proportion of sound energy striking the surface which is absorbed. Workrooms contain noise sources — machinery, equipment, services, public-address systems etc. — located throughout the volume. The sound radiated by these sources may consist of one (puretone), a few or wide range of (broadband source) frequencies. Sound radiation from a source may occur uniformly in a l l directions (omnidirectional source) or preferentially in certain directions (directional source). In the present work we consider broadband, omnidirectional sources. The amount of sound radiated by a source is described by its rate of energy emission — the acoustical power W in watts — or by its decibel equivalent — the sound power level, Lw = 10Zoc7(W/l(r 1 2 watts) . Noise is a concern when i t is excessive at receiver positions, which may be located anywhere in the volume. Of main interest is the re-ceived sound pressure p in Pa , the energy (proportional to p2) or the sound pressure level Lp = 10log(p2/A x 1 0 - l o P a 2 ) . Predictions of room noise levels are based on predictions of the room sound propaga-tion curve, SP(r) — the variation with distance from an omnidirectional point source of the sound pressure level, Lp(r), minus the source sound power level, Lw: SP(r) = Lp(r) - Lw (1.1) Once SP(r) is known it is possible to determine the resultant sound field for any num-ber of broadband sources by assuming them to be incoherent (random phase relations), determining the sound pressure level contribution of each from Lw, r and SP(r), and adding the various contributions energetically. Industrial workrooms are different from many other rooms in that they are not empty — they are 'fitted'. That is, they contain many obstacles (machines, stockpiles, benches Chapter 1. Introduction Figure 1.1: Sound-ray paths in a fitted region. etc. — the 'fittings' — also called furnishings or scattering obstacles) which scatter and absorb propagating sound. Fitt ings have a major effect on the magnitude and spatial distribution of noise levels in an industrial workroom, as well as on the rate of sound decay wi th t ime [1]. When sound rays are emitted from a source S of power W in a fitted region, the propagation paths are affected by the presence of the obstacles, as shown in Figure 1.1. The total sound energy received at the receiver R is the sum of the direct (or unscattered) contribution Edir IV -Qr Aircr2 (1.2) and al l of the indirect (or scattered) contributions for which the sound rays are reflected or scattered at least one time by room surfaces or by fittings. Models for predicting sound pressure levels in industrial workrooms exist which ac-count for the presence of fittings [2] [3]. There are three important factors which must be considered for fitted rooms as compared with empty rooms ; the fitting spatial dis-tr ibut ion (isotropic, localized to a layer on the floor etc.), the absorption coefficient of the fittings, and the "fitting density". The last two quantities would be expected to vary Chapter 1. Introduction 4 wi th frequency. Whi le the orders of magnitude of these two quantities are known, no method is available for determining these quantities directly [2] [4]. Furthermore, theoret-ical expressions for calculating fitting density assume small fittings and high frequency. "F i t t ing density" describes the average frequency at which sound rays encounter fit-tings. F i t t ing density Q — defined as the product of the number of fittings per unit volume and their average scattering cross-sectional area — is a very important parame-ter in predicting sound levels in fitted workrooms. Almost all existing prediction models use the Kuttruff fitting density formula [5] to calculate this parameter; the Kuttruff formula can be writ ten as e. = ^ £f (i.3) in which, V is the total room volume, in m 3 ; Si is surface area of a fitting, in m 2 . The fitting density has the unit of m _ 1 , and // = 1/Q in meters is the mean free path between the fittings. The Kuttruff formula is valid for very high frequencies and for small fittings, but it is used for al l but the lowest frequencies. The value of Q is typically 0.1 m _ 1 and varies between 0.05 m _ 1 in a sparsely fitted workrooms, up to about 0.15 m _ 1 in densely fitted workrooms wi th a low ceiling [6]. In summary, there is a great need for an accurate, validated method for calculating fitting density and fitting absorption coefficient — and their variations wi th frequency — in the case of large fittings. There is also a need for more efficient noise-level prediction models. A major objective of this research project is to develop and test methods for deter-mining the fitting density, Q, in industrial workrooms, so that SP levels can be more accurately predicted. This objective was originally to be achieved by measuring impulse responses (the variation of squared sound pressure p2 wi th t ime t received when the source generates an impulsive signal — see Figure 1.2) between a source and a receiver in empty and fitted test environments. The fitting density would be determined from Chapter 1. Introduction 5 P A 1 input output Figure 1.2: Illustration of impulse response. the attenuation of the direct sound peak in the fitted case, relative to the empty case. The direct sound would be isolated from the impulse response using gating methods. Unfortunately, it proved impossible to pursue this idea due to distortion of measured impulse responses by the imperfect test sound sources. Attempts at correcting for this distortion were unsuccessful. Thus another method was investigated. This involved de-termining — in fitted environments — the difference in steady-state sound pressure level measured at receiver positions when the direct path was not or was blocked by fittings. This method was unaffected by loudspeaker distortion and was successful. The method has been tested in an anechoic chamber, in a 1:8 scale model workroom and in a full-scale workroom. The second aim of this research project was to develop a workable, acceptably accu-rate, noise-level prediction model for use in industrial spaces. In the development of the new model, the unscattered and scattered energies are treated separately and summed. The unscattered part uses E q . 1.2. B y analysis and comparison with the Kuttruff [7], and Lindqvist [2] image-source models (see Section 2.2.2 and 2.2.5), and Ondet and Barbry [3] ray-tracing model (see Section 2.2.3 and 2.2.5), a modified image-source model is first Chapter 1. Introduction 6 derived for infinite spaces with fittings. Then this model is extended to bounded fitted spaces accounting for room size, room surface absorption and the fitting parameters. To accomplish these aims, the program of the research consisted of: 1. A review of the methods and theories for sound propagation and the prediction of sound pressure levels in fitted workrooms; 2. Theoretical development of a correction formula for calculating fitting density for the case in which the dimensions of the fittings are large compared with the room di-mensions, and of a formula for calculating fitting absorption coefficient using empty and fitted room surface absorption coefficients obtained by measuring reverberation times; 3. Establish the experimental methodology for 1:8 scale-model testing. Thus involved unsuccessful work aimed at correcting for loudspeaker distortion; 4. Development of a formula, Q(f), to calculate the variation of fitting density with frequency, and validation of this formula in an anechoic chamber with fittings, used as 1:8 scale model; 5. Validation of the formula, Q(f), in a fitted 1:8 scale model workroom; 6. Testing the formula, Q(f), in a real full-scale industrial workroom; 7. Development of a modified model for predicting sound pressure levels in work-rooms, based on the image-source method, and its validation by comparison with measurements. This thesis is organized as follows. Chapter 2 presents the theoretical background. In Chapter 3, a correction formula is derived for calculating the fitting density in the Chapter 1. Introduction 7 case of large fitting dimension, and a formula for calculating fitting absorption coefficient is derived. Chapter 4 discusses the general experimental method used in the following chapters. Chapter 5 reports on experiments done in an anechoic chamber without and wi th fittings. The change of fitting density with frequency is found, and a formula to express the relationship is derived using statistical methods. Chapter 6 discusses work to check the formula in a 1:8 scale model workroom wi th fittings. Chapter 7 discusses work to test the formula in a real industrial workroom. Chapter 8 gives the final conclusions and comments regarding future work. In the Appendix, a modified image-source model is developed for predicting sound pressure levels. It is validated using published measurements in a laboratory and in a scale model workroom. Chapter 2 Theoret ical Background This chapter presents a review of the relevant theory and methods relating to the program of research. In particular, the behaviour of sound in rooms is briefly discussed and an introduction is given to the modelling methods which have previously been used to predict sound distributions in industrial workrooms. This forms the background to work to develop methods for calculating fitting density and workroom noise levels. 2.1 Classical (Diffuse-field) Descr ipt ion of Sound Propagat ion in Rooms One of the predominant methods for predicting sound pressure levels in enclosed spaces is the Sabine theory of room acoustics. Developed by Sabine [8] in the early part of this century, this theory states that the intensity of the reflected sound wi l l not vary throughout a room because multiple reflections from the surfaces result in a uniform distribution of the sound energy. A sound field wi th these properties is called 'diffuse'. The Sabine equation is: £ p ( r ) = I . + 101og(^ + ! ) dB (2.1) in which, R = aS/(l — a.), in m 2 , is called the "room constant"; S is the total area of the room surfaces, in m 2 ; r is source/receiver distance, in m; and a is the average room-surface absorption coefficient. In a diffuse sound field the direct sound dominates the total sound close to the source, but at some distance from the source the reflected sound becomes dominant, resulting i n 8 Chapter 2. Theoretical Background 9 CQ •o * > S> S> Q. •o C o CO i Total sound pressure level s ^»^_ "N ^ ^ " » « ^ Diffuse part \ ^Directpar t 1>-log (distance) Figure 2.1: The total sound level in a diffuse field, showing the direct and diffuse contri-bution. the sound level remaining constant wi th increasing distance from the source, as shown in Figure 2.1. Classical diffuse-field theory is only appropriate for rooms which are quasi-cubic in shape, quasi-empty and wi th uniform surface absorption [7]. Unfortunately many classes of room and buildings common in modern architecture do not satisfy the Sabine require-ments. This category of 'non-Sabine' spaces includes many environments encountered in everyday life, such as factories, open-plan offices, supermarkets, school halls, airport buildings, tunnels and even domestic l iving rooms. In a non-diffuse sound field the sound energy does not reach a constant level at any distance from the source, but instead decreases continuously with distance. Sabine theory has been shown to be inappropriate for the prediction of sound pressure levels in such industr ial workrooms [4] [9] [10]. The following factors should be taken into account in predicting sound levels in rooms [9][11]:. room geometry; source and receiver positions; source sound power level; surface Chapter 2. Theoretical Background 10 sound absorption coefficients; fitting density; fitting spatial distribution; fitting sound absorption coefficient; air absorption; frequency variation. Frequency variation is usually described in octave bands. In most industrial applica-tions, the 125 — 4000 Hz octave bands are the most important. 2.2 Prediction Methods for Industrial Workrooms There have generally been three predictive modelling methods developed for work-rooms: 1. Physical scale models; 2. Mathemat ical modelling using geometrical acoustics • Image-source method • Ray-tracing method; 3. Empi r i ca l formulae. Geometrical acoustic models assume that sound propagates as dimensionless rays wi th energy but no phase, so diffraction and wave effects are not considered. A l l geometrical acoustic models make assumptions to simplify the calculations. The common assumptions are: short wavelength compared to the dimensions of the room or fittings; incoherent and omnidirectional sound sources; and room surface absorption coefficients which are independent of the angle of incidence of the sound. 2.2.1 Physical Scale Models The principles of three-dimensional physical scale modelling were first comprehen-sively propounded for use in concert hall design in 1934 [12]. Since that t ime such Chapter 2. Theoretical Background 11 models — first at 1:8, 1:10 or 1:20 scales, and more recently at 1:50 scale — have been used extensively to investigate workrooms. The application of scale-modelling principles to workrooms and the associated experimental methodology was discussed [13]. A 1:50 scale model of an idealized industrial workroom was used to investigate factory sound fields [14]. Attempts were made to bui ld an accurate 1:16 scale model of an existing workroom [15]. A scale model has the benefit that it gives more correct simulation of re-flections, especially for complex geometries where diffraction, diffusion and edge reflection are important. 2.2.2 The Image-source Method Figure 2.2: The image-source method: image source positions in the case of a single plane and i n an empty 2D rectangular room. The image method approximates the reflected sound path from a flat surface by creating a fictitious source — the image-source — by treating the reflecting surface as a mirror as shown in Figure 2.2. In the case of a single plane, the reflected sound is modelled as a sound path direct from the image source, S', to the receiver. In the case of Chapter 2. Theoretical Background 12 an empty room, multiple reflections are achieved by considering further image-sources. A t each reflection the strength of the image-source decreases in relation to the distance from the image-source to the receiver, and by a factor dependent on the absorption coefficient of the surfaces. In the case of a fitted room, the energy from a source to the receiver is divided into two parts; diffuse part — sound direct path blocked by fittings; undiffuse part — sound direct path not blocked by fittings. The undiffuse and diffuse parts are considered separately and summed. 2.2.3 The Ray- t rac ing M e t h o d Figure 2.3: Tracing a sound ray from sound source S to receiver R . The ray-tracing technique consists of 'emitt ing' sound rays, as shown in Figure 2.3, following their paths in the room and recording their successive reflections by surfaces. The sound source is modelled as a point in the room; it emits a number of rays which represent the radiated sound. The paths of these rays are followed unt i l the number of reflections reaches some specified reflection order. The ray direction from the source is random, resulting in some rays crossing a receiver cell directly, some reaching the receiver cell after reflection from fittings or room surfaces, and some not reaching the receiver at Chapter 2. Theoretical Background 13 al l , due to their termination. The energy associated with a ray is attenuated on each surface reflection by the reflection coefficient, 1 —a, where a is the absorption coefficient of the surface. In a fitted workroom, al l obstacles can be taken into consideration according to size and surface area, either as scatterers (machines, storage) or as reflectors (hoods, screens). A n advantage of the ray-tracing technique in predicting factory noise is that enclosed spaces of arbitrary shape can be easily represented, facilitating the modelling of internal walls, barriers or pitched roofs. 2.2.4 Empirical Models Empi r i ca l models are based on experimental observations and measurements to derive empirical formula or prediction curves of sound pressure levels or sound propagation [16][17]. Since they often do not take into account all relevant factors, most empirical models have l imi ted application. 2.2.5 Models Used in the Research Project A great array of models exist for predicting noise levels for industrial workrooms, many of which c la im "excellent agreement" with measurements [18]. Hodgson [11] [9], Windle [18], and Ondet and Barbry [19] have independently reviewed and validated models in published papers. The common conclusion is that the ray-tracing model developed by Ondet and Barbry is the most accurate model but involves significant run times; the Lindqvist model is similarly accurate but less attractive because of its l imited applicabili ty and excessive run times; the other models are less useful because they are less accuracy or of l imi ted application. The Ondet and Barbry model [3], R A Y C U B , is based on ray-tracing techniques. F i t -tings are randomly distributed within any number of pre-defined zones wi th some fitting Chapter 2. Theoretical Background 14 densities and absorption coemcients. Hence, this is the only model which is able to account for arbitrary distributions of fittings and to describe any shape of room with diverse absorption characteristics. Barriers may be considered within the room by de-scribing them as finite planes, although this does not include diffraction. The receiver is a cube of finite side length; the resulting sound pressure levels are thus averages over the cube. When using R A Y C U B the user must choose some input parameters such as the receiver size, the number of rays and the maximum reflection number of a ray on room surfaces and fittings. The Lindqvist model [2] [6], based on the image method, can be applied to any rectan-gular room and any source and receiver positions. In the analysis, the unscattered energy and the scattered energy are treated separately and then summed. The scattered energy is based on a rigorously derived fitted region impulse response. Fit t ings are assumed to be uniformly distributed over the floor area and to have an average absorption coefficient, as in most models. The fittings are described by the Kuttruff Q factor. In this research project, three prediction models are used: the Ondet and Barbry ray-tracing model, R A Y C U B ; the Lindqvist model; the new model based on the image method presented in the Appendix. 