Universidade Técnica De Lisboa Instituto Superior
UNIVERSIDADE TÉCNICA DE LISBOA
INSTITUTO SUPERIOR TÉCNICO
ACOUSTIC MODELLING FOR VIRTUAL SPACES
Diogo Gonçalo Franco Falcão Osório de Alarcão
(Licenciado)
Dissertação para obtenção do Grau de Doutor em
Engenharia Electrotécnica e de Computadores
Orientador: Doutor José Luís Bento Coelho
Júri
Presidente: Reitor da Universidade Técnica de Lisboa
Vogais: Doutor Luís Manuel Braga da Costa Campos Doutor Jorge Viçoso Patrício Doutor José Luís Bento Coelho Doutor João Manuel Domingues Perdigão Doutor Afonso Manuel dos Santos Barbosa
Novembro 2005 UNIVERSIDADE TÉCNICA DE LISBOA
INSTITUTO SUPERIOR TÉCNICO
ACOUSTIC MODELLING FOR VIRTUAL SPACES
Diogo Gonçalo Franco Falcão Osório de Alarcão
(Licenciado)
Dissertação para obtenção do Grau de Doutor em
Engenharia Electrotécnica e de Computadores
Orientador: Doutor José Luís Bento Coelho
Júri
Presidente: Reitor da Universidade Técnica de Lisboa
Vogais: Doutor Luís Manuel Braga da Costa Campos Doutor Jorge Viçoso Patrício Doutor José Luís Bento Coelho Doutor João Manuel Domingues Perdigão Doutor Afonso Manuel dos Santos Barbosa
Novembro 2005
ii
To my son Rodrigo and to my daughter Inês
iii Abstract
This thesis presents a new model for the sound energy propagation inside enclosed spaces. The new model is developed starting from the physical first principles and from rigorous mathematical definitions. The theoretical foundation allows the rigorous definition of the energy-based quantities used for describing sound energy propagation inside enclosures and the definition of a general energy balance equation for sound particles, applicable to any room. In addition, new theoretical tools for the study of sound energy fields in rooms are introduced, which are based on the language of functional analysis. The governing equations are thus expressed in terms of linear operators with very convenient properties, which are mathematically detailed.
In addition, a new combined method that solves the derived equations is presented. This new combined method uses an extended mirror image source method solving for the propagation of the specularly reflected energy components inside enclosures, and a time-dependent hierarchical radiosity method solving for the propagation of the diffusely reflected energy components. New algorithmic refinements are introduced in the computer implementation of this combined method. Finally, it is shown that the new proposed method is fast, flexible and accurate enough to be applied as an alternative method for room acoustics simulation.
Key words: room acoustics, sound particles, linear integral operators, diffuse and specular reflections, extended image source method, time-dependent hierarchical radiosity.
iv Resumo
A presente Tese descreve um novo modelo da propagação da energia sonora em recintos fechados. O novo modelo é desenvolvido a partir de primeiros princípios físicos e de definições matemáticas rigorosas. As fundações teóricas obtidas permitem definir de forma rigorosa as quantidades energéticas usadas na descrição da propagação de energia sonora dentro de salas bem como a definição de uma equação geral de balanço de energia para partículas sonoras, aplicável a uma sala qualquer. São introduzidas novas ferramentas teóricas para o estudo de campos de energia sonora em salas, baseadas na linguagem da análise funcional. As equação que governam a propagação de energia sonora são pois expressas em termos de operadores lineares com propriedades convenientes.
Adicionalmente, é apresentado um novo método combinado para a resolução das equações derivadas. Este método combinado utiliza o método das imagens estendido para a solução da propagação das componentes reflectidas especularmente e o método de radiosidade hiérarquica, dependente do tempo, para obter a solução da propagação das componentes reflectidas difusamente. São ainda apresentados vários aperfeiçoamentos utilizados nos algoritmos implementados do método combinado.
Finalmente, mostra-se que o novo método combinado é de computação rápida, fléxivel e suficientemente preciso estabelecendo-se num método alternativo para a simulação de acústica de salas.
Palavras chave: acústica de salas, partículas sonoras, operadores integrais lineares, reflexões difusas e especulares, método das fontes imagem estendido, radiosidade hierárquica dependente do tempo.
v ACKNOWLEDGMENTS
I would like to sincerely thank my advisor Professor J. L. Bento Coelho for giving me the opportunity of conducting this work under his supervison. I thank him for the continuous encouragement and unhesitating support during this endeavour and for the the help in the preparation of this thesis.
