ROOM SOUND FIELD PREDICTION BY ACOUSTICAL RADIOSITY by Eva-Marie Nosal B.Sc. (Pure Mathematics), The University of Calgary, 2000 B.Sc. (Applied Mathematics), The University of Calgary, 2000 B.Mus. (Piano performance), The University of Calgary, 2000 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 2002 © Eva-Marie Nosal, 2002 ABSTRACT Acoustical radiosity is a technique based on assumptions of diffuse reflection and incoherent phase relationships that has been used to predict room sound fields. In this research, the background to acoustical radiosity is given, the integral equation (on which the technique is based) is derived, and a numerical solution is detailed for convex rooms of arbitrary shape. Several validations are made by comparison of the numerical solution to (1) analytical solutions for a sphere; (2) results from a ray tracing algorithm in cubical enclosures, and; (3) measurements in three real rooms. ii TABLE OF CONTENTS Abstract ii Table of Contents iii List of Tables vii List of Figures viii Acknowledgments xi Dedication xii 1 Introduction …………………………………………………………………… 1 1.1 Room sound field prediction 1.2 History and literature review 1.3 Why acoustical radiosity? 1.4 Organization of thesis 2 Theoretical development 2.1 Assumptions ……………………………………………………………. 9 2.1.1 Diffuse reflection 2.1.2 Incoherent phase relationships 2.1.3 Other assumptions 2.2 Diffuse reflection ………………………………………………………. 10 2.2.1 Lambert’s (Cosine) Law 2.2.2 Intensity from radiation density 2.2.3 Intensity from incident intensity 2.3 Integral equation ………………………………………………………. 14 2.3.1 Explanation and derivation 2.3.2 Simplifications 2.3.2.1 Impulsive sound sources 2.3.2.2 Steady sound sources iii 2.3.3 Direct radiation density 2.3.4 Sound pressure at the receiver 2.3.5 Note on view independence 2.4 Analytical solutions of the integral equation …………………………... 19 2.4.1 Sphere 2.4.2 Flat room 3 Numerical solution 3.1 Discretization ………………………………………………………….. 22 3.1.1 Enclosure discretization 3.1.2 Direct contribution 3.1.3 Sound pressure at the receiver 3.1.4 Time discretization 3.2 Form factors …………………………………………………………… 25 3.2.1 Literature on form factors 3.2.2 Form factor algebra 3.2.3 Analytical form factors for rectangular rooms 3.2.4 HeliosFF 3.2.4.1Two-level hierarchy 3.2.4.2 Cubic tetrahedral algorithm 3.3 Integrals over solid angles ……………………………………………... 30 3.3.1 Possible approaches 3.3.2 Spherical triangle method 3.4 Algorithm ……………………………………………………………… 33 3.4.1 Basic 3.4.2 Averaging 3.4.3 Sound pressure at the receiver 4 Predictions from the solution 4.1 Impulse response ……………………………………………………… 37 4.1.1 Definition 4.1.2 Prediction 4.1.3 Consistency check 4.1.4 Integrating the impulse response 4.2 Signal response ……………………………………………………….. 40 4.3 Echogram 4.4 Steady state sound pressure level 4.5 Sound decay curve 4.6 Reverberation and early decay time iv 4.7 Other parameters ………………………………………………………. 43 4.7.1 Clarity 4.7.2 Definition 4.7.3 Center time 4.7.4 Strength 4.7.5 Others 5 Validation and experimentation 5.1 Validation of the numerical solution ………………………………….. 45 5.2 Comparison preliminaries …………………………………………….. 46 5.2.1 Parameters, echograms, and discretized echograms 5.2.2 Auralization 5.2.3 Ray tracing with RAYCUB 5.2.4 Predicted impulse response length 5.3 Predictions for a cubic room …………………………………………… 53 5.3.1 Initial predictions with varying patch sizes 5.3.1.1 Patch size 5.3.1.2 Absorption distribution 5.3.1.3 Computational efficiency 5.3.2 Predictions with varying time resolution 5.3.3 Predictions with varying time limits 5.3.4 Predictions using form factors by HeliosFF 5.3.5 Comparisons to ray tracing 5.3.6 More on ray tracing with diffuse reflection 5.4 Experiment …………………………………………………………….. 82 5.4.1 Test environments 5.4.1.1 Squash court 5.4.1.2 Environmental Room 5.4.1.3 Hebb 12 5.4.2 Measurements 5.4.3 Air-absorption exponents 5.4.4 Source power 5.4.5 Room surface absorption coefficients 5.5 Comparison between measurement and prediction ……………………. 89 5.5.1 Echograms 5.5.2 Discretized echograms 5.5.3 Acoustical parameters 5.5.4 Conclusions 6 Conclusion ……………………………………………………………………. 109 v Bibliography ………………………………………………………………………….. 113 Appendix A: Definitions ……………………………………………………………. 117 Appendix B: Relationships …………………………………………………………. 125 B.1 Reverberation and decay time B.2 Clarity and definition B.3 Absorption, reflection, and transmission coefficients B.4 Air absorption coefficients and exponents Appendix C: Theorems ……………………………………………………………. 127 C.1 Girard’s theorem C.2 Generalization of Girard’s theorem to arbitrary polygons C.3 Theorem Appendix D: Codes …………………………………………………………………. 132 vi LIST OF TABLES 5.1 Numerical and analytical predictions for three spheres ………………………. 45 5.2 Time and memory requirement for predictions on three spheres ……………... 46 5.3 Distribution of absorption in the cubical rooms ………………………………. 53 5.4 Parameter predictions for the four cubical rooms …………………………….. 