2.3 Modelling of Sound Wave Propagation in Fitted Regions When a sound ray is emitted from a source in a fitted region, its propagation is affected by the presence of the obstacles, as shown in Figure 2.4. When there are a large numbers of obstacles wi th different shapes and orientations it becomes impossible to describe separately the influence of each of them. Therefore, a statistical approach to the problem becomes preferable. This procedure, proposed by Kuttruff [5], has been adopted by several authors [2] [3] [10] [20]. Briefly, this approach is based on the following Chapter 2. Theoretical Background 15 O O Figure 2.4: A single sound-ray path in a fitted region. hypotheses: • The obstacles are considered as point-like scattering objects reradiating an om-nidirectional sound wave. The validity of this hypothesis depends on the ratio between the wavelength Ai of the emitted energy and the typical dimension Df oi the obstacle. As a general rule, one must have A < Df [3]; • The scattering effect follows the Poisson process. This means that, if the energy emitted by the source is discretized into sound particles, and if the fact of one of these particles meeting an obstacle is a random event A\, then the sequence of random events Ai,A2, A3,..., Ai,..., follows a Poisson distribution, p(2/) = ^ e _ A , y = 0 , 1 , 2 , . . . (2.2) However, it must be supposed that the number of obstacles encountered by a particle between times t and t + dt is independent of the number of obstacles encountered before t ime t. Under this assumption, the probability Wk that a sound particle hits k obstacles Chapter 2. Theoretical Background 16 after t ime tk can be expressed as, = exp(-Qctk)(Qctk)k k\ The cumulative distribution function F(r) associated with this distribution is [5] 1 — exp(-Qr), r > 0, F(r) = V - (2.3) 0, r < 0. The number of the fittings per unit volume can be expressed by n = N0/V where No designates the number of obstacles contained in volume V. Let Sf be the average scattering cross section of an obstacle. This is difficult to determine when the obstacles have complex shapes. In this case, the simplest method is to approximate the shape by a sphere wi th diameter d and with the same external area Sv = ird2. The average scattering cross sectional area, for high frequency, is then equal to the visible cross section area of the equivalent sphere, given by Sf = ird2/A. Therefore we have, Sf = Sv/A Consequently, in volume V, the average scattering cross sectional area per unit vol-ume, called "the fitting density" and also known as "the scattering frequency", is given by the expression, 1 N0 q Qo = nSf = -J2-^ m " 1 (2.4) This formula is called the Kuttruff fitting density formula. Again , it is valid for very high frequency and for obstacles which are small and spherically shaped, but it is used in most cases by most predictions without regard to its applicability. Note that the quantity If = 1/Qo in m is the fitting mean free path length between fittings. Chapter 2. Theoretical Background 17 Lindqvist [2] gave a correction formula for correcting for the possibility of overlap of fittings in the Poisson process. F i t t ing density from the Kuttruff formula wi l l increase about 5% after correction by the Lindqvist formula when the fittings are large. Hodgson [4] measured factory noise levels and compared the measurements wi th ray-tracing predictions using different fitting density values. A best-fit fitting density was obtained which was about 40% greater than that from the Kuttruff formula. Kurze [20] noted that the Poisson distribution results in an absorption exactly de-scribed by Sabine's formula. However in many cases, the fitting density from E q . 2.4 is lower than that expected. Kurze gave some suggestions for applying somewhat different statistical models for fitting density (section 3.1 gives more detail). A k i l and Oldham [21] concluded that it is the product of fitting density and fitting absorption coefficient that determines the sound propagation characteristics in a fitted workroom. They found that the difference in sound pressure level in rooms wi th fittings and without fittings is almost constant and that the SP characteristics are nearly identical if one parameter is halved and the other is doubled. In summary, methods exist for calculating fitting density but which assume small dimension and high frequency. In the next chapter, a correction formula is developed for calculating the fitting density in case of large fittings. Chapter 3 Theoretical Developments We have seen that compared with empty rooms, there are two important parameters besides the fitting spatial distribution used in predicting sound pressure levels in fitted rooms. One is the fitting absorption coefficient, the other is the fitting density. Methods for calculating fitting density were reviewed in Chapters 1 and 2. Recall that existing methods assume small fittings and high frequency. In this chapter, first a correction formula for calculating fitting density in cases when the fitting dimensions are large is derived, and the magnitude of the correction is investigated; then a correction formula for calculating fitting absorption coefficient is developed using the empty and fitted room surface absorption coefficients obtained by measuring the reverberation times. 3.1 Correction of the Fitting Density for Large Fitting Dimension In the Kuttruff formula, the fitting size was assumed small compared wi th the fitting mean free path. In real factories, the machine sizes may be same order of magnitude as the fitting mean free path; the fitting density predicted by the Kuttruff formula wi l l often be inaccurate. It is therefore necessary to correct this formula for large fitting size. 3.1.1 Introductory Discussion Kurze [20] noted that in many cases, the fitting density found using E q . 2.4 is smaller than expected, so higher surface absorption coefficients must be used in prediction to obtain agreement wi th measurements. Kurze suggested the use of somewhat different 18 Chapter 3. Theoretical Developments 19 statistical relations. The shifted distribution is one possible candidate; in the range 7"0 < r < oo, the mean free path If should be where r 0 is some distance. This means that the fitting mean free path is smaller than that calculated from E q . 2.4 by an amount equal to the distance ro- In the case of numerous of fittings of different dimensions, it should be the average radius of the fittings. Another possibility for arriving at a reduced mean free path length is to use the gamma distribution. The mean free path becomes '/ = 7T-in which p is the parameter in the gamma function T(p). For p — 1, this equation is a special case of E q . 2.4; for p > 1, If is less than l/Qo, so it provides better consistency between absorption coefficients and decay rates of reverberation curves [20]. Hodgson [4] measured the sound pressure levels in a factory and compared the mea-surements wi th predictions by a ray-tracing model. B y using different values of fitting density and fitting absorption coefficient, Hodgson found the best-fit fitting density in the fitted region to be 40% larger than the value given by the Kuttruff formula. Lindqvist [2] gave a formula for correcting for the possibility of overlap of fittings in the Poisson process. Lindqvist supposes there is a Poisson distribution of fittings wi th density Q and considers two identical spheres of radius R, which overlap as shown in Figure 3.1. The probability density of no second fitting being found within a distance r from the centre of a fitting is / ( r ) = Qe-^ (3.1) The probabili ty function for no fitting within a distance of 2R is thus Chapter 3. Theoretical Developments 20 2S, Figure 3.1: Two overlapping spheres of radius R. r2R FC2R) = / Qe~Qrdr = 1 -Jo In this way, an approximate correction formula is found as follows: -2QR (3.2) QL = Q0-(I + ^ - 1 - Q I R 2 ) (3.3) in which, Q0 is the Kuttruff fitting density. As an example, if QQR = 0.1, the correction w i l l mean an increase in the fitting cross-section, and therefore in the fitting density, of 8 %. 3.1.2 Correction for Large Fitting Density When the mean free path, lf0 = 1/Qo, is the same order of magnitude as, or one order greater than, the fitting dimension, we should subtract the fitting dimension from the mean free path, as shown in Figure 3.2. That is, 1 1 Q - Q o ~ 2 R (3.4) Chapter 3. Theoretical Developments 21 \ \ Figure 3.2: Three spheres of radius R used to estimate the mean free path reduction. Considering the dimensions of typical fittings, we have to include another correction. In Figure 3.2, there is a th i rd fitting (3) near the other two (1 and 2). If the thi rd fitting is dimensionless, it w i l l not block the sound propagation from fitting 1 to fitting 2. However, if the fitting has non-zero dimension, it blocks the path from fitting 1 to 2. The real propagation path wi l l be from fitting 1 to 3; it is less than that from fitting 1 to 2, the mean free path IJQ. Suppose that there is a Poisson distribution of fittings with density QQ, and consider three identical spheres of radius R, shown in Figure 3.2. From E q . 2.3, the probability function for no fitting wi thin distance r is e~®°r. This is only true for dimensionless fittings. The probability function for having a fitting within distance r is 1 — e _ < ^ ° r , and the probability for the fitting blocking the free path is 2R/2irr after considering the fitting dimension as shown in Figure 3.2. For a two dimensional problem, for any position from the centre of a fitting in a region of a circle from 0 to 27T and distance r from 2R to 1/Qo — 2R, the probability for the fitting with radius R blocking the original mean free path is (1 - e-Q°T)2R/(2Trr). For the region of 0 < r < 2R and 1/Q0 - 2R < r < 1/Q0, the fittings w i l l have the possibility to overlap which have been corrected by the Lindqvist Chapter 3. Theoretical Developments 22 formula E q . 3.3. The further reduction of the mean free path can be written as 1 /•2ir+ p-±--2R o/? Replacing exp(-Qr) by (1 — Qr), an approximation can be obtained A±- = 2R- 8R2Q0 (3.6) The total reduction of the mean free path from Eqs. 3.4 and 3.6 is Ak - 4-R - 8.R2<2o (3.7) Replacing 2R by the fitting mean dimension Df, the corrected fitting density is Q " ~ i - 22? /+ 2Q0D} ( 3 - 8 ) Combining the new correction with E q . 3.3 by Lindqvist , the fitting density can be writ ten as Q' = ~J~ on , o i^ n2 ( 3 - 9 ) ^ - 2Df + 2Q0Dj in which, a is the correction factor given by Lindqvist , 37T 8 and Qo is calculated by the Kuttruff formula E q . 2.4. The effect of the new correction depends on parameters Df and Qo. As an example, if Qo — 0.1 m _ 1 and Df = 1 m , the fitting mean free path wi l l decrease from 10 m to 8.2 Chapter 3. Theoretical Developments 23 Table 3.1: Some typical corrected fitting densities (m 1 ) . Qo Q, Df = 0.5 m Q, Df = 1 m Q, Df = 2m Q, Df = 5 m 0.05 0.051 0.053 0.054 0.051 0.055 0.056 0.052 0.061 0.064 0.055 0.080 0.094 0.10 0.102 0.110 0.113 0.104 0.122 0.128 0.108 0.147 0.165 0.118 0.200 0.288 0.15 0.155 0.174 0.181 0.159 0.201 0.218 0.167 0.259 0.314 0.187 0.240 0.351 m , and the fitting density wi l l increase to 0.122 m _ 1 , which is 22% larger than Qo- As is the case for E q . 2.4, E q . 3.9 is only true for high frequency. Table 3.1 shows some typical fitting densities corrected by Eqs. 3.9 and 3.3, in which the first column is the Kuttruff Qo, columns 2 to 5 are corrected values for Df = 0.5, 1, 2, and 5 m respectively, the left value in each column is QL from the Lindqvist formula E q . 3.3, the middle value is Q" from E q . 3.8 and the right value is Q' from E q . 3.9. From the table, we can see that the differences are small for small Q0 and Df, but that the differences are large when Df and Qo are large. The value of Q' from E q . 3.9 is 134 % greater than Q0 for the extreme case of Q0 = 0.15 m - 1 and Df = 5 m . It can be explained by considering a hypothetical region fitted with n spherical fittings of diameter 5 m . The dimensions of the region are 100 x 100 x 100 m. The surface area of a sphere is Sv = TTD2 = 78.54 m 2 ; from the Kuttruff formula n = WQ0/SV = 7639.4. If the spheres are uniformly distributed in three dimensions, the distance between two sphere centers is 100/7639.4 1 / 3 = 5.08 m , as shown in Figure 3.3. This is just slightly larger than the sphere diameter, and the mean free path between the fittings is much smaller than l/Q0 = 1/0.15 = 6.67 m , so 1/Q' = 1/0.351 = 2.85 m by Eq.3.9 seems reasonable. This analysis assumes a uniform distribution of fittings; it is quite clear that these spheres cannot in fact be distributed randomly in this region. In the following chapters, this equation is validated by experiment, and a formula to express the variation of fitting density with frequency is derived and validated. Chapter 3. Theoretical Developments 24 5.08 m •* Figure 3.3: Dimension of spheres and distance between them. 3.2 A Correction Formula for Calculating Fitting Absorption Coefficient A t present there are no method available for measuring fitting absorption coefficient directly [4] [18]. Two methods are often used to obtain this parameter for predicting sound propagation in fitted workrooms: one is the best-fit method using prediction model and assuming fitting density known; the other method is the empirical estimation. A corrected formula for determining this parameter is derived in this section. Normal ly surface absorption coefficients are determined from measured reverberation times, using diffuse-field theory. This is only accurate when changes in reverberation t ime result uniquely from changes in absorption — for example in empty rooms. If the introduction of fittings into a room simply increased the room absorption, it would be possible to determine the fitting absorption coefficient, a / , from the reverberation t ime in the fitted room, and the resulting "fitted-room" average surface absorption coefficient using the formula: CifT(Sr + Sf) - cterSr ctf = (3.11) in which, ctfT is absorption coefficient of fitted room (included fittings), cter is that of Chapter 3. Theoretical Developments 25 empty room, Sr is room surface area, and 5/ is fitting surface area. However, fittings — even when of low absorption — reduce reverberation times sig-nificantly due to scattering effects. Use of E q . 3.11 gives excessive fitting absorption coefficient — for example, maybe 2 or 3 times larger than expected. The reason that fittings affect sound propagation is that they reduce the mean free path between sound reflections in the fitted room. In other words, the rate of sound reflection by the surfaces and fittings is much greater in the fitted room than in the empty room. This phenomenon increases the rate of sound decay, so E q . 3.11 must be corrected. In the following, a corrected version of E q . 3.11 to calculate fitting absorption coeffi-cient is derived. We define the effective mean free path D r , e / / as that including fitting scattering effects. In empty rooms, the effective mean free path of sound propagation is equal to the room mean free path, -D r,e// — DT = 4V/S, where V is room volume and S is room surface area. In fitted rooms, the mean free path wi l l be reduced by a factor, exp( — QDT). The effective absorption of the room surfaces wi l l change from STaT to STaT/exp( — QDr). For the same reason, the effective absorption of the fittings w i l l change from SfOtf to SfCif/exp(—QDT). Using these factors, E q . 3.11 can be corrected as follows: afT{ST + Sf) - aerST/exp(-QDT) a> = 3,1^-QD,) ( 3 - 1 2 ) This formula can determine fitting absorption coefficient using the empty and fitted room surface absorption obtained by measuring the reverberation times. This method is independent of any prediction model. In Chapter 7, this method is validated using the measurements in a machine shop. In the next chapter, we establish the experimental methodology for the scale-model tests which wi l l be used to validated the new fitting density and absorption coefficient models. Chapter 4 Measurement Methods This chapter presents the general experimental methodology used in the experimental work done to validate the new theoretical models, and briefly introduces the instrumenta-t ion involved. Tests were performed in an anechoic chamber and in a test enclosure, both considered as 1:8 scale models, using the M a x i m u m Length Sequence System Analyzer ( M L S S A ) . Thus it is of interest to review scale modelling principles and the operation of the M L S S A system. The unsuccessful attempts to resolve a problem l imi t ing the accuracy of impulse response measurement is reported in this chapter also. 4.1 Principles of Scale Modelling and Choice of Scale Factor In order to model the sound field in an enclosure at l : n scale (n is the scale factor), al l dimensions are scaled by 1/n. To avoid confusion between scale-model and full-scale values, al l quantities referring to the scale model are denoted by a prime, so the corresponding full-scale lengths become I = nl'. If the sound velocity in air, c = c', the t ime sound takes to travel some distance is t = nt'. Furthermore, the ratios between the model and full-scale wavelengths and frequencies are: A = n\' and / = f'/n. For accurate scaling of al l wave/enclosure effects, the model wavelength-to-dimension ratios must equal those at full scale; thus, the frequencies of the test signals must be scaled by n. Further, the frequency dependence of air absorption m in the model must be scaled by n , which means m = m'/n. Surface absorption coefficients remain the same: a = a'. The dimensions of al l fittings are scaled by 1/n, in which case wavelength-to-dimension 26 Chapter 4. Measurement Methods 27 ratios are preserved, and the fitting density, Q, is scaled by n : Q = Q'/n [13]. If a model is constructed according to the above scaling laws, then both L P and L W wi l l scale by n. This is because distance scales by 1/n; the source produces a certain mean-square pressure at a distance r full scale, but at r / n in the model[13]. Sound propagation i n a scale model suffers excess attenuation resulting from the difference between m' and nm. This could be compensated for approximately using prediction results. However this was not done in this work since our only objective was to compare prediction and scale-model experiment. The full scale sound pressure levels and sound power levels are related to the model values by mul t ip lying by n , the scale factor. Thus: LWFS = LWM — 20log(n) ; Lpps = LPM — 20log(n) The sound propagation, SP, is: SPFS — LPFS — Lwps-We have the following relationship in a free field: LwFs = LpFS(rFS) + 20log(rFS) + 11 dB (4.1) If rFS = 1 m , then SPi?s{r) becomes: SPFS(r) = LpM{rM) - 20log(n) - LpM(rM = 0.125m) + 20log(n) - 11 So the formula to calculate the SP in full-scale values is, SP(r) = Lp{rM) - Lp{rM = 0.125) - 11 dB (4.2) The scale factor is l:n, wi th n = 8 used in all experiments reported here. The reason for choosing n = 8 is that the mean dimension of the available bottles, which were used as scale-model fitting to model real scattering objects in factories, correspond to the Chapter 4. Measurement Methods 28 typical mean dimensions of large full-scale fittings in workrooms. The anechoic chamber corresponds to a sufficiently large region. Furthermore, n = 8 has the advantage that model-scale octave-band frequencies, when scaled to full-size equivalents, correspond to standard octave-bands. In most industrial applications, the 125 — 4000 Hz octave bands are most important. These correspond to a scale-model frequency range of 1000 — 32000 Hz in a 1:8 scale model, the test range involved in all tests. 4.2 Measurement System Measurements were made in different test environments using the measurement sys-tem illustrated in Figure 4.1. The M L S S A system generates an electrical test signal which is amplified by a power amplifier; a loudspeaker converts the amplified electrical signal to sound energy. The sound waves propagate in the test environment, and the receiver microphone responds to the sound pressure and converts it to an electrical signal; this signal returns to the M L S S A system after being amplified by the measuring amplifier. The instruments used were as follows: • Personal computer, P C III, containing the M L S S A system; • Microphone, Bruel Sz Kjaer type 4138, with frequency response from 20 Hz — 200 kHz ; • Microphone power supply, Bruel & Kjaer type 2804; • Measuring amplifier, Bruel & Kjaer type 2609; • Power amplifier, model No. 2250 M B ; • Three different loudspeakers were used to cover the wide test frequency range. Chapter 4. Measurement Methods 29 Microphone Power Supply B&K 2804 Measuring Amplifier B&K 2609 Computer PC III MLSSA system Power Amplifier Model No 2250 MB Figure 4.1: Measurement system showing the test environment and the measuring in-struments. Chapter 4. Measurement Methods 30 — Speaker 1: diameter 75 m m , maximum power 30 W , frequency range 250 Hz — 25 k H z , wi th a cone, shown in Figure 4.2; — Speaker 2: diameter 72 m m , maximum power 20 W , frequency response 170 Hz - 25 k H z , without cone; — Speaker 3: diameter 105 m m , maximum power 70 W , frequency response l k — 70 k H z , wi th the cone. 4.3 M L S S A System The M L S S A system is a powerful system analyzer based on newly developed tech-niques using maximum-length sequences ( M L S ) [22]. M L S S A performs transfer-function analysis on any linear systems, but its primary application is in acoustical measurements. The M L S stimulus is periodic and exactly repeatable and has a low crest factor for high signal energy content. Thus MLS-s t imula ted measurements provide very high signal-to-noise ratios [22]. The M L S S A system measures the impulse response, the most funda-mental descriptor of any linear system. The response of the system to the M L S stimulus is measured. The system impulse response is obtained from the cross-correlation of the M L S and the response. From the impulse response a wide range of important functions are derived through computer-aided post-processing. The transfer function, for instance, is obtained by applying a Fast Fourier Transform ( F F T ) to a segment of the impulse response. M L S S A allows pre-averaging of the raw data over up 16 M L S periods; this can increase the signal-to-noise ratio of a measurement. Further averaging is performed by using the Go Average command, which averages impulse responses, not the raw data. In al l measurements, 2 Go Average x 16 pre-averages were used. Chapter 4. Measurement Methods 31 o E E J Loud speaker 4 480 mm Figure 4.2: The cone used to reduce loudspeaker directivity. For speakers 1 and 2 the main M L S S A operational parameters were as follows: A c -quisition length = 16384 samples (270.3 msec); Acquisi t ion sample rate = 60.61 kHz ; Anti-al iasing filter bandwidth = 20 kHz; Stimulus amplitude = ±0 .8613 volts for speaker 1 and ±0 .3281 volts for speaker 2; F F T window = rectangular; F F T size = 16384 points; F F T resolution = 3.70 Hz; F F T mode = transfer function. The M L S S A parameters for speaker 3 were: Acquisi t ion length = 32768 samples (278.5 msec); Acquis i t ion sample rate = 117.6 kHz; Anti-aliasing filter bandwidth = 40 k H z ; Stimulus amplitude = ±1 .579 volts; F F T window = rectangular; F F T size = 32768 points; F F T resolution = 3.59 Hz; F F T mode = transfer function. 4.4 Scale-model Sound Source The ideal sound source for the experiments should be powerful enough, small dimen-sion, omnidirectional, and should work over the test frequency range, from 125 to 32k octave bands. Unfortunately no single, unmodified commercial loudspeaker meets these requirements. Chapter 4. Measurement Methods 32 90° ~WcP Figure 4.3: Measuring loudspeaker directivity. 270 270 Figure 4.4: Directivities of speaker 1 with the cone in 4 octave bands. Chapter 4. Measurement Methods 33 Figure 4.5: Directivities of speaker 2 in 4 octave bands. Three different loudspeakers were used as sound sources — speakers 1 and 3 included a cone. The cone was used to reduce the sound directivity of the loudspeaker. The cone, shown in Figure 4.2, was made of fibre glass, and was covered wi th a layer of 5 - 8 m m modelling clay to reduce radiation from the walls. To test the directivity of the loudspeakers, sound pressure levels in octave bands were measured in an empty anechoic chamber as shown in Figure 4.3. Figures 4.4 and 4.5 show the directivities of speaker 1 with the cone and of speaker 2, respectively. The distance from the speaker to the microphone was 1 m. The speaker and microphone were located 1 m above the wire net floor. Chapter 4. Measurement Methods 34 Loudspeaker 2 (no cone) has significant directivity in all octave bands, especially at low and high frequencies, as shown i n Figure 4.5. A t 250 H z and angle 90° the sound pressure level was 12 dB lower than in front. Normally, the pressure level in front was more than 3 dB greater than that of other positions. Loudspeakers of 1 and 3 wi th the cone have less directivity i n octave bands as shown i n Figure 4.4 (similar curves were obtained for speaker 3 wi th the cone), especially at high frequencies. Normally, the pressure level in front was no more than 2 dB greater than that of other positions, but at 500 H z and angle 120° the sound pressure level was 7 d B lower than the average. Using the cone can efficiently reduce the sound directivity of a loudspeaker. In al l experiments, in order to reduce the effect of the loudspeaker directivity, the speaker was always turned to face the receiver microphone. The sound power of the loudspeaker was calculated using sound pressure levels mea-sured in the free field with r = 0.125 m and the same relative positions of speaker and receiver used i n experiments. 4.5 Correction for Loudspeaker Distortion The method which was originally to be investigated for determining fitting density involved measuring the impulse response between a loudspeaker and a microphone in empty and fitted environments. This involved correcting for distortion of input signal by the loudspeaker. This section discusses unsuccessful attempts to resolve a problem l imi t ing the accuracy of impulse response measurement. We tested two methods for dealing wi th the problem of loudspeaker distortion — the correlation method and the equalization method. The bias error of loudspeaker distortion is discussed in the end of this section. Chapter 4. Measurement Methods 35 4.5.1 Correlation and Spectral Analysis Method Correlation and spectral analysis methods are especially useful in cases when the functions to be compared are stochastic or at least have such a complicated structure that a mutual relationship cannot be recognised by simple inspection. It is just this aspect which frequently applies to room acoustics, as for instance in the impulse response [7]. Let Si(t) and ^ ( i ) be two stationary time functions, such as two signals generated by loudspeakers; their cross-correlation function <7i2(r) is: 5 l 2 ( r ) = l i m i - fT° Sl(t)s2(t + r)dt (4.3) T0->oo i Q Jo When Si = 5 2 , <7n is the autocorrelation function. The power spectral density function GXy{f) can be obtained from the correlation function by applying Fourier transforms [23]. x(t) X(f) hxy(t). H x y ( f ) y(t) Y(f) Figure 4.6: Ideal single input/single output linear system. A n ideal single-input/single-output system, shown in Figure 4.6, has input x(t) and output y(t). The system has unit impulse-response function hxy(t); this is the output when the input is a perfect impulse b~(t). In the frequency domain these quantities are X ( / ) , Y(f) and H(f) obtained from the Fourier transforms of the t ime function. In the t ime domain, the convolution operation — in frequency domain, simple mult ipl icat ion — gives the relationship between the three quantities: /oo x(r)h(t - T)CLT -oo = x(t)*h(t) (4.4) Y(f) = X(f)Hxy(f) (4.5) Chapter 4. Measurement Methods 36 If we do not consider the deviation of the output, in the terms of the autospectral functions [23] [24] Gyy(f) = \Hxy(f)\2Gxx(f) (4.6) where | . r7 x y ( / ) | is the system gain factor, and G is the one-sided autospectral density function. The complex frequency response function H(f) cannot be determined using the real autospectra Gxx(f) and Gyy(f). Instead, this is given by H*y{f) = ^ (4.7) or *.</) = (4.8) A n important relationship involving the magnitude of the cross-spectral density func-t ion across a linear system leads to the (ordinary) coherence function, defined as [23] [24]: ^ v ( / ) = n , t n m 0 * ^ ( / ) < 1 (4-9) \Gxy(fW G„U)Gm{f) This function is a powerful tool i n evaluating the quality of both the frequency response function estimates and the estimates for the true output spectral density G ^ ( / ) - As the true output G'^f) equals \Hxy(f)\2Gxx(f), it follows from E q . 4.8 that 7 l V Gyy(f) 1 i U J Thus the coherence function can be interpreted as the fractional portion of output y{t) which is linearly due to input x(i) at frequency / . This leads to the concept of the coherent output spectrum which is in fact merely a way of estimating the true output spectrum G' (f). Defined in this way the coherent output spectrum is usually computed Chapter 4. Measurement Methods 37 by Gyy'U)=ilvU)Gyy{f) (4.11) The output noise G'nn(f), which is that part of the output which does not follow from linear operations on x{t) at frequency / , becomes Gnn(f) = [1 ~ lUf)}Gyy(f) (4.12) If a part of source contributing to the output is incoherent with al l the other contributions, this part can be treated as noise. Then the source can be identified, which means that its contribution to be the total output can be separated, using the coherent output spectrum, Eq.4 .11 . The above methods can be used to try to separate the signal distortion by the loud-speaker: this w i l l be attempted for one measurement made in the present work. The acoustic system includes the power amplifier, loudspeaker (with cone) and the room as shown in Figure 4.7. The system input is the M L S S A output and the output is the signal measured at r = 1 m for the case of 81 bottles (see Section 4.6) using loudspeaker 1; the two signals are shown as impulse responses in Figure 4.8. The M L S S A output can be measured by connecting its output to its input directly. x(t) ! Power Loud Cone X(f) i Amplifier Speaker Sound Propagation in Room System Figure 4.7: Schematic diagram of the acoustic system. y(t) Y(f) The ordinary coherence function between the input and output signals is calculated using E q . 4.9 and is shown in Figure 4.9. The cross-correlation and auto-correlation functions of the input and output signals are shown in Figure 4.10. Chapter 4. Measurement Methods 38 0.5 0.0 Impulse response(volts) (a) i i i i i i i i i i i Time(msec) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.005 0.000 -0.005 Impulse response(volts) (b) ' ' Time(msec) 10 20 30 40 50 Figure 4.8: (a) input and (b) output signals of the acoustic system. 1.0002 - Linear magnitude(volts/volts) 1.0000 f 0.9998 0.9996 0.9994 - i • i 103 10" Frequency(Hz) Figure 4.9: The calculated ordinary coherence function between the input and output signals for the system shown in F i g . 4.7. Chapter 4. Measurement Methods Cross-correlation function volts'volts • 5 . 0 0 0 0 E - 7 h - 3 0 - 2 0 Cross-correlation function volts'volts - 2 0 - 1 0 (a) Time(msec) 1 0 . 2 0 3 0 4 0 (b) Time(msec) 1 0 2 0 3 0 4 0 0 . 0 0 0 1 0 Auto-correlation function - votts*volts (c) 0 . 0 0 0 0 5 -0 O O O O O Time(msec) - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 Auto-correlation function volts'volts (d) Time(msec) Figure 4.10: The cross-correlation and auto-correlation functions, (a) gxy, (b) gyx, gxx, (d) gyy, of the input and output signals of the system shown in Fig. 4.7. Chapter 4. Measurement Methods 40 Transfer function magnitude \ ;B case 1 dB volts/volts (0.1 oct) if!,' Al il 111 case 2 -30 * il!VH i i case 3 lu r i • VM i l l !i k IA IWi 11 -A\ i / l I"' \ / ! '? '1 ii *\jT*! i CT, / \ \ A -40 / A w 1 V * I M . / i / h all / l / 1 u 6 1 V ' V -50 / V A / Ni-' J \ -60 . . . 1 103 104 Frequency(Hz) Figure 4.11: Comparison of true transfer functions for 3 cases (see text). Using the ordinary coherence function, 7 j u , the true output G'yy(f) is calculated using E q . 4.11, and the true transfer function is calculated by E q . 4.8. Figure 4.11 gives the comparison between the true transfer function and that measured. In Figure 4.11, case 1 is the measured transfer function obtained from the F F T of the measured impulse response, case 2 is that calculated using E q . 4.7, and case 3 is that calculated by Eqs. 4.9, 4.11 and 4.8 to suppress the incoherent noise. From the figure it can be seen that the results for case 1 and 2 are similar wi th small differences at high frequency; case 3 is very close to cases 1 and 2, since the ordinary coherence function, ^ly, shown in Figure 4.9, is very close to 1 for al l frequencies. Figure 4.11 shows that the incoherent noise is very small , and that the sound reflections in the cone cannot be separated by this method. Chapter 4. Measurement Methods 41 4.5.2 Equalization Method The second method is by equalization, using the M L S S A system. The principle of equalization in the M L S S A system is the complex ratio of a measurement to a reference measurement (representing the loudspeaker distortion) which can be a measured in the empty room or measured near the source i n the fitted room. Unfortunately equalization did not work for this case; the sound reflections in the cone were not completely separated by equalization, the signal to noise ratio decreased significantly, and the results lost the original meaning. 4.5.3 Bias Error of Loundspeaker Distortion x(t) Ideal System y'(t) p Bias Error y(t) X(f) h(t), H(f) Y'(f) e(t), E(f) Y(f) Figure 4.12: Demonstration of the error in the system. Next , another way to deal with the bias error of loudspeaker distortion is discussed. Assume the bias error arising mainly from the cone is a function of frequency and t ime for impulse response, as shown in Figure 4.12, but that it is constant in any octave band. The assumption is reasonable since the cone gives an attenuation for sound propagation which depends on the frequency, and for a given frequency it should be constant. Let e(t) to be the transfer function of all bias errors, as shown in Figure 4.12. y(t) is the measured output, y'(t) is the true or corrected output. In energy, Ey = EeTTOTE'y. The sound pressure level, Lp — 10log(EerroT) + L'p, in which only Ey and Lp can be measured or calculated, and EeTTOr and E'y are unknown. In what follows in the next chapters, the ratio of two energies — or the difference of the two sound pressure levels — is used, so the bias error disappears from these expressions and only the random error is left: Chapter 4. Measurement Methods 42 d = 27.5 cm E o o Figure 4.13: The dimensions of the bottles used as 1:8 scale fittings. Ev(l) _ EeTrorE'y(l) _ E'y(l) Ey(2) EerrorE'y(2) E'y{2) (4.13) Lp(l)-Lp(2) = 10log(Eerror) + L'p(l)-10log(EeTTOT)-L'p(2) = L'p{l)-L'(2) (4.14) 4.6 Scale-Model Fittings For modelling sound propagation in fitted regions, empty 18.9 1 mineral-water bottles were used as fittings. These hard plastic bottles were 40 cm high and 27.5 cm in diameter as shown in Figure 4.13 (3.2 m high and 2.2 m in diameter in full scale). The reason for choosing these bottles is that the size is suitable for scale modelling wi th a scale factor of 1:8 when modelling the case in which the fitting dimension is large, and the absorption coefficient of the bottle surface is low. The surface area, Sv, of one bottle is approximately 0.46 m 2 , and the mean dimension is 0.3 m. The chamber volume, V is 46.2 m 3 . Using the Kuttruff formula of E q . 2.4, the fitting density is Chapter 4. Measurement Methods 43 Qo = nSv 1~V n x 0.46 4 x 46.2 rn - l (4.15) in which n is the number of the bottles. Thus, • n = 81 bottles for QQ = 0.2 m _ 1 (0.025 m " 1 full scale); • n = 162 bottles for Q0 = 0.4 m _ 1 (0.05 m _ 1 full scale); • n = 243 bottles for Q0 = 0.6 m _ 1 (0.075 n r 1 full scale). Tests were performed for these three fitting densities which are typical of low and moderately densely fitted workrooms. It was not feasible to test higher densities. A few tests were also done with more absorbent fittings. These were made by covering the bottles wi th cotton sheeting material. 4.7 Test Environments 4.7.1 Anechoic Chamber A n anechoic chamber was used as a test environment to approximate an infinite region. It was considered to be a 1:8 scale model. The dimensions of the anechoic chamber are 4.7 x 4.1 x 2.4 m high, which in full scale values are 37.6 x 32.8 x 19.2 m high. The chamber walls, floor and ceiling were made of glass-fibre wedges. A wire net trampoline allowed access to the chamber. 