62 5.5 Air absorption exponents at 23° C, 50% relative humidity, and normal atmospheric pressure in 10-3 m-1 ……………………………………… 85 5.6 Sound Power Levels (dB) of the source with settings for (1) the Environmental Room and Hebb 12 and (2) the squash court …………………. 87 5.7 Surface absorption coefficients ……………………………………………….. 89 D.1 List and description of MATALB M-files ……………………………………. 132 vii LIST OF FIGURES 1.1 Specular, semi-diffuse, and diffuse reflection ………………………………….. 3 1.2 Conversion of specularly to diffusely reflected sound energy …………………. 6 2.1 Projected length ………………………………………………………………... 11 2.2 Parametrization of a sphere ……………………………………………………. 12 2.3 Illustration for the derivation of the integral equation ………………………… 15 3.1 Form factor geometry …………………………………………………………. 25 3.2 Illustrations for the spherical triangle method ………………………………… 31 5.1 t versus reverberation time for various ε ………………………………… 52 finalmin 5.2 Discretization of the cubical rooms …………………………………………… 54 5.3 Echograms for Cube 1 from predictions with 24, 96, 294, and 600 patches …………………………………………………………………………. 56 5.4 Range reduced from Figure 5.4 for more detail ……………………………….. 57 5.5 Range reduced and domain increased from Figure 5.5 for more detail …………………………………………………………………………… 58 5.6 Discretized echograms for Cube 1 ……………………………………………. 60 5.7 Parameters predictions versus number of patches for Cube 1 ………………… 61 5.8 CPU time and memory requirements versus number of patches for Cube 2 …………………………………………………………………………. 64 5.9 Echograms for Cube 1 with varying time resolution …………………………. 66 5.10 Parameter predictions versus 1/∆t for Cube 4 ……………………………….. 67 5.11 CPU time and memory requirement vs. discretization frequency for Cube 1 …………………………………………………………………….. 68 5.12 Parameter predictions versus texact (s) for Cube 2 (with t final = 2 s) …………… 70 5.13 Parameter predictions versus t final (s) for Cube 1 (with texact = 0.6 s) …………. 71 viii 5.14 Echograms for Cube 2 from radiosity (600 patches), ray tracing with diffuse reflection, and ray tracing with specular reflection predictions ……………………………………………………………………… 73 5.15 Range reduced from Figure 5.14 for more detail ……………………………… 74 5.16 Discretized echograms for Cubes 1 and 2 by radiosity and ray tracing (diffuse and specular) with time resolution of 0.05 s ………………….. 75 5.17 Discretized echograms for Cubes 3 and 4 by radiosity and ray tracing (diffuse and specular) with time resolution of 0.05 s ………………….. 76 5.18 Parameter predictions by radiosity and ray tracing with diffuse, 50% diffuse/50% specular, and specular reflection for all four cubes ………… 77 5.19 Echograms from radiosity and ray tracing with diffuse reflection in the squash court ………………………………………………………………... 80 5.20 Discretized echogram from radiosity and ray tracing with diffuse reflection in the squash court ………………………………………………….. 81 5.21 Dimensions of Hebb 12 ……………………………………………………….. 83 5.22 Experimental setup ……………………………………………………………. 84 5.23 Source and receiver positions in the three measured rooms ………………….. 91 5.24 Measured and predicted echograms in the squash court at 1 kHz …………….. 92 5.25 Measured and predicted echograms in the Environmental Room at 1 kHz ………………………………………………………………………….. 93 5.26 Measured and predicted echograms in Hebb 12 at 1 kHz …………………….. 94 5.27 Discretized echogram for the squash court at 1 kHz ………………………….. 96 5.28 Discretized echogram for the Environmental Room at 1 kHz ………………… 97 5.29 Discretized echograms for Hebb 12 1 at kHz …………………………………. 98 5.30 Parameter values as a function of position in the squash court 1 at 1 kHz …………………………………………………………………………. 100 5.31 Parameter values as a function of frequency (Hz) in the squash court ………………………………………………………………………….. 103 5.32 Parameter values as a function of frequency (Hz) in the Environmental Room ………………………………………………………… 104 ix 5.33 Parameter values as a function of frequency (Hz) in Hebb 12 ………………. 105 A.1 Double lune with angle α …………………………………………………….. 120 A.2 Great circle and spherical triangle on a sphere ……………………………….. 121 C.1 Extending the edges of a spherical triangle to form 3 great circles ………….. 128 C.2 Three double lunes defined by a spherical triangle …………………………... 128 C.3 Covering the sphere by three lunes ………………………………………….. 128 x ACKNOWLEDGMENTS First and foremost, I’d like to thank my supervisor and mentor, Murray Hodgson, for his ideas, endless support, and unwavering confidence in my work. Murray gave me the freedom I needed to be creative and pushed just enough to get me over any blocks that inevitable came my way. He encouraged and helped me to present and publish my work. His unique perspectives, original ideas, and boundless enthusiasm for acoustics, combined with a vast experience in the field and superior communication skills, make Murray a wonderful person to work with. I will always remember our long discussions and napkin drawings with much fondness. The opportunity for me (in the math department) to work with Murray Hodgson (in mechanical engineering) was provided by the Institute of Applied Mathematics (IAM) at UBC.