4.7.1.1 E m p t y Chamber In order to confirm that the empty anechoic chamber is a good approximation to an infinite empty region (free field), the empty chamber was measured using the three loud-speakers. Using speakers 1 and 3, octave-band Lp measurements in the empty chamber Chapter 4.. Measurement Methods 44 were made at distances of 0.125, 0.25, 0.5, 0.625, 1, and 1.25 m. W i t h speaker 2 measure-ments were made at 0.5, 1 and 1.25 m distances only. Average levels were determined from measurements at between 4 and 8 positions at each distance. Table 4.1: Background noise in the empty anechoic chamber Octave band(Hz) 125 250 500 l k 2k 4k 8k 16k 32k LP (dB) 35.9 33.5 33.1 33.8 35.3 37.7 40.3 42.5 44.7 The background noise was measured in octave bands. It varied from 33 dB at low frequency to about 44 d B at high frequency, as shown in Table 4.1, which is good enough for al l the application cases, except with speaker 3 at 1000 Hz at which the sound pressure level is only 4 dB greater than the background noise when the source/receiver distance greater than 1 m . Figure 4.14 shows the variation of octave-band sound pressure level with distance of source/receiver , measured using speaker 3. The loudspeaker was located at the middle of the chamber, 1 m above the wire net floor; the microphone had the same height as the speaker. From Figure 4.14, we can see that most octave bands display the - 6 d B / d d expected in a free field. However, there is some deviation from this behaviour when the microphone is near the wall . This shows that the chamber is not perfectly free field, and that there is some reflection from the walls, especially at low frequency. To avoid this shortcoming, in subsequent experiments both source and receiver were located at least 1 meter from any surfaces in the chamber. 4.7.1.2 Fitted Anechoic Chamber In order to study sound propagation in an infinite fitted region, a test environment must be found which is large enough to represent a sufficiently good approximation to Chapter 4. Measurement Methods 45 1 0° Distance(m) Figure 4.14: Octave-anechoic chamber. band sound pressure level variation with distance in the empty SP(dB) \ (a) CD 2 , 0- 5 CO ( b ) 0 0 -5 : -5 -10 -10 -15 >v--15 i IO"1 10° Distance(m) 10"1 io" Distance(m) Figure 4.15: Comparison SP curves, (a) Q = 0.1 m - 1 , (b) Q = 0.5 m _ 1 , (—) in an fitted very large region; ( ) in the fitted anechoic chamber. Chapter 4. Measurement Methods 46 such a region. The anechoic chamber, considered as a 1:8 scale model, and suitably fitted, was used. R A Y C U B was used to compare the fitted chamber wi th an approximately infinite fitted region. A n infinite region was approximated using a room size of 100 X 100 x 100 m . Figure 4.15 shows the comparison of SP curves for the chamber wi th those for the very large region using Q = 0.1 and 0.5 m _ 1 . The source is at the middle of both regions. It was found that the differences of sound pressure levels for the two regions are very small — less than 1 dB — if the source is at the middle of the chamber and the source/receiver distance is less than half of the room mean free path, MFP = 4V/S, in which V is the room volume and S is the room surface area. For the anechoic chamber, MFP = 2.3 m . The differences are large when the source or the receiver is near any room surface; the differences increase with fitting density and with the source/receiver distance by up to 3 dB at r = 2 m wi th Q = 0.5 m 4 . To avoid problems associated wi th the finite size of the chamber, in subsequent mea-surements in the chamber the maximum source/receiver distance was 1.25 m and the source was always located in the middle of the chamber. Used in this way, the anechoic chamber is an adequate approximation to an infinite region. 4.7.2 Scale-model Workroom Tests were also done in a more realistic environment — a 1:8 scale model workroom. Figure 4.16 is a photograph of the model. It was 3.75 m long, 1.875 m wide, and 1.875 m or 0.625 m high, as shown in Figure 4.17. In full-scale dimensions, the workroom was 30 m long, 15 m wide, and 15 m or 5 m high. The vertical walls and ceiling were made of varnished plywood, and the floor was of concrete. The average absorption coefficient of the surfaces was measured in a diffuse sound field to be approximately 0.05 in a l l test octave bands. As a check on the accuracy of these surface absorption coefficients in the non-diffuse Chapter 4. Measurement Methods 47 Figure 4.16: Photograph of the scale model. 3.75 m Figure 4.17: Dimensions of the scale model space. Chapter 4. Measurement Methods 48 sound field of the scale-model workroom, the empty room was measured to obtain the sound propagation curves; then R A Y C U B was used to model the room and to predict the SP curves for different surface absorption coefficients unti l a best-fit wi th the mea-surement was obtained. A t first it was assumed that the surface absorption coefficients were 0.05 for al l surfaces and all frequencies, but the SP curves measured in the empty workroom were a l i t t le different from those predicted by R A Y C U B using this coefficient at high frequencies. - Comparison between prediction and experiment showed that the co-efficients of the ceiling should be 0.05 for 2 kHz and 4 kHz , 0.1 for 8 k H z , 16 k H z and 32 k H z , since the ceiling was thinner than the vertical walls. The coefficient of other surfaces was 0.05 for al l frequencies. These coefficients were used in al l subsequent predictions. The experimental methods described above wi l l now be applied to the measurement of sound propagation in empty and fitted environments in order to validate the new theoretical models. Chapter 5 Experiments in the Anechoic Chamber with Fittings This chapter presents details of the experiments in the anechoic chamber to determine the fitting density in fitted regions, and to derive a relationship for the variation of fitting density wi th frequency. 5.1 Sound Propagation in the Presence of a Single Obstacle Free-field sound propagation in the presence of a single obstacle was measured in the anechoic chamber using a single mineral-water bottle. The following cases can be defined according to the relative positions of the obstacle, microphone and loudspeaker, as shown in Figure 5.1: a. Case 1 — No fitting; b. Case 2 — A fitting is located 5 cm from the source on the source/receiver axis, so that the direct sound is blocked; c. Case 3 — A fitting is located midway between the source and receiver, blocking the direct sound; d. Case 4 — A fitting is located 10 cm from microphone on the source/receiver axis, blocking the direct sound; e. Case 5 — A bottle is located midway between the source and receiver, 5 cm away from the straight line between microphone and speaker, not blocking the direct sound; f. Case 6 — A bottle located midway between the microphone and the speaker just blocks the direct sound; that is, one side of the bottle just touches the straight line 49 Chapter 5. Experiments in the Anechoic Chamber with Fittings 50 5 cm 10 cm ——H Figure 5.1: The positions of the obstacle. between microphone and speaker; g. Case 7 — A bottle is about 2 cm behind the microphone. Octave-band steady-state sound pressure levels were measured in al l of the above cases. Table 5.1 shows the changes in level relative to case 1. The second line gives the ratio of average obstacle dimension to sound wavelength. From Table 5.1, we can see that the high frequency energy is greatly reduced when the direct sound is blocked, but that the low frequency energy is hardly affected; that is, the fitting produces a strong acoustic shadow only at high frequency. In case 5, the low frequency energy is reduced even though the direct path is not blocked, and the high frequency energy is apparently not changed. In case 6, approximately half the energy is lost at all frequencies. If the sound reflected from the obstacle is in phase with the direct sound, the sound pressure level should increase by about 6 d B . From case 7, we can see most frequencies increase by about 3 d B , which supports the mean-square method often used in acoustics when summing direct and reflected sound. A t 500 Hz the sound level increase by 4.5 d B , which Chapter 5. Experiments in the Anechoic Chamber with Fittings 51 Table 5.1: Changes in SPL measured in the presence of a single obstacle (in dB) . octave-band 250 Hz 500 Hz 1 k H z 2 kHz 4 kHz 8 k H z 16 k H z Df/X 0.220 0.440 0.880 1.760 3.521 7.042 14.84 case 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 case 2 -0.3 -0.7 -2.2 -3.5 -6.9 -16.9 -19.2 case 3 -0.2 -0.3 -1.0 -4.1 -9.6 -15.4 -16.0 case 4 -2.9 -3.2 -5.9 -12.6 -17.4 -18.4 -19.6 case 5 -0.6 -2.1 0.0 0.3 0.4 0.4 -0.2 case 6 -0.7 -2.3 -4.9 -2.8 -2.8 -2.2 -2.0 case 7 2.7 4.5 3.3 -2.0 1.7 2.5 3.3 means the reflected sound is almost in phase with the direct sound; however it appears to be out of phase at 2 k H z since the sound pressure level decreases by 2 d B . In order to avoid sound pressure levels being too much affected by only one fitting in fitted rooms, in the following measurements it was necessary to avoid locating the microphone too near (less 10 cm) to any individual fitting. 5.2 Measurement and Analysis For modelling sound propagation in large fitted regions, the anechoic chamber was fitted wi th empty mineral-water bottles — 81, 162 or 243 bottles were used. A l l the bottles had caps on them to avoid them acting as resonator absorbers. The bottles were arranged randomly in the chamber in three dimensions. The coordi-nates of the bottles were determined from a set of computer-generated random numbers. The bottles were hung by strings from a wire mesh covering the ceiling. The one end of the string held the bottle cap and the other end was fixed to the acoustically transparent wire netting on the ceiling. A l l bottles had their tops pointing towards the ceiling — thus bottle orientation was not random. It was found that when the number of bottles is large, it was impossible to arrange the bottles completely randomly, since sometimes two Chapter 5. Experiments in the Anechoic Chamber with Fittings In- 0.005 I (a) o CL in CD 0.000 -0.005 -0.010 10 Time(msec) 20 •u> 0.005 0.000 i-(b) -0.005 -0.010 10 15 Time(msec) 20 In- 0.005 (C) 0.000 -0.005 -0.010 10 1 5 Time(msec) 2 " Figure 5.2: Impulse responses at r=lm, (a) Q=0m 1 ; (b) Q=0.2m 1; (c) Q=0.6m~ Chapter 5. Experiments in the Anechoic Chamber with Fittings 53 m -40 Filtered Cumulative Energy - dB(2000 Hz, 1.0 oct) -50 .' * 3 J -60 1 /'*' if Empty room Q = 0.2 Q = 0.4 Q = 0.6 -70 ] I i 0 10 20 Time(msec) Figure 5.3: The filtered cumulative energy curves. or three bottles are too close. In this case, some bottles were moved to nearby positions. A t each source/receiver distance 8 to 20 positions were measured wi th more positions for larger distances; average sound pressure levels in octave bands were calculated from these values. It was observed that at low frequencies the individual sound pressure levels were 5 to 10 dB higher or lower than the average value when the microphone was near one or two bottles. Such measurements were not taken into account since they are special cases and they only affect the variation not the mean values if the sample is large. Figure 5.2 shows the impulse responses measured at a distance of 1 m in the room when empty and when fitted wi th 81 or 243 bottles using speaker 3. The effect of fitting — distributing energy more uniformly in time — can be seen. Note though that the prominent secondary peaks are due to reflections in the loudspeaker cone. Figure 5.3 shows the 2000 Hz octave-band cumulative energy curves, the integration Chapter 5. Experiments in the Anechoic Chamber with Fittings 54 from 0 to time t of the impulse responses, for the empty chamber, and the fitted chamber wi th 81 bottles, 162 bottles and 243 bottles. From these curves, we can see that the steady-state energy increases with fitting density for this case. Since the attempts to resolve the distortion of measured impulse responses by the imperfect loudspeaker were unsuccessful (see Section 4.5), a method using stead-state sound octave-band pressure levels is used to determine fitting density in the next section. 5.3 Calculation of the Fitting Density from the Measured Data In this section, we develop a method for calculating the fitting density from the steady-state sound pressure level data measured in a fitted infinite region; note that this method is independent of any prediction model. 5.3.1 Sound Energy In sound propagation measurement in fitted regions we can differentiate between sound pressure levels measured in two cases — when the direct sound is not blocked, and when it is blocked by fittings. As discussed in section 2.2.2, from the theory of room acoustics, the total sound energy is the sum of two parts — the scattered part and the unscattered part. The unscattered sound energy density, E u n s c t , was expressed by Kuttruff [7] as: W Eunact = -r^-e-rQ = EQ=0e-rQ (5.1) in which, W is the sound power and c is the sound speed in air. The sound energy can be related to the sound pressure level Lp using, (5.2) Chapter 5. Experiments in the Anechoic Chamber with Fittings 55 Thus E q . 5.1 can be rewritten as 10L> -ct/10 I O L P . « = ° / 1 0 'unset — (5.3) c c Following are definitions of some relevant sound energies: • EQ-O- The sound energy measured in a free field with Q = 0 m _ 1 , which means that there is only direct sound from the source and no reflected sound; • Etotai- The total sound energy in a fitted room. This is the sum of the unscattered energy and the scattered energy; i n measurement i t is the energy average of a l l the measurement positions with the same source/receiver distance; • Eact: The scattered part of the total sound energy; • Eun3Ct: The unscattered part of total sound energy; • Ebiock'- The sound energy for the case that there are one or more fittings blocking the direct sound. Ebiock is not the same as Eact. This is because the fittings are not dimensionless. They block not only the direct sound, but also some scattered sound; thus Ebiock is less than E3Ct; • Enon^biock- The sound energy for the case when there is no fitting blocking the direct sound. Following are some relationships between these sound energy components. Etotai — Eun3Ct + E, 'set (5.4) E3Ct — Enon-block ~ EQ=Q (5.5) From Eqs. 5.4 and 5.5, the unscattered sound energy can be expressed as Chapter 5. Experiments in the Anechoic Chamber with Fittings 56 EUnsct — Etotai — E3ct = EQ=0 — Enon-biock + Etotai (5.6) Since EunBCt = EQ=oe~rCi, dividing both sides of E q . 5.6 by EQ=O, results in —rO i Enon—block EiQial . . e v = 1 (5.7) £>Q=0 Thus, the fitting density Q is only dependent on three measurable quantities which al l vary wi th frequency. Thus the variation of Q wi th frequency is given by Qff) = _I/ n[i _ E™-biockU)-Etot«iU)] (5 g) r EQ=o{}) Table 5.2 gives values for Q(f) calculated from the results of the measurements de-scribed in the last section using E q . 5.8. In Figure 5.4, the Q(f) are plotted. In the anechoic chamber, 81, 162 and 243 bottles were used as fittings. Table 5.3 gives the cor-responding fitting densities calculated using the Kuttruff formula, Qo, from the Lindqvist correction E q . 3.3, QL, and by the new correction E q . 3.9, Q'. These values are also shown i n Figure 5.4. We can see that the fitting densities are much higher than the Kutt ruff fitting density Q0 at high frequencies, much lower at low frequencies, and only equal to the Kuttruff values at some intermediate frequencies. The Lindqvist correction formula has about the same results as the Kuttruff formula, since QL only increase the fitting density values less than 8% in these cases. The values obtained by the new correction formula, E q . 3.9, agree well wi th the measured at high frequencies as expected. From Figure 5.4, we see that the fitting density varies significantly wi th frequency. The Q', QL and Qo a r e constants and only valid for very high frequency. In the next section, a model wi l l be found to describe the variation of fitting density wi th frequency. Chapter 5. Experiments in the Anechoic Chamber with Fittings 57 Table 5.2: Q(f) determined from the measured data using E q . 5.8. Qo ( m " 1 ) 250 500 l k 2k 4k 8k 16k 32k C A S E 0.2 — — 0.093 0.144 0.203 0.229 0.249 0.259 r=0.5 m speaker 3 — — 0.099 0.142 0.160 0.181 0.191 0.199 r=0.625 m — — 0.105 0.128 0.169 0.219 0.227 0.235 r = l m — — 0.104 0.138 0.191 0.226 0.244 0.253 r=1.25 m 0.2 — — 0.098 0.138 0.173 0.195 0.206 0.209 r=0.5 m speaker 3 — — 0.104 0.128 0.155 0.256 0.262 0.276 r=0.625 m absorbent — 0.088 0.131 0.185 0.210 0.207 0.218 r = l m fittings — — 0.093 0.124 0.176 0.208 0.222 0.230 r=1.25 m 0.2 0.024 0.056 0.114 0.148 0.169 0.210 0.224 — r=0.5 m speaker 1 0.052 0.081 0.109 0.140 0.172 0.197 0.222 — r=0.625 m 0.048 0.067 0.112 0.145 0.171 0.190 0.226 — r = l m 0.047 0.074 0.115 0.135 0.174 0.211 0.219 — r=1.25 m 0.2 0.034 0.074 0.121 0.115 0.196 0.201 0.256 — r=0.5 m speaker 2 0.042 0.058 0.132 0.162 0.198 0.223 0.260 — r = l m 0.027 0.044 0.106 0.133 0.138 0.198 0.228 — r=1.25 m 0.4 — — 0.241 0.311 0.401 0.544 0.587 0.621 r=0.5 m speaker 3 — i — 0.221 0.352 0.394 0.424 0.576 0.622 r=0.625 m — — 0.228 0.324 0.420 0.548 0.608 0.661 r = l m — — 0.231 0.301 0.401 0.531 0.604 0.623 r=1.25 m 0.4 0.068 0.150 0.215 0.342 0.375 0.411 0.509 — r=0.5 m speaker 1 0.095 0.134 0.254 0.309 0.383 0.427 0.468 — r=0.625 m 0.088 0.133 0.271 0.318 0.434 0.563 0.616 — r = l m 0.076 0.139 0.231 0.330 0.452 0.478 0.502 — r=1.25 m 0.4 0.050 0.148 0.264 0.336 0.446 0.537 0.548 — r=0.5 m speaker 2 0.091 0.151 0.258 0.327 0.462 0.522 0.610 — r = l m 0.078 0.148 0.243 0.373 0.477 0.582 0.645 — r=1.25 m 0.6 0.136 0.224 0.342 0.577 0.623 0.872 0.910 — r=0.5 m speaker 2 0.161 0.259 0.359 0.492 0.668 0.831 1.004 — r = l m 0.142 0.249 0.414 0.530 0.757 0.958 1.059 — r=1.25 m 0.6 — — 0.357 0.561 0.760 0.845 0.963 1.100 r=0.5 m speaker 3 — — 0.408 0.543 0.735 0.864 0.993 0.938 r=0.625 m — — 0.391 0.579 0.677 0.825 0.961 1.088 r = l m — — 0.330 0.536 0.628 0.817 0.975 1.091 r=1.25 m Chapter 5. Experiments in the Anechoic Chamber with Fittings 58 Figure 5.4: Variat ion of fitting density wi th frequency, (a) Q0 — 0.2 m _ 1 ; (b) Q0 — 0.4 m _ 1 ; (c) Qo — 0.6 m - 1 , calculated from the experimental data using E q . 5.8. Also shown are the fitting densities Q0, QL and Q'. Chapter 5. Experiments in the Anechoic Chamber with Fittings 59 Table 5.3: The fitting density by three methods (in m *). Bottles by Kuttruff, Q0 by Lindqvist , QL by E q . 3.9, Q' 81 0.2 0.205 0.232 162 0.4 0.42 0.541 243 0.6 0.643 0.944 5.3.2 A Model for Q(f) From the last section, £ ? ( / ) / - > o o approximately equals the corrected Q'\ at low fre-quencies the fitting density is much smaller than Q'. Let us derive a model for Q(f)-According to Figure 5.4, a non-linear model must be used to express the relationship between fitting density and frequency. After considering several models, the following model was chosen: w = r m h (5-9) in which, / is frequency; /o is a fundamental frequency, which depends on the fitting mean dimension, Df wi th /o = cjDf \ c is sound speed in air. Df can be calculated from the dimensions L\, L2 and L3 of the fittings using following formula: — - - ( — + — + — ) (5 10) Df 3 L\ L2 L3 In our experiments wi th water bottles, Df = 0.3 m, and c = 344 m/s at temperature 20 C° , so /o = 1136 Hz . It remains to find estimates for A and B in E q . 5.9. First use a statistical method to find the estimates A and B. Divid ing both sides of E q . 5.9 by £?«>(/) it can be rewritten as follows: A + A f Chapter 5. Experiments in the Anechoic Chamber with Fittings 60 Let and 1 A B a = A b = A _f_ Y Q°° fo ' Q(f) The non-linear E q . 5.11 becomes a linear equation in the form: Y = a + bX (5.12) The estimated values a and b for a and b by least squares are as follows: b = LXyl Lxx a-Y-iX in which Lxx = £ ( X , - - Xf Lyy = ^{Yi - Yf Lxy = £ ( X , - X)(Yt - Y) The sample correlation coefficient r is: _ Lxy \JXX Lyy The various quantities are calculated using the measured data, and they are: a — 0.9353; b = 1.3033; X = 0.8794; Y = 2.0814; Lxx = 329.2; Lyy = 617.3; Lxy = 429.1; r = 0.9518; N = 215. From these values, A = 1.0692 and B = 1.3935, so the best-fit model derived from the measured data is: Chapter 5. Experiments in the Anechoic Chamber with Fittings 61 1.0692g o o ( / ) W ) = 1 + 1.3935/o// ( 5 " 1 3 ) 5.3.3 Correlation Test of Q(f) This section is to test the linear association of the variables X and Y in E q . 5.12 — the dependency between the two variables, using a t test [25]. Test H0 : 6 = 0 versus Ha : 6 > 0, wi th a — 0.05. This means that if Ho is rejected, we conclude that b > 0 and that the fitting density tends to increase wi th an increase of frequency. Since we desire to test whether there is evidence that b > 0 with significance level a = 0.05, the appropriate test statistic is XX Here 5 = ^ / L y v / ( i V — 2). Thus the value of the appropriate test statistic for testing H0 : 6 = 0 versus H„ : 6 > 0 is 0.9353 - 0 = 9.9683 1.7024^/1/329.2 Since the statistic above is based on N — 2 = 213 degrees of freedom, and the appropriate rejection region is t > 20.05 = 1.645, we reject H0 in favour of Ha at a = 0.05 level of significance; thus the evidence strongly suggests that the fitting density is dependent on frequency. From a practical point of view, the fitting density is expected to increase wi th an increase of frequency. 5.3.4 Final Model for Q(f) According to E q . 5.9 the best-fit variation of Q wi th frequency as determined from the measurement data can be approximated by Chapter 5. Experiments in the Anechoic Chamber with Fittings 62 1 ° 4 frequency(Hz) Figure 5.5: Comparison of two models for Q(f). 1.07goo(/) Q U ) - 1 + 1.4/o// ( 5 - 1 4 ) In fact, we would expect. that when / —> oo, then = Qoo, so A = 1. Since the magnitude factor is reduced from 1.07 to 1, let B — 1.2 to avoid calculated values reducing at low and middle frequencies. The two models are compared in Figure 5.5, from which we can see that the only differences occur at high frequencies. At low and middle frequencies the two models are very close. Since the simpler model is most practical we choose the following final model: In Figure 5.6, the measured data for Q(f)/Qco is plotted with the curves of Eqs. 5.14 and 5.15. From the figure, we can see that using Eq. 5.15 to express the relationship of fitting density with frequency is reasonable. Chapter 5. Experiments in the Anechoic Chamber with Fittings 63 c f a : j $ 1.0 r * • Y 0.5 ± + 0.0 , , I 1 10 3 1 0 f requency(Hz) Figure 5.6: Comparison between Q(f), as (+ +) measured, and as predicted (—) by Eq.5.15 and (---) by Eq.5.14. Note that since fo depends on the fitting dimension, this model in principle allows that parameter to be taken into consideration. Unfortunately, it was not feasible to perform experiments wi th other fitting dimensions in order to verify the relationship. 5.4 Comparison of Experiment with Predictions by Existing Models This section discusses comparisons between the measurements in the anechoic cham-ber used as a 1:8 scale model and predictions by existing predicting models discussed in Chapter 2 — R A Y C U B , the Lindqvist model and the new model. The full scale parameter values are: R o o m size — 37.6 m in the x direction, 32.8 m in the y direction and 19.2 m high; Sound source position is at (18.0 m , 16.0 m, 9.6 m); Receiver positions are 9.6 m high, 16.0 m in the y direction, and 1, 2, 4, 5, 8 and 10 Chapter 5. Experiments in the Anechoic Chamber with Fittings 64 Table 5.4: The parameters used in prediction (full scale values). frequency (Hz) 500 1 k 2 k 4 k air absorption exponents ( N p / m ) 0.00075 0.0025 0.0095 0.028 fitting density Q0 ( m _ 1 ) , 23 bottles fitting density Q0 ( m _ 1 ) , 46 bottles fitting density Qo ( m _ 1 ) , 69 bottles 0.025 0.05 0.075 0.025 0.05 0.075 0.025 0.05 0.075 0.025 0.05 0.075 fitting density Q(f) ( m - 1 ) , 69 bottles 0.088 0.101 0.109 0.113 absorption coeff. of room surfaces 1.0 1.0 1.0 1.0 absorption coeff. of fittings 0.0 0.0 0.0 0.0 as above but bottles with absorbent 0.1 0.1 0.1 0.1 m away from the sound source in the x direction. Measurement conditions were typically 23 C° and 50% relative humidity, and the air absorption exponents ( N p / m ) and the other parameters used in prediction are listed i n Table 5.4 in full-scale values. The values of the other parameters used in the prediction were: 1.0 for the absorption coefficients of the room surfaces, and 0.0 for those of the fittings, but 0.1 for the case in which the bottles were absorbent. The measured SP and predicted SP by models were given in full scale values, in which the values at 4000 Hz were lower than those at other frequencies, since the air absorption losses were not compensated. In R A Y C U B , the receiver size was 0.5 m , the ray number was 500,000, and the max imum trajectory was 30. Figure 5.7 to 5.10 show the measured SP curves compared with the predicted by R A Y C U B using the Kuttruff formula to calculate the fitting densities. We can see from these figures that the predictions agree well wi th the measured for the cases in which the fitting densities were low — fitted wi th 81 or 162 bottles. In these cases, the differences between the measured and the predicted levels are less than 0.5 dB for all frequencies and all distances. For the case fitted wi th 243 bottles, the predicted curves are only agreement well at frequencies 500 and 1000 Hz , at higher frequencies there were significant differences Chapter 5. Experiments in the Anechoic Chamber with Fittings 65 between the measured and predicted. The predictions underestimated about 0.5 dB at 4000 Hz for most distances, the reason was that constant fitting density was used. When fitted wi th 81 or 162 bottles, the fitting densities by the Kuttruff formula were very similar to those by the correction formula — E q . 5.15, but when the room was densely fitted wi th 243 bottles, the differences were large at high frequencies. The following part of this section uses the Q(f) calculated by E q . 5.15 to predict the SP values using the ray-tracing, Lindqvist and new models, and compares them with the measured values; only the result for 243 bottles is presented. Plo t t ing the results in Figure 5.11 to 5.14, the agreement of the predicting model wi th the measured is seen to be much better than that using the Kuttruff formula. We can see from these figures that the ray-tracing model gives the best agreement with measurement — differences wi th in 0.3 dB — that the Lindqvist model seems to underestimate the sound pressure level by up to 1 dB at distances far from the source, and that the new model is similar to the ray-tracing model but underestimates by up 0.3 dB at the distances near the source. In summary, measurements of sound propagation in the empty and fitted anechoic chamber have led to, and validated, an empirical model — E q . 5.15 — for calculating fitting density — and its variation with frequency — in the case of large fittings. In the next chapter we extend the validation to a scale-model workroom. Chapter 5. Experiments in the Anechoic Chamber with Fittings 66 -35 I L _ , , , , , , — . . I 1 0 1 0 Distance(m) Figure 5.7: Octave-band sound propagation in the anechoic chamber used as a 1:8 scale model room wi th randomly distributed 81 bottles (Q0 = 0.025 m _ 1 ) , as measured and predicted by the ray-tracing model. P(dB) -10 • • 500Hz 1 kHz measured measured W T 2 kHz measured -15 4 kHz — 500Hz - - 1 kHz measured RAYCUB RAYCUB -20 "^V. 2 kHz 4 kHz RAYCUB RAYCUB -25 -30 s -35 i 10° 1 ° 1 Distance(m) Figure 5.8: A s for F i g . 5.7 but the bottles covered with absorbent (Q0 = 0.025 m _ 1 ) . er 5. Experiments in the Anechoic Chamber with Fittings m -10 • 500Hz measured QL • 1 kHz measured CO T 2 kHz measured -15 - • 4 kHz • 500Hz - 1 kHz measured RAYCUB RAYCUB -20 - 2 kHz - 4 kHz RAYCUB RAYCUB -25 -30 -35 i 10° 1 0 ' Distance(m) Figure 5.9: As for F i g . 5.7 but with 162 bottles (Q = 0.05 m " 1 ) . Figure 5.10: As for F i g . 5.7 but with 243 bottles (Q = 0.075 m " 1 ) . Chapter 5. Experiments in the Anechoic Chamber with Fittings m -10 2. CL • measured CO -15 RAYCUB Lindqvist new model -20 X v -25 \ ^ X -30 N -35 • • • I 10° 1° 1 Distance(m) Figure 5.11: Comparison of 500 Hz octave-band SP curves for 243 bottles, Q(f) — 0. m _ 1 , as measured and as predicted by R A Y C U B , Lindqvist and new models. Figure 5.12: As for F i g . 5.11 but for 1000 Hz , Q(f) = 0.101 m " 1 . er 5. Experiments in the Anechoic Chamber with Fittings Figure 5.13: As for F i g . 5.11 but for 2000 Hz , Q(f) = 0.109 m - 1 . Figure 5.14: As for F i g . 5.11 but for 4000 Hz , Q(f) = 0.113 m - 1 . Chapter 6 Experiments in a Scale Model Workroom This chapter reports the results of sound propagation measurements made in a 1:8 scale model workroom (described in section 4.7.2) when empty and fitted. The main objective was to validate the correction formula for fitting density — E q . 3.9 — and the model of fitting density variation wi th frequency — E q . 5.15. Three configurations of fittings were tested: randomly distributed throughout the whole room (see section 6.1); randomly distributed in a layer on the floor (see section 6.2); uniformly distributed i n a layer on the floor (see section 6.3). 6.1 Randomly Distributed Fittings The 1:8 scale model workroom is 3.75 m long, 1.875 m wide, and 1.875 m high, as shown in Figure 4.17. In full scale dimensions, the room is 30 m F S long, 15 m F S wide, and 15 m F S high. Measurement environmental conditions were: temperature 24 C° , relative humidi ty 60%. In this chapter, only loudspeaker 3 was used. In this experiment, the loudspeaker was located centrally in the x and y directions and 7.2 m F S above the floor (all the parameters in full scale values). Receivers were at the same height, at the same position in the y direction as the loudspeaker, and at 1, 2, 4, 5, 8 and 10 m F S from the loudspeaker in the x direction. First the empty room was measured to find the surface absorption coefficients using best-fit method; the resulting coefficients along with the other parameters in full scale values used i n the predictions, are listed in Table 6.1. 70 Chapter 6. Experiments in a Scale Model Workroom 71 Table 6.1: The parameters in full scale values used in prediction for the scale-model workroom wi th randomly distributed fittings. frequency 250 Hz 500 Hz 1 kHz 2 k H z 4 k H z air absorption exponents ( N p / m ) 0.0003 0.0007 0.0021 0.0075 0.026 absorption coeff. of ceiling 0.05 0.05 0.1 0.1 0.1 absorption coeff. of other surfaces 0.05 0.05 0.05 0.05 0.05 absorption coeff. of fittings 0.0 0.0 0.0 0.0 0.0 fitting density Qo ( m _ 1 ) , 23 bottles fitting density Qo ( m _ 1 ) j 46 bottles fitting density Qo ( m _ 1 ) , 69 bottles 0.025 0.05 0.075 0.025 0.05 0.075 0.025 0.05 0.075 0.025 0.05 0.075 0.025 0.05 0.075 fitting density Q(f) ( m _ 1 ) , 69 bottles 0.07 0.088 0.101 0.109 0.113 Figure 6.1 and 6.2 show the measured SP curves in the empty room and the predic-tions by R A Y C U B and by the new image-source model in full scale values. For R A Y C U B , the receiver size was 1 m , the number of rays was 200,000, and trajectory number was 50. For the image-source model, the image factor was 50 since the surface absorption co-efficients are small . For the empty room, the sound energy distribution results only from the non-diffuse part in which the Kuttruff equation is used for both the Lindqvist and the new models. From these curves, we can see the difference between the ray-tracing and image-source models for the empty room; since the surface absorptions were corrected using R A Y C U B , the predicted values by R A Y C U B are quite close to those measured. It can be seen from these figures that there is significant difference between the two models in predicting sound pressure levels in the empty room; L'ps predicted by the image-source model are always lower than those predicted by ray-tracing, especially at frequencies wi th low surface absorption coefficients. Next the fittings were distributed randomly in the model in three dimensions. The fittings — the bottles — are the same as described in Chapter 4 with 23, 46 and 69 bottles used — corresponding to fitting densities Qo = 0.025, 0.05 and 0.075 m _ 1 in full Chapter 6. Experiments in a Scale Model Workroom 72 101 Distance(m) Figure 6.1: Octave-band sound propagation in the empty 1:8 scale model workroom, as measured and predicted by the ray-tracing model. m Figure 6.2: Octave-band sound propagation in the empty 1:8 scale model workroom, as measured and predicted the new image-source model. Chapter 6. Experiments in a Scale Model Workroom 73 m T3 ST -10 co -15 -20 -25 • 250Hz measured ^ 500Hz measured • 1 kHz measured • 2 kHz measured 4 kHz measured 250Hz RAYCUB 500Hz RAYCUB 1 kHz RAYCUB 2 kHz RAYCUB 4 kHz RAYCUB To5 "~ Distance(m) 101 Figure 6.3: Octave-band sound propagation in the 1:8 scale model workroom wi th 23 randomly distributed bottles ((Jo = 0.025 m _ 1 ) , as measured and predicted by the ray-tracing model. scale values, as calculated by the Kuttruff formula. The loudspeaker and receiver had the same positions as in empty room. For every distance, 4 measurements were averaged. Since the surface absorption is very low, the differences in sound pressure levels in the blocked and non-blocked cases were small — typically from 0.5 to 1.5 d B . In Figures 6.3 to 6.5, the SP curves predicted using the Kuttruff fitting density Q0 are plotted and compared to those measured. From these figures, we can see that the prediction models agree well wi th the measured levels when the fitting densities are small . A t higher fitting density, the predictions agree well wi th the measured values at low frequencies, but not at high frequencies. For the case of 69 bottles the fitting densities Q(f) calculated by E q . 5.15 are listed in Table 6.1. Figures 6.6 and 6.7 give the comparison between predictions by the ray-tracing model and the new model using Q(f)- Since the surface absorption coefficients are low, er 6. Experiments in a Scale Model Workroom CD T3 ST -10 -15 -20 -25 T • X... 250Hz measured 500Hz measured 1 kHz measured 2 kHz measured 4 kHz measured 250Hz RAYCUB 500Hz RAYCUB 1 kHz RAYCUB 2 kHz RAYCUB 4 kHz RAYCUB 10" Distance(m) 101 Figure 6.4: As for F i g . 6.3 but with 46 bottles (Q0 = 0.05 m " 1 ) . m "a Q T -10 C O -15 -20 h -25 h • 250Hz measured A 500Hz measured • 1 kHz measured • 2 kHz measured A 4 kHz measured 250Hz RAYCUB 500Hz RAYCUB .. i k H z RAYCUB 2 kHz RAYCUB 4 kHz RAYCUB lo 5 Distance(m) 101 Figure 6.5: As for F i g . 6.3 but with 69 bottles (Q0 = 0.075 m - 1 ) . Chapter 6. Experiments in a Scale Model Workroom 75 m •o ST -10 co -15 -20 -25 • 250Mz measured A 500Hz measured • 1 kHz measured • 2 kHz measured •4 4 kHz measured 250Hz RAYCUB 500Hz RAYCUB 1 kHz RAYCUB 2 kHz RAYCUB 4 kHz RAYCUB _ J 10° Distance(m) 101 Figure 6.6: S P ( d B ) with 69 randomly distributed bottles, as measured and predicted by the ray-tracing model using Q(f)-Figure 6.7: As for F i g . 6.6 but predicted by the new model. Chapter 6. Experiments in a Scale Model Workroom 76 CD •o 1.0 0.5 r 2000 Hz 0.0 -0.5 + 0 _+____ 0 + 0 + 0 0 0 -1.0 Dis tance(m) 10" 10' CQ 1.0 0.5 4000 Hz 0.0 -0.5 + 0 H 0 __+__ 0 + 0 ^ 0 -1.0 1— Distance(m) 10" 10' Figure 6.8: Comparing the differences predicted by ray-tracing model: using Qo (0); using Q(f) (+). the new model seems to have underestimated the sound pressure levels up to 2 dB at low frequencies and at the receiver positions far from the source. The Lindqvist model has the same problem. Since the sound absorptions of the surfaces and fittings are low, the effect of the fittings on sound pressure levels is very small; using the differences between the measured and predicted SP values is a better way to see the difference between using Q0 and Q(f)- The differences between the predicted and measured SPs at 2000 and 4000 Hz are plotted in Figure 6.8. A t low frequencies — 250, 500 and 1000 Hz — Q(f) is very close to Q0, so the differences are about same; at higher frequencies, the differences using Q(f) are nearer zero than those using Q0- However in general the differences are very small and the results not conclusive. Chapter 6. Experiments in a Scale Model Workroom 77 O O O 0.275m receivers o o o o o o o o o o o / o o o o o ' o o o A' Q O Q O O Q Q Q d 3.75m E LO 00 loud speaker with the cone Figure 6.9: Uniform distribution of 32 bottles on the floor of the scale-model workroom. 6.2 Fittings on the Floor In the next experiment, loudspeaker was located near the floor of the enclosure. The source was 1 m F S above the floor, centred in the y direction and 2.5 m F S away from one wall in the x direction, as shown in Figure 6.9. The receiver positions were centred in the y direction, 1.5 m F S above the floor, and 1, 2, 5, 10, 15, 20 and 25 m F S from the speaker i n the x direction. The speaker faced i n the y direction, not facing the receiver positions, as shown in Figure 6.9. The surface absorption coefficients are shown in Table 6.2 along wi th the other parameters used for prediction, in full scale values. Figure 6.10 plots the measured SP curves and the curves predicted by R A Y C U B for the empty scale-model workroom in full scale values, and the agreements are good. Next the fittings were located in a layer on the floor — with an empty region above — using 16 or 32 bottles. The bottles were upside-down on the floor, wi th height 4 m F S , so that the area touching the floor was so small that it could be ignored when Chapter 6. Experiments in a Scale Model Workroom 78 Table 6.2: The parameters used in prediction for fitting on the floor frequency 2 kHz 4 kHz 8 k H z 16 k H z 32 k H z air absorption exponents N p / m 0.0003 0.0007 0.0021 0.0075 0.026 absorption coeff. of the ceiling 0.05 0.05 0.1 0.1 0.1 other surface absorption coff. 0.05 0.05 0.05 0.05 0.05 fitting density Q0 ( m _ 1 ) , 16 bottles fitting density Qo ( m _ 1 ) , 32 bottles 0.0625 0.125 0.0625 0.125 0.0625 0.125 0.0625 0.125 0.0625 0.125 fitting density Q(f) ( m " 1 ) , 32 bottles 0.157 0.197 0.225 0.244 0.254 calculating fitting density. The room was modelled with a 4.16 m F S high lower fitted zone wi th an empty zone above. The fitting densities for the fitted zone were 0.0625 or 0.125 m _ 1 calculated by the Kuttruff formula in full scale values. The bottle distribution was random for the case of 16 bottles, and either random or uniform for the case of 32 bottles. The uniform distribution of 32 bottles in the scale model room is shown i n Figure 6.9. It was found by measurement that there is not much difference between SP's measured wi th the random and uniform distributions of the bottles. The difference was normally less than 0.5 d B , and never exceeded 1.0 d B . In Figures 6.11 and 6.12 the measured SP curves are compared wi th the SP curves predicted by R A Y C U B using the Kuttruff fitting density, which is 0.0625 m _ 1 for 16 bottles and 0.125 m _ 1 for 32 bottles, in the fitted zone (in full scale values), we can see that the agreements are well for cases in which the fitting density is lower or the frequencies are lower, but not at high frequencies with higher fitting density. Next, the measurement was compared to predictions using the fitting density Q(f) to validate Eqs. 3.9 and 5.15. The fitting densities Q(f) of the 32 bottles in the fitted zone calculated by E q . 5.15 are listed in Table 6.2. Figures 6.13 to 6.17 plot the measured SP and the SP predicted by R A Y C U B using Q(f). In these figures, the measured values Chapter 6. Experiments in a Scale Model Workroom 79 Distance(m) Figure 6.10: S P ( d B ) measured in the empty room compared with that predicted by the ray-tracing model. were for the case of 32 bottles on the floor with an empty zone above, the curves desig-nated R A Y C U B ( a ) represent the measured configuration, R A Y C U B ( b ) was for the same configuration as R A Y C U B ( a ) but using Q0, R A Y C U B ( c ) was for the same configuration except that the 32 bottles were randomly distributed in 3D; for R A Y C U B ( d ) the fittings were on the floor but the fitting absorption coefficients are the non-zero values given below. The measured values agree well wi th prediction when the distance is less than 20 m F S , but prediction always gives higher values than measured when the distance is larger than 20 m F S (same problem using the Kuttruff Q0)- Analysis suggested that this problem maybe be related to the absorption of the fittings. The absorption coefficient was originally chosen to be zero at a l l frequencies, but this may not be true — especially at high frequencies. When the distances are large, this problem must be considered, so Chapter 6. Experiments in a Scale Model Workroom 80 CD s -101 OL CO -15 -20 -25 -30 1 • 250 Hz measured A 500 Hz measured • 1 kHz measured • 2 kHz measured •4 4 kHz measured 250 Hz RAYCUB 500 Hz RAYCUB 1 kHz RAYCUB 2 kHz RAYCUB 4 kHz RAYCUB I i 10° 101 Distance(m) Figure 6.11: Octave-band sound propagation in the 1:8 scale model workroom wi th 16 randomly distributed bottles on the floor (Qo = 0.0625 m _ 1 ) , as measured and predicted by the ray-tracing model. CD 3. -10 o_ co -15 -20 -25 -30 • 250 Hz measured A 500 Hz measured • 1 kHz measured • 2 kHz measured 4 kHz measured 250 Hz RAYCUB 500 Hz RAYCUB 1 kHz RAYCUB 2 kHz RAYCUB 4 kHz RAYCUB I i 10° 101 Distance(m) Figure 6.12: As for F i g . 6.11 but with 32 bottles (Q = 0.125 m " 1 ) . Chapter 6. Experiments in a Scale Model Workroom 81 coefficients of 0.01 for 250 Hz , 0.02 for 500 Hz and 1 kHz , 0.03 for 2 kHz and 0.04 for 4 k H z in full scale values were used, since the problem was more serious at high frequencies. W i t h these parameter values, the predictions are a l i t t le better, as shown i n Figures 6.13 to 6.17. To compare the difference between fittings on the floor and randomly distributed in 3D, Figures 6.13 to 6.17 also plot the prediction by R A Y C U B for the fittings distributed in 3D. For 32 bottles randomly distributed in 3D, Q0 = 0.0348 m _ 1 in full scale values — from Eq.5.15 Q = 0.0428 m " 1 and <?(250) = 0.0254, Q(500) = 0.0319, Q{lk) = 0.0365, Q(2k) = 0.0394 and Q(4k) = 0.041 m " 1 . From these figures, we can see that the fitting distribution affects the sound pressure levels, but that the effect is small at low frequencies in this case; predictions R A Y C U B ( a ) and R A Y C U B ( c ) are very similar, and both agree well wi th the measured values. Sur-prizingly, R A Y C U B ( c ) is more similar to measurement for receiver positions far from the source than is R A Y C U B ( a ) . A t high frequencies the effect is large — up to 2 dB at 4 k H z and r = 25 m F S — since at this frequency the fitting density in the fitted zone is much greater than at low frequencies. In this case, the predicted values are sensitive to the fitting absorption coefficient; R A Y C U B ( d ) results are closer to the measured values at positions far from the source but underestimate the SP near the source, so the coeffi-cients 0.01 to 0.04 used in R A Y C U B ( d ) are reasonable. This also shows that the etf = 0 used in the last chapter is correct, since in that case there is no reflection of sound from the walls, so the sound pressure level is not sensitive to the coefficient (less than 0.2 d B difference between the prediction models using df = 0 and ctj — 0.05 ). As in Figure 6.8, Figure 6.18 plots the differences between the SP predicted by R A Y -C U B and the measured SP at frequencies 2000 and 4000 Hz , for the differences were very small at low frequencies since the fitting densities were similar. It can be seen that the differences obtained using Q(f) are much smaller than those using Qo especially at 4000 Hz . For low frequencies — 250 and 500 Hz — Q(f) and Q0 predict similar values, so the Chapter 6. Experiments in a Scale Model Workroom 82 Figure 6.13: Comparison of 250 Hz octave-band SP curves for 32 bottles on the floor, as measured and as predicted by ray-tracing for four configurations (see text). differences are about the same. 6.3 Fittings on the Floor and Absorbent Ceiling In order to observe predicted SP values which are significantly different using Q0 and Q(f), a further experiment was done in the scale-model workroom. The ceiling was lowered to 5 m F S high (0.625 m in model scale value), and absorbent material (glass fibre 5 m m in thickness) was applied to it to increase the ceiling absorption coefficient. The positions of the loudspeaker and receivers were the same as in section 6.2. The fitting distribution was similar to that in section 6.2, but 31 bottles were used and the bottle behind the loudspeaker was removed, as shown in Figure 6.9. The fitting density calculated by the Kuttruff formula was 0.1 m _ 1 , and the correction value calculated by E q . 5.15 was 0.183 m _ 1 in full scale values. Chapter 6. Experiments in a Scale Model Workroom 83 Figure 6.14: As for F ig . 6.13 but for 500 Hz . CO -20 1—4 ' ' • ' 1 ' ' ' ' ' 1 0 10 Distance(m) Figure 6.15: As for F i g . 6.13 but for 1000 Hz . er 6. Experiments in a Scaie Model Workroom Figure 6.16: As for F ig . 6.13 but for 2000 Hz . Figure 6.17: As for F i g . 6.13 but for 4000 Hz . Chapter 6. Experiments in a Scale Model Workroom 85 Figure 6.18: Comparing the difference of SP with 32 bottles on the floor predicted by ray-tracing using Q0 (o) and using Q(f) (+). First the empty room was measured and predicted to find the ceiling absorption coefficients using the best-fit method; the results with the other full scale parameter values are shown in Table 6.3. Measurement conditions were typically 28.5 C° and 50% relative humidity. Figure 6.19 shows the SP curves measured in the empty room and the predictions by R A Y C U B in full scale values; the measured SP curves agreed well wi th those predicted by ray-tracing model. For R A Y C U B , the receiver size was 1 m , the number of rays was 200,000, and trajectory number was 30. Figures 6.20 to 6.24 compare the measured results with those predicted by R A Y C U B using Qo and Q(f), as well as by the Lindqvist and new models using Q(f). Because of the absorbent material on the ceiling, and the lower ceiling, the differences of SP values predicted between using Q0 and Q(f) were large — up to 3.5 dB at 4000 Hz and r = 25 m F S — especially at high frequencies and with the receiver far from the source. Chapter 6. Experiments in a Scale Model Workroom 86 Table 6.3: The parameters in full scale values used in prediction for the scale-model workroom wi th uniformly distributed fittings on the floor. frequency (Hz) 250 500 1 k 2 k 4 k air absorption exponents ( N p / m ) 0.00035 0.00073 0.0021 0.0075 0.025 absorption coeff. of ceiling 0.4 0.45 0.4 0.4 0.3 absorption coeff. of other surfaces 0.05 ' 0.05 0.05 0.05 0.05 absorption coeff. of fittings 0.01 0.02 0.02 0.03 0.04 fitting density Qo ( m - 1 ) 0.1 0.1 0.1 0.1 0.1 fitting density Q(f) ( m - 1 ) 0.109 0.137 0.157 0.169 0.176 From these figures, we can see that at 250 Hz (full value) both predicted SP curves by R A Y C U B are similar to the measured one which is not smooth — probably due to modal effects. A t the other frequencies — 500, 1000, 2000 and 4000 Hz — there are significant differences between the SP curves predicted using Q0 and Q(f), and the SP curves predicted using Q(f) are in excellent agreement with the measured SP curves at al l distances. Differences of more than 1 dB occur only at 250 Hz . The new correction formula — E q . 3.9 — and the model to calculate Q(f) — E q . 5.15 — are strongly supported by these experiment results. The SP curves predicted by the new model are similar to those predicted by R A Y C U B , being only up to 1 dB lower at receiver positions far from the source. They are closer to the curves measured and predicted by R A Y C U B than are those by the Lindqvist model. In summary, in the measurements of this chapter the SP values predicted using the new correction formula give more accurate results than do those using the Kutt ruff formula. The new prediction model developed in Appendix works well in most cases, but underestimates sound pressure levels up to 2 dB when the receiver is far from the source and the average surface absorption coefficient is low. The next chapter discusses experimental validation in a real workroom. Chapter 6. Experiments in a Scale Model Workroom 87 • 250 Hz measured A 500 Hz measured Distance(m) Figure 6.19: S P ( d B ) — empty 5 m high scale-model workroom with absorbent ceiling, compared wi th R A Y C U B . Figure 6.20: Comparison of 250 Hz octave-band SP curves for 31 uniform distributed bottles on the floor, as measured and as predicted by ray-tracing for two configurations (see text) and by the Lindqvist and the new models. er 6. Experiments in a Scale Model Workroom Figure 6.21: As for F i g . 6.20 but for 500 Hz . Figure 6.22: As for F i g . 6.20 but for 1000 Hz . Chapter 6. Experiments in a Scale Model Workroom 89 (dB) -10 D_ CO vs . X . -15 -20 -25 • measured RAYCUB -- QO N X - \ RAYCUB--Q(f) \ \ \ Lindqvist -30 l new model I . 10° 10 Distance(m) Figure 6.23: As for F i g . 6.20 but for 2000 Hz . Figure 6.24: As for F ig . 6.20 but for 4000 Hz . Chapter 7 Validation of Q(f) by a Measurement in a Machine Shop In this chapter, measurements in a full-scale room are used to validate the correction formula for fitting density developed in Chapters 3 and 5. The correction formula for calculating fitting absorption coefficient is also validated using the measurements. The measured data were from Hodgson's work [4], which studied ray-tracing predic-tion of SP in a fitted machine shop. The dimensions of the room were 46 x 15 x 7.2 m. The floor of the room was of concrete, its walls were made of unpainted blockwork, and its ceiling was of typical steel-deck construction (consisting of corrugated metal inside, insulation, a vapor barrier and gravel outside). The roof was supported by metal truss-work. The average octave-band absorption coefficients of the surfaces were estimated from previous measurements of the reverberation time in nominally empty buildings of the same construction, and have been found to vary little from one building to another [26]. The surface absorption coefficients were 0.12 at 250 Hz, 0.1 at 500 Hz, 0.08 at 1000 Hz, and 0.06 at 2000 and 4000 Hz. The fact that the coefficient tends to decrease with increasing frequency is considered normal in building with suspended-panel roofs; the relatively high absorption at low frequencies is due to their vibration and transmission characteristics [4]. Air absorption exponents in Np/m were 0.0003 at 250 Hz, 0.0005 at 500 Hz, 0.001 at 1000 Hz, 0.003 at 2000 Hz and 0.006 at 4000 Hz. The machine shop contained many fittings (total 63) distributed fairly uniformly over the floor area. The fittings included machine tools and other equipment, work benches, cabinets, and stock piles with an average fitting height of 1.5 m. The total fitting surface 90 Chapter 7. Validation of Q(f) by a Measurement in a Machine Shop 91 Table 7.1: Values of parameters used for ray-tracing prediction of SP in the machine shop. frequency (Hz) 250 500 1000 2000 4000 air absorption exponent ( N p / m ) 0.0003 0.0005 0.001 0.003 0.006 empty room surface absorption coeff. 0.12 0.10 0.08 0.06 0.06 fitted room surface absorption coeff. 0.21 0.18 0.15 0.14 0.14 fitting density in upper zone ( m _ 1 ) 0.03 0.03 0.03 0.03 0.03 fitting density in lower zone ( m _ 1 ) 0.23 0.23 0.23 0.23 0.23 fitting absorption coeff. in upper zone 0.05 0.05 0.05 0.05 0.05 fitting absorption coeff. in lower zone 0.1 0.1 0.1 0.1 0.1 area as determined from the dimensions of rectangular boxes that would just enclose the fittings was 675.5 m 2 , so Q0 = 0.16 m _ 1 in the fitted zone by the Kutt ruff formula. The fitting density of the upper region, which was essentially empty but contained a mobile crane, lighting fixtures, and the roof trusswork, was estimated to be 0.03 m _ 1 with absorption coefficient 0.05. Some parameters used in prediction are listed in Table 7.1. Measurements of sound propagation were made in the machine shop, in octave-band from 250 to 4000 Hz . The sound source was located at 5 m from one end wall at m i d width, and 1.5 m above the floor. The receivers were the same height as the source at distances of 1, 2, 5, 10, 15, 20, 25, and 30 m from the source along the room center line. B y comparing SP's predicted by ray-tracing (in which the room surface absorption coefficient i n empty room was used) wi th measured, and using the best-fit method, it was found that the fitting density in the fitted region was 0.23 m _ 1 wi th fitting absorption coefficient 0.1 for al l frequencies. The agreement was excellent at al l frequencies. Differences of more than 1 dB occurred only at large distances and low frequency. Let us now apply the correction fitting formula to the above data. The mean fitting Chapter 7. Validation of Q(f) by a Measurement in a Machine Shop 92 Table 7.2: The parameters used for prediction using Q(f). frequency (Hz) 250 500 1000 2000 4000 fitting density Q(f) ( m - 1 ) , ! ) / = 1.024m 0.093 0.134 0.172 0.201 0.219 fitting density Q(f) ( m - 1 ) , ! ) / = 1.30m 0.104 0.150 0.192 0.224 0.245 average fitting density ( m _ 1 ) 0.10 0.142 0.182 0.213 0.232 fitting absorption coeff. 0.2 0.15 0.12 0.1 0.1 dimension was calculated from the dimensions of all the fittings: Dt^t.Dfi (7.1) in which, n = 63 is the number of fittings, Si is the surface area of the i th fitting . Using this method, the Df = 1.024 m , Q = 0.241 m _ 1 . We can also use a weighted method to calculate D'f as follows: Vf~Y.DfiSilY.Si (7.2) t i B y this method, the D'f = 1.3 m , Q = 0.269 m _ 1 . The Q(f) values calculated by these two method are listed in Table 7.2. The fitting densities Q(f) from the first method were less than those from the second, since the small Dfi make a small contribution to the fitting density Q0, but seriously reduce the total mean fitting dimension Df. The values from the second method are affected by several extra large fittings, thus the average values calculated by the two methods are used. The corrected fitting density Q was 0.255 m _ 1 , which was very similar the value of 0.23 m _ 1 found by the best-fit method [4]; this supports the validity of the correction formula — E q . 3.9. The next step was to use the best-fit method to validate the variation of fitting density wi th frequency Q(f) given in Table 7.2. B y comparing different values of fitting absorption coefficient, it was found that the best-fit values were 0.2 at 250 Hz , 0.15 at 500 Hz , 0.12 at 1 k H z , 0.1 at 2 k H z and 4 k H z . The predicted SP values are compared with those measured in Figures 7.1 to 7.5, Chapter 7. Validation of Q(f) by a Measurement in a Machine Shop 93 m T3 1 °' Distance(m) Figure 7.1: 250 Hz S P ( d B ) curves measured and predicted by R A Y C U B using Q(f) = 0.1 m _ 1 and Q / = 0.2. in which the curves designated R A Y C U B - c are modelling the same configuration but using constant fitting density 0.23 m - 1 and fitting absorption coefficient 0.1. Note at 4000 H z the two curves are about the same, since the difference of the fitting densities is only 0 .002 m - 1 . The agreement is excellent at al l five frequencies and at al l distances. The agreement is as good as that obtained by Hodgson using constant fitting absorption coefficient 0.1 and constant fitting density 0.23 m _ 1 [4]. Since fitting absorption coefficient cannot at present be measured directly, it is not possible to say which set of prediction parameters represents reality. Next we try to calculate the fitting absorption coefficient using the correction formula — E q . 3 .12 — developed in section 3.2. Note that Hodgson presented "fitted-room" surface absorption coefficients for the machine shop listed in Table 7.1. They were 0.21 at 250 Hz , 0.18 at 500 Hz , 0.15 at 1000 Hz , and 0.14 at 2000 and 4000 Hz [4]. Chapter 7. Validation of Q(f) by a Measurement in a Machine Shop 94 SP(dB) -10 • measured RAYCUB RAYCUB-c -15 -20 -25 1 , . . . . . 10° 10' Distance(m) Figure 7.2: 500 Hz S P ( d B ) curves measured and predicted by R A Y C U B using Q{f) = 0.142 rn"1 and af = 0.15. Figure 7.3: 1000 Hz £ P ( d B ) curves measured and predicted by R A Y C U B using Q(f) = 0.182 m " 1 and af = 0.12. Chapter 7. Validation of Q(f) by a Measurement in a Machine Shop 95 Figure 7.4: 2000 Hz S P ( d B ) curves measured and predicted by R A Y C U B using Q(f) = 0.213 m - 1 and af = 0.1. m T3 Figure 7.5: 4000 Hz S P ( d B ) curves measured and predicted by R A Y C U B using Q(f) = 0.232 m " 1 and af = 0.1. Chapter 7. Validation of Q(f) by a Measurement in a Machine Shop 96 For the machine shop in this chapter, the room volume was divided to two zones — lower zone, designated wi th subscript 1, with fitting density Q(f), and an upper zone, designated wi th subscript 2, wi th fitting density 0.03 m _ 1 . In the case of two zones, E q . 3.12 can be rewritten as Q=/r(ffr + 5/i) - aerSr/exp(-Q'(f)DT) - ctf2Sf2/exp(-Q2DT2) 0 1 1 1 Sn/expf-Qi'fiDn) { L 6 ) in which, Q'(f) is the average fitting density of the room, Q'(f) = (<3i(/)Vi + Q2V2)/V; Dr is the mean room free path, Dr = AV/S — 8.8 m; Sf2 is the upper zone fitting surface, Sf2 = AV2Q2 = 472 m 2 ; DTl is the mean free path in the lower fitted zone, Dri — 4 V i / 5 r i = 2.65 m; Dr2 is the mean free path in the upper zone, Dr2 = 4:V2/Sr2 = 7.58 m . The term ctfr(Sr + is the true surface absorption; since the true fitting surface area in the upper zone was small and difficult to estimate, and the fitting absorption coefficient was small , this term does not include the upper zone. The calculated fitting absorption coefficients for the fitted zone using these parameters are 0.21 at 250 Hz , 0.14 at 500 Hz , 0.09 at 1000 Hz , 0.11 at 2000 Hz , and 0.10 at 4000 Hz . These values are similar to those determined by a best-fit method listed in Table 7.2, except that the value at 1000 Hz was 25% lower than expected for unknown reasons. If we use the constant fitting density Q = 0.23 m _ 1 found by Hodgson, the results become 0.061 at 250 H z , 0.059 at 500 Hz , 0.056 at 1000 Hz , and 0.10 at 2000 and 4000 Hz . The values at 2000 and 4000 Hz are exactly equal to the results obtained by the best-fit method [4], but the first three values are significantly smaller. This suggests that fitting densities must vary wi th frequency. From this case, we can see that E q . 3.12 is a useful formula for calculating fitting ab-sorption coefficient from empty and fitted room surface absorption coefficients obtained from measured reverberation times. The results agree well wi th those found using the Chapter 7. Validation of Q(f) by a Measurement in a Machine Shop 97 best-fit method. According to the new way of describing the fittings, the fitting ab-sorption coefficients decrease with increasing frequency, and the fitting densities increase wi th frequency in this case. According to Hodgson's description, the fitting density and absorption coefficient are constant. It is not possible to say which description of the fittings is physically correct — they both gave similar SP predictions. This may support the work of A k i l and Oldham [21]. In summary, the measurements in the full-scale machine shop support the validity of the correction formula for calculating fitting density — E q . 3.9 — and the model describing the variation of fitting density with frequency — E q . 5.15. The agreement was excellent at al l frequencies and distances. Chapter 8 Summary and Conclusion The objective of this research is to make a contribution to industrial acoustics wi th the a im of predicting and reducing noise levels in workrooms. The main a im of the research was to develop and test a method for calculating the fitting density — a very important parameter used in predicting sound propagation — and the variation of this parameter wi th frequency. In Chapter 3, a correction formula, E q . 3.9, for calculating the fitting density in the case of large fitting dimensions was derived, and the magnitude of the correction was investigated. In Chapter 5, the variation of fitting density with frequency was found from sound-propagation experiments in the anechoic chamber, and a model — E q . 5.15 — to describe the variation, Q(f), was derived using statistical methods. The results show that the fitting density calculated by the Kuttruff formula is only valid in some middle frequency range — at low frequency fitting densities are smaller than those calculated by the Kuttruff formula, and at high frequencies they are much greater. In Chapters 6 and 7, the fitting density models — E q . 3.9 and 5.15 — were validated by experiments in a scale model workroom and in a full-scale machine shop. In the scale model workroom, four configurations of geometry and fitting distribution were tested: randomly distributed throughout the whole room; randomly or uniformly distributed in a layer on the floor wi th an empty zone above; uniformly distributed on the floor and wi th a lower, absorbent ceiling. It was found that there is not much difference between SP's measured wi th random and uniform distributions of the fittings on the floor. The 98 Chapter 8. Summary and Conclusion 99 difference was normally less than 0.5 d B , and never exceeded 1.0 d B . The measurements both in the scale model and the full scale workroom are strongly support the new fitting density model. The sound propagation curves predicted using fitting densities calculated by Eqs. 3.9 and 5.15 are in excellent agreement with the measured values at al l distances and at al l frequencies. Differences of more than 1 dB occur only at low frequency when the receiver is far from the source. It was shown that the SP curves predicted using the Kutt ruff formula only agree well wi th measurement for the cases of sparsely fitted rooms, or densely fitted rooms at low frequencies, since in these cases the fitting density values calculated by the two methods are about same. However, for the case of densely fitted rooms wi th large fittings, the fitting density values calculated using the Kuttruff formula are too small to agree with those measured, especially at high frequencies. Experiments were taken both in an anechoic chamber and a scale model workroom using the M L S S A system. The scale modelling principles and physical scale techniques were reviewed and developed in Chapters 4 and 6. Chapter 4 also discussed the unsuc-cessful attempts to correct the distortion of measured impulse responses by the imperfect loudspeaker. It was found that scale model method is a useful tool to find sound complex propagation phenomena, for example, the variation of fitting density with frequency only can be found by scale-model methods or by real measurement in full scale workrooms. Since measurements in full scale workrooms are more expensive, and the room geometry and fitting spatial distributions are not easy to change, scale-model methods give an alternative way to perform a kind of real measurements. In Chapter 3, a corrected formula — E q . 3.12 — for calculating fitting absorption coefficient using empty and fitted room surface absorption coefficients was derived. In Chapter 7, this formula was validated in comparison with measurements in a machine shop. The results using E q . 3.12 are very similar with the values obtained by best-fit method. However, this method — E q . 3.12 — stil l needs to be validated by more Chapter 8. Summary\and Conclusion 100 measurements. It is should be noted that in deriving Q(f) the fitting dimension is considered by cal-culating a fundamental frequency related to dimension. Thus it takes fitting dimensions into account. However, only one kind of fitting was used in the project. Since it was not feasible to perform experiments with other fitting dimensions in order to verify the rela-tionship between fitting density and dimension, this model needs further validation — for example by measurement in the anechoic chamber or scale-model workroom with fittings of widely different dimensions but the same fitting density, and by more measurements in real workrooms. Three prediction models were used in the project — ray-tracing, the Lindqvist image-source model and the-new model. In the limited published validation attempts it was found that the ray-tracing model is the most useful model, especially for rooms having complex shape, and that it gives accurate predictions in all cases. The Lindqvist model is similarly accurate but less attractive because of its long run times. The new model developed here, and expressed by Eqs. A.4 and A.5, was based on the image-source method and was validated for hypothetical workrooms, a real workroom and a scale model workroom. This model was found to be a fast, workable and accurate method for predicting [sound pressure levels in fitted workrooms. Its main shortcoming is that it only applies to rectangular rooms, as do most image-source models. For other room shapes — for example, rooms having sawtooth shaped ceilings — the room shape must be approximated as rectangular, thereby reducing prediction accuracy. Further work to improve the new model could involve its extension to deal with fittings located only on floor, with barriers, and with sound source directivity. A method to efficiently shorten the run time — using an approximate formula to calculate the contribution of higher-order image sources distribution — could be developed. In summary, this research has made four major contributions to the field of industrial Chapter 8. Summary and Conclusion 101 acoustics: • A correction formula for calculating fitting density in case of large fittings; • The variation of fitting density with frequency and a model for determining the relationship; • A correction formula for calculating fitting absorption coefficient using empty and fitted room surface absorption coefficients obtained by measuring reverberation times; • A n improved prediction model for fitted workrooms based on image-source methods which provided a fast, workable and accurate alternative to existing fitted room models. To predict sound propagation in factories is a complex process. Besides improving existing models (mainly the ray-tracing model) and developing new models, it would be of great use to find a method to accurately determine the absorption coefficients of fittings — which would be expected to vary wi th frequency — thus increasing the accuracy of prediction by al l prediction models. The corrected formula — E q . 3.12 — is a method for calculating fitting absorption coefficient using room surface absorption coefficients in empty and fitted room obtained by measuring room reverberation times, but it requires further validation. Nomenclature c Sound speed in air m/s . d Diameter in m. Mean dimension of fittings in m . R o o m mean free path in m . E Sound energy in J . f Frequency in Hz . fo Fundamental frequency relative with fitting dimension FS Parameter relative to full scale value. 9 Correlation function. G Spectral density. h, H Transfer function. I Length in m . h Mean free path between fittings in m. F i t t ing dimensions in three directions in m . LP Sound pressure level in d B . Lw Sound power level in d B . m A i r absorption exponent in N p / m . M Parameter relative to scale value. MFP Room mean free path in m . n Scale factor. V sound pressure in Pa . Q Fi t t ing density in m _ 1 . 102 Nomen clat ure 103 Qo F i t t ing density from the Kuttruff formula in m _ 1 . QL F i t t ing density from the Lindqvist correction formula in m Q(f) F i t t ing density as function of frequency in m _ 1 . r Distance between sound source and receiver in m . R Room constant in m 2 ; Sphere radius in m. S Surface area in m 2 . Sf Cross section area of fitting in m 2 . SP Sound propagation in d B . SPL Sound pressure level in d B . Sv External surface area of fitting in m 2 . t T ime in second. x, y, z Coordinate directions. V Volume in m 3 . W Sound power in W . a Sound absorption coefficient of room surface. af Sound absorption coefficient of fitting surface. ctm Mean Sound absorption coefficient of room surfaces. aT Sound absorption coefficient of room surface. A Sound 1 wavelength i n m . 7 Coherence function. Bibliography [1] M . Hodgson, "Measurements of the Influence of Fittings and Roof P i t ch on the Sound F ie ld in Panel Roof Factories," Appl ied Acoustics 16, 1983, pp. 369-391. [2] E . A . Lindqvist , "Noise Attenuation in Large Factory Spaces," Acoustica, Vol.50, 1982, pp. 313-328. [3] A . Ondet and J . Barbry, "Modell ing of Sound Propagation in F i t ted Workshops Using Ray-tracing," Journal of Acoustical Society of America , 85(2), 1989, pp. 787-796. [4] M . Hodgson, "Case History: Factory Noise Prediction Using Ray Tracing — Experimental Validat ion and the Effectiveness of Noise Control Measures," Noise Control Engineering Journal 33, 1989, pp. 97-104. [5] H . Kuttruff, "Sound Decay in Reverberation Chambers wi th Diffusing Ele-ments," Journal of Acoustical Society of Amer ica 69, 1981, pp. 1716-1723. [6] E . A . Lindqvist , "Noise Attenuation in Factories," Appl ied Acoustics 16, 1983, pp. 183-214. [7] H . Kuttruff, "Room Acoustics," The third edition, Elsevier Science Publishers L T D . , 1991. [8] W . C . Sabine, "Reverberation," The American Architect and Engineering Record, 1900, pp. 3-42. [9] M . Hodgson, "On the Accuracy of Models for Predicting Sound Propagation in F i t t ed Rooms," Journal of Acoustical Society of America , 88(2), 1990, pp. 871-878. [10] M . Hodgson, "Physical and Theoretical Models as Tools for the Study of Factory Sound Fields," P h . D . Thesis, University of Southampton, 1983. [11] M . Hodgson, "Review and Crit ique of Exist ing Simplified Models for Predict ing Factory Noise Levels," Canadian Acoustics 19(1), 1991, pp. 15-23. [12] Spandock, F . , "Akustische Modellversuche", A n n . Phys. 20 (1934), pp. 345-360. [13] M . Hodgson and R . J . Orlowski, "Acoustic Scale Modell ing of Factories Part 1," Journal of Sound and Vibra t ion , 113(1), 1987, pp. 29-46. 104 Bibliography 105 [14] M . Hodgson and R . J . Orlowski, "Acoustic Scale Modell ing of Factories Part 2," Journal of Sound and Vibra t ion , 113(2), 1987, pp. 257-271. [15] R . J . Orlowski and M . Hodgson, "Acoustic Scale Modell ing of Factories, Part 3," Journal of Sound and Vibra t ion , 121(3), 1988, pp. 525-545. [16] R . Friberg, "Noise Reduction in Industrial Ha l l Obtained by Acoustical Treat-ment of Ceilings and Walls ," Noise Control and Vibra t ion Reduction 6, 1975, pp. 75-79. [17] J . K . Thompson, L . D . Mi tche l l and C . J . Hurst, " A Modified R o o m Acous-tic Approach to Determine Sound Pressure Levels in Irregularly Proportioned Factory Spaces," Proceedings of Inter-Noise 76, Washington D . C , pp. 465-468. [18] R . M . Windle , " A n Independent Comparison and Validation of Noise Predict ion Techniques Inside Factories," Proceedings of the Institute of Acoustics, Vol.16, part 2, 1994, pp. 423-432. [19] A . Ondet and J . Barbry, "Note in connection with a Comparison of Different Models for Predicting Sound Level in Fi t ted Industrial Rooms," Letters to the Edi tor , Journal of Sound and Vibra t ion , Vol.143, 1990, pp. 343-350. [20] U . J . Kurze , "Scattering of Sound in Industrial Spaces," Journal of Sound and Vibra t ion , Vol.89(3), 1985, pp. 365-377. [21] H . A . A k i l and D . J . Oldham, "Determination of the Scattering Parameters of Factory Fit t ings," Proceedings of the Institute of Acoustics, Vol.25, part(3), 1993. [22] " M L S S A Reference Manual ," Version 9.0, D R A Laboratories, 1994. [23] Benat, J . S. and Piersol ,A. G . , "Engineering Applications of Correlation and Spectral Analysis ," 1980, Wiley-Interscience Publication. [24] Benat, J . S. and Piersol ,A. G . , "Random Data: Analysis and Measurement Procedures," second Edi t ion . New York, 1986, Wiley-Interscience Publ icat ion. [25] W i l l i a m Mendenhall , Richard L . Scheaffer and Dennis D . Wackerly, "Mathemat-ical Statist ics 'with Applications," third edition, P W S Publishers, 1986. [26] M . Hodgson, "Towards a Proven Method for Predicting Factory Sound Propa-gation," Proceedings of Inter-Noise'86 (Cambridge, M A , 1986), pp. 1319-1323. [27] S. Dance, "The Development of Computer models for the prediction of Sound Distr ibut ion in Fi t ted non-diffuse Spaces," P h . D . Thesis, South Bank Univer-sity, London, U K , 1993. Appendix A An Improved Prediction Model for Fitted Rooms In this appendix, an improved image-source model for predicting sound pressure levels in fitted rooms is developed and validated. A . l Model Development A.1.1 Infinite Region In developing the model, first an infinite region containing randomly-distributed fit-tings is considered, and an expression for the sound energy density, E(r,t), is developed as a function of distance and of t ime relative to the emission by the source of an ideal impulsive signal. In common wi th most geometrical models [2] [3], the are following as-i sumptions are made: 1. There is a random distribution of fittings in three dimensional space; 2. The sound energy is scattered uniformly in all directions from these objects. The validity of the first assumption depends on the factory to which the model is to applied. If the factory space is small with a few large fittings, or if as in many factories, the fittings are on the floor, or if there is in a factory of any size a regular array of identical machines, the distribution of fittings cannot reasonably be assumed to be random. The validity of the second assumption depends on the dimensions of the fittings in relation to the wavelength of the sound. It is only true if the fitting dimension is larger 106 Appendix A. An Improved Prediction Model for Fitted Rooms 107 than the wavelength. Fortunately, the sizes of machines and other objects in factories are often large, so it is reasonable to assume a uniform angular distribution of scattered energy except at low frequencies. As is common wi th image-source methods, the diffuse and non-diffuse energy contri-butions are considered separately and summed. The non-diffuse energy component is due to a sound particle not hit t ing any obstacles between the sound source and the receiver, and the diffuse component represents striking at least one obstacle. The non-diffuse en-ergy is described by the Kuttruff equation [7], as in image-source models. The sound energy of the non-diffuse contribution is, W E^_d = ^—^ exp{-(Q +m)r] ( A . l ) in which, W is the sound source power, in watts, c is the sound speed in air, in m/s , r is the distance from the source to the receiver, Q is the fitting density calculated using the Kutt ruff formula, and ra is the air absorption exponent in N p / m . The diffuse part of the Kuttruff model [7] is a simple model only for infinite space. That is Edx} = J^W(^t)^2exp[-(AQ + m)ct}exP(-^-)dt (A.2) in which, A = —ln(l — af), wi th a / the absorption coefficient of the fittings. The Lindqvist model is complex and needs long run time, so it is not easy to improve. Compared wi th the Lindqvist and ray-tracing models, the Kuttruff model generally works well, but gives a higher value for the case that the receiver is far from the source when fitting density is large and gives lower values for the others. It was found that using Q to the power 1.4 and (ci) to the power 5/3 instead of the term (Q/ct)3/2 in the Kuttruff model can obtain better results, and that using a term (3.2 — l.br/ct) made the Appendix A. An Improved Prediction Model for Fitted Rooms 108 impulse response curves more similar to the predictions by the ray-tracing model. In the new model, the diffuse energy distribution is modified from the Kuttruff model and is improved as follows: /•oo \WQ1A r 3r20 E<« = L M ^ F 5 * 3 ' 2 - ^)<M-(AQ + m)ct}<M-^)dt (A.3) Figure A . l gives the comparison of this model with the Kuttruff model, the Lindqvist model, and Ondet and Barbry's ray-tracing model ( R A Y C U B ) , in an infinite space wi th fitting densities 0.05, 0.1 and 0.15 m _ 1 , and with af = 0. Since the ray-tracing model only applies to finite spaces, an infinite region was approximated using a room size of 200 x 200 x 200 m 3 . For this case, the receiver size was 1 m near the source, and 5 m far away the source; the max imum number of reflections was 100, and the number of rays was 1,000,000. The results from the new model agree well wi th those of from the ray-tracing model and the Lindqvist model. A. 1.2 Finite Bounded Region In the case of finite bounded region the image-source method considers the absorption of the walls, floor and ceiling, the fitting density, the fitting absorption and the positions of the source and receiver. The fittings are assumed to be distributed randomly in three dimensions. The model can be readily applied to any rectangular room and any source and receiver positions. The non-diffuse energy contribution is the same used by the Lindqvist model, that is We-(Q+™)ri 6 2Wi ( r ) = E 4 7 r r 2 c na - * P r ( p ) (^ ) i n which, ap is the absorption coefficient of the room surface p, n^(p) represents the number of times that the sound propagation is reflected by surface p in the i t h image Appendix A. An Improved Prediction Model for Fitted Rooms 109 m -10 2. CL CO -20 -30 (a) Q = 0.05 1/m • Kuttruff RAYCUB Lindqvist new model -40 ^ 10u 10' i o2 Distance (m) 63" -10 a. CO -20 (b)Q = 0.1 1/m -30 ^ ^ ^ ^ ^ -40 i 10u 10' Distance (m) co -10 2, o_ CO -20 (C)Q = 0.15 1/m -30 -40 • 10u 10' ' i n 2 Distance (m) Figure A . l : Comparison of SP predicted by the new model and other models for an infinite fitted space, ctf = 0. Appendix A. An Improved Prediction Model for Fitted Rooms 110 order, is the distance from the receiver to the real source or to an image source. The diffuse energy contribution is divided into two parts. One comes only from the reflection by the fittings in finite region; the second is the energy distribution reflected by room surfaces. The total diffuse energy contribution used in the new model equals to: in which B = -0.175(1 + af)yjQrct/Dr - AQ(ct - r.) + (0.4 + am)r/3DT there Dr is the room mean free path, in m; r is the direct distance from the real source to the receiver, in m; ctm is the average absorption coefficient of the room surfaces; a / is the mean absorption coefficient of fittings, and A = — l n ( l — a-f). The form of the new model is more similar to the Lindqvist model than the others, but they are two completely different models. The summation of the Poisson function in the integral used by Lindqvist is not used in the new model, this makes the new model runs much faster than the Lindqvist model. The term that sound energy loss by reflection from room surfaces relative image source is introduced to the diffuse energy contribution, which can more accurately simulate the real cases. The model was programmed in "Sun c"; the program listing is in Section A . 3 . The next section gives comparisons with other models and with experiment. Appendix A. An Improved Prediction Model for Fitted Rooms 111 100 m y 0 7 • x Sound source Sound receivers Figure A . 2 : Hypothetical workroom. A.2 Experimental Validation of the Model A.2.1 Hypothetical Workroom Firs t , the model was compared with the Ondet and Barbry ray-tracing model, R A Y -C U B , and the Lindqvist model for several hypothetical workrooms. Figure A . 2 shows one of the configurations, for which Figure A .3 gives the comparison results. It is a rectangular room wi th dimensions 100 m long, 40 m wide and 5 m high. The source and al l receiver positions are at same height of 1.5 m above the floor, and centred i n the y direction. The source is at 20 m from one wall in the x direction; the receivers are 1, 2, 3, 4, 5, 10, 15, . . . , 55, 60 m from the source in the x direction. The room-surface absorption coefficients are 0.05 for the floor, 0.3 for the ceiling and 0.1 for all the vertical walls. F i t t ing densities are 0.5, 0.1 or 0.15 m _ 1 with the fitting absorption coefficient equal to 0.1. The air absorption exponent is 0.001 N p / m . These parameter values are typical of real workrooms [4] [10]. In the ray-tracing predictions, the receiver size was 1 m for most positions (5 m Appendix A. An Improved Prediction Model for Fitted Rooms 112 m -10 CL co .20 -30 -40 -50 -60 (a) Q = 0.05 1/m 10" RAYCUB Lindqvist new model 10' Distance (m) Distance (m) Figure A .3 : Comparison of the SP predicted by three models in the hypothetical work-room shown in F i g . A.2 , for three fitting densities. Appendix A. An Improved Prediction Model for Fitted Rooms 113 30 m • ii E oo v ' N / \ ( \ ( \ f \ ( > \ / \ / V ^ ? V • \ V \ ) - / .^ ? / \ — , — , — , — — ^ — \ Sound source \ Sound receiver Figure A .4 : Floor plan of the warehouse showing dimensions and source and receiver positions. < r <40 m), but 0.5 m for r < 5 m and 2.5 m for r > 40 m; the maximum trajectory number was 30, the ray number was 500,000. For image-source predictions, the max imum image order was 15. From Figure A . 3 , we can see that the new model agrees well wi th the other two for this case. After comparison for other cases, it is concluded that the new model agrees well wi th the Ondet and Barbry ray-tracing model and with the Lindqvist image-source model for most typical cases, but that there are significant differences for cases in which the fitting absorption coefficient is greater than 0.5 and the receivers are far from the source. In these cases, the new model gives lower sound pressure levels than those from the other two, by up to 3 d B . This is not considered a serious l imitat ion since fitting absorption coefficients seldom exceed 0.2. A.2.2 Real Workroom The second comparison is wi th experiments in a warehouse tested by Ondet and Bar-bry [3] [19]. The prediction parameters for the warehouse are as follows: room dimensions — 30 x 8 x 3.85 m high; sound absorption coefficients — 0.1 for the four walls, 0.05 for Appendix A. An Improved Prediction Model for Fitted Rooms 114 SP (dB) -10 -15 • -20 -25 • measured - RAYCUB Lindqvist new model • -30 1 101 Distance (m) Figure A.5: Comparison of SP curves predicted by three models with that measured in the warehouse, for Q = 0.135 m _ 1 . m -10 1 CO -15 -20 -25 -30 measured RAYCUB Lindqvist new model 10' Distance (m) Figure A.6: Comparison of SP curves predicted by three models with measurement in the warehouse, for Q = 0.068 m _ 1 . Appendix A. An Improved Prediction Model for Fitted Rooms 115 0.1 0.2 0.3 Figure A . 7 : Comparison of the energy impulse responses predicted by the new model and by R A Y C U B in the warehouse, wi th r = 20 m , Q = 0.135 m _ 1 . the floor and 0.15 for the ceiling; fitting density — Q = 0.135 m _ 1 ; fitting absorption coefficient — 0.3; measurement frequency — 1000 Hz; source height — 0.85 m; receiver height — 1.5 m . The source and receiver positions are shown i n Figure A . 4 . The comparisons between the SP curves predicted by the ray-tracing, the Lindqvist and the new model are given in Figure A . 5 . Figure A.6 gives the comparison in same conditions, but wi th Q = 0.068 m _ 1 [27]. F rom Figures A . 5 and A . 6 , it can be see that the new model agrees well the measurements especially for the case of Q — 0.135 m _ 1 . Figure A . 7 gives the energy impulse response in the warehouse with r = 20 m , and Q = 0.135 m _ 1 , as predicted by the new model and by R A Y C U B ; the shapes are very similar. Appendix A. An Improved Prediction Model for Fitted Rooms 116 SP (dB) -20 -25 • ^ - ^ ^ — -30 -35 • measured I RAYCUB Lindqvist new model -40 . . . . . . i i 10' Distance (m) 1 0 Figure A . 8 : Comparison of SP curves predicted by three models with measurement in the scale model workroom, for Q = 0.025 m _ 1 . A.2.3 Scale Model Workroom Comparisons were also made between prediction and experiment in a 1:50 scale model workroom [11]. The parameters (in full scale values) for the scale model are as fol-lows. Sound absorption coefficient: 0.07 for all room surfaces and fittings; Measure-ment frequency: 630 Hz; F i t t ing density: 0.025 m - 1 and 0.05 m - 1 ; Room dimension: 110 x 55 x 5.5 m high; Sound source position: at half height and width and at 5 m from one end wall; Receiver position: at half height and width, at distances of 5, 10, 20, 50, 70, 90 m from the source. The comparison between prediction and experiment is given in Figures A .8 and A . 9 . From the two figures, we can see that the new model agrees well wi th the experiments for most measurement positions, but overestimates for positions near the sound source. In summary, from the l imited comparison cases considered, it can be concluded that Appendix A. An Improved Prediction Model for Fitted Rooms 117 (dB) o_ co -20 -25 -30 • measured -35 . RAYCUB _ Lindqvist new model S \ X \ \ \ , \ N. \ X \ N \ \ X -40 i • i 101 Distance (m) 1 0 Figure A .9: Comparison of SP curves predicted by three models with measurement in the scale model workroom, for Q = 0.05 m - 1 . the new model presented in this chapter works well for most cases. Of considerable practical importance is the fact that it's run time is about 10 times less than that of the Lindqvist model; it is much faster than the ray-tracing model for cases when the number of receiver positions is less than 10. Furthermore, the accuracy compares well wi th the other two models. The only shortcoming is that it only applies to rectangular rooms, as do most image-source models. For other room shapes — for example, rooms having sawtooth shaped ceilings — the room shape must be approximated as rectangular, thereby reducing prediction accuracy. Appendix A. An Improved Prediction Model for Fitted Rooms 1 A.3 The Program of the New Model /* program * mism.c * w r i t t e n by Ke L i in SUN.C, Jul y 15, 1995. This program i s used to c a l c u l a t e the sound pressure l e v e l d i s t r i b u t i o n in a f i t t e d room, using a modified image-source model, with one source and up to 20 r e c e i v e r s , and 8 frequency octave bands with d i f f e r e n t absorption c o e f f i c i e n t s of surfaces and f i t t i n g s and f i t t i n g d e n s ity f o r each frequency. The input data f i l e i s mism-in.dat, output i s mism-out.dat. */ # in c l u d e <stdio.h> /*# inc l u d e <stdlib.h>*/ /*# inc l u d e <sys/ieeefp.h> # i n c l u d e f l o a t i n g p o i n t .h>*/ # in c l u d e <math.h> # d e f i n e SIZER 20 # d e f i n e SIZEF 8 FILE * f i n , * f o u t ; f l o a t pi=3.14159265, c=340.,roomx, roomy, roomz; f l o a t ssx, ssy, ssz, rex[SIZER], rey[SIZER], rez[SIZER]; f l o a t abF [S.IZEF] , ab [SIZEF] [6] , mab,q,m,abf ,Lw [SIZEF] ,Aci ; f l o a t q [SIZEF]., M [SIZEF] , V, SW ,LS , Af ,R0 , t i n t v , t o t a l t ; f l o a t ppow() , E_ndif(), Ediffus() ,absorpO ; f l o a t SPL[SIZEF][SIZER],SP[SIZEF][SIZER],f_v[SIZEF],dist[SIZER]; i n t i , n i , rece_n,f_n; i n t main(viod) { int f,i,ix,iy,iz,swichx=0,swichy=0,swichz=0; f l o a t imagx, imagy, imagz,rx,ry,rz; f l o a t r,rr2,edir,edif,menab; f l o a t pxl=1.0,px2=1.0,pyl=1.0,py2=1.0,pzl=1.0,pz2=1.0; i = d a t a i n ( v i o d ) ; /* data input, by c a l l i n g d a t a i n */ p r i n t f ( " *********** working ***********\n"); V=roomx*roomy*roomz; SW=2*(roomy*roomz+roomy*roomx+roomx*roomz); LS=4*V/SW;, /* main c a l c u l a t i o n s */ f o r (f=0; f < f_n; f++) /* do loop f o r each octave band */ { q=q[f]; m=M[f]; abf=abF[f] ; mab=((ab[f] [0]+ab[f] [1])*roomy*roomz+(ab[f] [2]+ab[f] [3])* Appendix A. An Improved Prediction Model for Fitted Rooms roomx*roomz+(ab[f][4]+ab[f][5] )*roomx*roomy)/SW; Af=absorp(abf); for (i=0; i < rece_n; i++) /* do loop for receivers */ { rx=rex[i] ; ry=rey[i] ; rz=rez[i] ; edir=0.0; edif=0.0; R0=sqrt((ssx-rx)*(ssx-rx)+(ssy-ry)*(ssy-ry)+(ssz-rz)*(ssz-rz)) for (ix=l; ix <= n i ; ) { i f (swichx==0) { i f (ix'/02 == 1) imagx = ssx-(ix-l)*roomx; else imagx = -(ix-2)*roomx - ssx; pxl=ppow(ix,ab[f][0],swichx); swichx = 1; px2=ppow(ix,ab[f][1].swichx);} else { i f (ix==l) { swichx=0; ix++; continue;} i f (ix°/02 == 1) imagx = (ix-l)*roomx + ssx; else imagx = ix*roomx - ssx; swichx = 0; px2=ppow(ix,ab[f][1],swichx);} for (iy=l; iy+ix <=ni; ) { i f (swichy ==0) { i f ( i y % 2 ==1) imagy = ssy-(iy-l)*roomy; else imagy = -(iy-2)*roomy - ssy; pyl=ppow(iy,ab[f][2],swichy); swichy = 1; py2=ppow(iy,ab[f][3],swichy);} else { i f (iy==l) {swichy=0; iy++; continue;} i f (iy°/,2 == 1) imagy = (iy-l)*roomy + ssy; else imagy = iy*roomy - ssy; swichy = 0; py2=ppow(iy,ab[f][3].swichy);} for (iz=l; iz+iy+ix <=ni; ) { i f (swichz ==0) { i f ( i z % 2 ==1) imagz =ssz -(iz-1)*roomz; else imagz=-(iz-2)*roomz -ssz; pzl=ppow(iz,ab[f][4],swichz); swichz =1; pz2=ppow(iz,ab[f][5],swichz);} else { i f (iz==l) {swichz=0; iz++; continue;} i f (iz%2 ==l) imagz = (iz-l)*roomz + ssz; else imagz = iz*roomz - ssz; swichz =0; pz2=ppow(iz,ab[f][5].swichz);} rr2=(imagx-rx)*(imagx-rx)+(imagy-ry)*(imagy-ry)+(imagz-rz)* (imagz-rz); r = sqrt(rr2); menab=pxl*px2*pyl*py2*pzl*pz2; i f (menab < l.e-15); Appendix A. An Improved Prediction Model for Fitted Rooms 120 else { edir += E_ndif(r)*menab; edif += Ediffus(r)*menab; } i f (swichz==0) iz++; } i f (swichy==0) iy++; } i f (swichx==0) ix++; } SP[f][i]= 10*loglO((edir+edif*c)/4./pi); SPL [f] [i] = Lw[f] + SP[f] [i] ; dist[i]=RO; } } /* output the f i n a l results */ fprintf(fout,"\n\n"); f p r i n t f ( f o u t , " Sound pressure levels in dB\n"); fprintf(fout."distance"); for (i=0; i<f_n; i++) fp r i n t f ( f o u t , " °/,gHz" ,f _v [i] ) ; fprintf(fout,"\n\n"); for (i=0; i < rece_n; i++) { f printf (f out, '7,8. 3f " , dist [i] ) ; for (f=0; f<f_n; f++) f p r i n t f (f out,'7.8. If ",SPL[f] [i]) ; fprintf(fout,"\n") ; } fprintf(fout,"\n\n"); f p r i n t f ( f o u t , " Sound Propagation in dB\n"); fprintf(fout,"distance"); for (i=0; i < f_n; i++) f p r i n t f (f out," °/.gHz" ,f _v[i]) ; fprintf(fout,"\n\n"); for (i=0; i < rece_n; i++) { f p r i n t f (fout,"°/.8.3f", d i s t [ i ] ) ; for (f=0; f<f_n; f++) f p r i n t f (f out," (/.8. If", SP[f] [i]) ; fprintf(fout,"\n"); } fclose(fout); printf ("************ END ************\n") • return; } Appendix A. An Improved Prediction Model for Fitted Rooms 121 /* input data, from data f i l e * in-nf.dat * */ int datainO { f l o a t abxl,abx2,abyl,aby2,abzl,abz2; f i n = f opeii("mism-in.dat" , "r") ; fout = fopen("mism-out.dat","w") ; fprintf(fout,"Program mism.c output data f i l e . \ n \ n " ) ; fscanf ( f i n , "'/of'/of/of " ,&roomx,&roomy,&roomz) ; f p r i n t f (fout, "room dimension in x = c/0g\n" , roomx) ; f p r i n t f (fout, "room dimension in y = °/.g\n" , roomy) ; f p r i n t f (fout, "room dimension in z = °/„g\n\n" .roomz) ; fscanf ( f i n , "'/0f °/0f °/of" ,&ssx,&ssy,&ssz); fscanf ( f i n , "'/.d" ,&rece_n) ; for(i=0; i< rece_n; i++) fscanf ( f i n , "°/.f °/.f %f " ,&rex[i] ,&rey[i] ,&rez[i]) ; f p r i n t f (fout, "sound source at x=,/0g\n" , ssx) ; f p r i n t f (fout, "sound source at y=,/0g\n" , ssy) ; f p r i n t f (fout, "sound source at z=t/0g\n\n" , ssz) ; for(i=0; i< rece.n; i++) { f p r i n t f (fout, "sound receiver at x=°/,g\n" ,rex[i] ) ; f p r i n t f (fout, "sound receiver at y=,/0g\n" ,rey [i] ) ; f p r i n t f (fout, "sound receiver at z=,/,g\n\n" , r e z [ i ] ) ; } f scanf (fin,*7,d",&f_n); for(i=0; i < f i n ; i++) f scanf ( f i n , "°/0f" ,&f_v[i] ) ; for(i=0; i < f_n; i++) { fscanf (f in, "°/.f'/.f y.f °/.f c/„f °/0f " ,&abxl ,&abx2,&abyl,&aby2,&abzl,&abz2) ; ab [i] [0] =abxl; ab [i] [l] =abx2; ab [i] [2] =abyl; ab [i][3]=aby2;ab[i][4]=abzl; ab [i] [5]=abz2; } for(i=0; i < f i n ; i++) fscanf ( f i n , '7.f %f °/.f %f " ,&q[i] ,&abF[i] ,&M[i] ,&Lw[i]) ; for(i=0; i < f_n; i++) { f p r i n t f (fout, "at frequency °/0gHz\n" ,f _v [i] ) ; f p r i n t f (fout," the wall absorption cof. in x- =°/,g\n" ,ab[i] [0] ) ; f p r i n t f (fout," the wall absorption cof. in x+ =,/,g\n" ,ab[i] [1] ) ; f p r i n t f (fout," the wall absorption cof. in y- =yog\n" ,ab[i] [2] ) ; f p r i n t f (fout," the wall absorption cof. in y+ =y,g\n" ,ab[i] [3] ) ; f p r i n t f (fout," the wall absorption cof. in z- ='/og\n" ,ab[i] [4] ) ; f p r i n t f (fout," the wall absorption cof. in z+ =c/,g\n" ,ab[i] [5] ) ; f p r i n t f (fout, "the f i t t i n g density =°/.g\n" , Q [i] ) ; f p r i n t f (fout, "the f i t t i n g absorption coeff. = y,g\n" , abF [i] ) ; f p r i n t f (fout, "the a i r absorption m=°/,g\n" ,M[i] ) ; Appendix A. An Improved Prediction Model for Fitted Rooms 122 fprintf(fout,"the sound power of source = %g dB\n\n",Lw[i]);} fscanf (fin,"c/.dc/.f,,,&ni,&Aci) ; fprintf(fout,"image factor = yod\n",ni); f printf (f out, "the integral accuracy = 0/og\n\n" , Aci) ; f c l o s e ( f i n ) ; return; } f l o a t absorp(x) /* function for calculating absorption */ fl o a t x; { i f ( x > 0.9999) return 4.; else return -log(l.-x); } f l o a t ppow(i,ai,sw) /* function to calculate power */ int i,sw; floa t a i ; { flo a t i i . s i ; i f (i==l) return 1.0; i i = i ; si=sw; i f (ai >= 0.9999) return 0.0; else return pow((1.-ai),(ii-1.-si)); } f l o a t E_ndif(r) /* function to calculate non-diffuse energy */ fl o a t r; { f l o a t e_ndif; e_ndif = exp(-(q+m)*r)/(r*r); return e_ndif; } f l o a t Ediffus(r) /* function to calculate diffuse energy */ fl o a t r; { f l o a t edf,ediff,delt=0.04,t; int i,n=50; i f (q==0.0) return 0.0; ediff=0.0; for(i=0; i < n; i++) { t=r/c+i*deit+delt/2.; edf=exp(-m*c*t)*exp(-3.*r*r*q/(4.*c*t))*(3.2-1.5*r/c/t)/ Appendix A. An Improved Prediction Model for Fitted Rooms 123 pow(c*t,5./3.)*(exp(-(Af+0.1)*q*c*t)*((1.25+mab)/(1+2*R0/LS))+ exp (-Af*q*(c*t-r))*pow(l-mab,(c*t-r)/LS)*exp(-0.175*(1+abf)* sqrt(q*R0/LS*c*t)) *exp((0.4+mab)*R0/LS/3.)*(1.0-abf*abf)); ediff+=edf; i f ( i < 20); else{ i f ( e d f / e d i f f < Aci) i=50;} } return ediff*pow(q,1.4)*delt; } /* end of the program */ A Sample of Input File — mism-in.dat 30.0 15.0 5 0 — - dimensions of room in x, y, and z directions. 2.5 7 5 1 0 — -- coordinates of the source. 7 — •- number of the receivers (nl). 3.5 7 5 1 5 — •- coordinates of the receivers, nl line s . 4.5 7 5 1 5 7.5 7 5 1 5 12.5 7 5 1 5 17.5 7 5 1 5 22.5 7 5 1 5 27.5 7 5 1 5 5 number of the octave bands (n2). 250 500 1000 2000 4000 values of the frequencies. 0.05 0 05 0 05 0.05 0.05 0.4 absorption coefs. 0.05 0 05 0 05 0.05 0.05 0.45 of the 6 surfaces in 0.05 0 05 0 05 0.05 0.05 0.4 order x0,x+,y0,y+,z0,z+. 0.05 0 05 0 05 0.05 0.05 0.4 One line for each ovtave 0.05 0 05 0 05 0.05 0.05 0.3 band, n2 lines. 0.109 0 01 0 00035 100 f i t t i n g density, absorption 0.137 0 02 0 00073 100 coefficient of the f i t t i n g s , 0.157 0 02 0 0021 100 ai r absorption conponents Np/m, 0.169 0 03 0 0075 100 and the source power level dB. 0.176 0 04 0 025 100 One line for each frequency. 15 image-source factor. 0.005 integral accuracy. 

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