I must also thank my family, specially my wife Marta, for her love, understanding and patience at every step along the way. A very special thanks to my parents, especially to my mother, who was the first person that pressed me to start such a difficult task, giving me confidence and encouragement. Finally, to my young son and daughter, I thank them for their great patience when, many times, their father was wholly absorbed in his research and could not give them all the attention they deserve.
This work was partially financially supported by FCT – Portuguese Foundation for Science and Technology under the III QCA of the EU.
v i TABLE OF CONTENTS
Chapter 1 – Introduction...... 1 Chapter 2 – Basic Theory Concepts...... 7 2.1 Introduction...... 7 2.2 Physical Descriptors...... 8 2.2.1 Characterisation of Acoustic Disturbances...... 9 2.2.2 The Wave Equation...... 11 2.2.3 Intensity and Energy Density ...... 16 2.2.4 Harmonic and Non-harmonic Sound Waves...... 20 2.3 Sound Sources and Receivers...... 22 2.4 Sound Radiation...... 24 2.5 Sound Receivers...... 25 2.6 Propagation of Sound Waves...... 26 2.6.1 The Sound Field in Front of a Wall...... 27 2.6.2 Diffraction of Sound Waves ...... 39 2.6.3 The Attenuation of Sound in Air...... 41 2.6.4 Sound Absorbers...... 43 2.6.4.1 Membrane Absorbers...... 43 2.6.4.2 Porous Absorbers ...... 45 2.6.4.3 Resonant Absorbers ...... 46 2.6.4.4 People and Furniture...... 47 2.7 Sound Perception ...... 47 2.7.1 Fundamental Properties of the Human Ear ...... 48 2.7.1.1 Intensity Perception of the Auditory System. Thresholds...... 49 2.7.1.2 Equal Loudness Contours...... 50 2.7.1.3 Frequency Perception of the Auditory System...... 51 2.7.1.4 Critical Bandwidths and Masking...... 52 2.7.1.5 Time Perception...... 53 2.7.1.6 Spatial Sound Perception...... 53 Chapter 3 – Sound Fields in Enclosures...... 57 3.1 Introduction...... 57 3.2 The Wave Equation and Boundary Conditions...... 58 3.3 Natural Modes for the Rectangular Room ...... 59 3.4 Steady-state and Transient Sound Fields inside Enclosures...... 65 3.5 Reverberation Time...... 69 3.5.1 Time Distribution of Reflections...... 71 3.5.2 Spatial-Directional Characteristics of Reverberation. Diffuse Sound Fields. .. 72 3.5.3 Quantitative Measures for Sound Fields inside Enclosures...... 73 3.5.4 Subjective Evaluation of Room Acoustics...... 78 Chapter 4 – Acoustic Modelling of the Sound Field in Enclosures...... 81 4.1 Wave-based Methods...... 82 4.1.1 Analytic Methods ...... 84 4.1.1.1 Modal Decomposition ...... 84 4.1.1.2 Integral-based Methods. The Helmoltz-Huygens-Kirchhoff Theorem....85
v ii 4.1.2 Finite Element Formulations...... 86 4.1.2.1 The Finite Element Method (FEM)...... 86 4.1.2.2 The Boundary Element Method (BEM) ...... 87 4.1.2.3 Finite Difference Methods (FDM)...... 88 4.2 Geometrical Acoustics Based Methods...... 88 4.2.1 The Diffuse Field Theory...... 90 4.2.2 The Mirror Image Source Method (MISM)...... 95 4.2.3 The Ray Tracing Method...... 99 4.2.4 Beam, Cone and Pyramid Tracing ...... 102 4.2.5 Finite Sound Ray Integration Method (FSRIM) ...... 105 4.2.6 The Radiant Exchange Method. The Radiosity Method...... 105 4.2.7 Hybrid Methods ...... 109 Chapter 5 – A New Model for Sound Energy Propagation Inside Enclosures...... 111 5.1 Introduction...... 111 5.2 Sound Particles and Sound Energy ...... 114 5.2.1 Measure Spaces, Measures and Domains...... 114 5.2.2 Sound Particles and the Phase Space ...... 118 5.2.3 Particle Measures and the Phase Space Density...... 120 5.2.4 The Trajectory Space and Particle Events...... 123 5.2.5 The Trajectory Space Flux...... 126 5.3 Transport Equations...... 128 5.3.1 A Transport Equation for Particles...... 129 5.3.2 Boundary Conditions ...... 135 5.3.3 The Integral Version of the Transport Equation for Particles...... 137 5.3.4 Definition of Energy Quantities for describing Sound Fields...... 143 5.3.5 A General Transport Equation for Sound Energy...... 150 5.3.6 Steady-state Regime ...... 155 5.3.7 Time-dependent Regime...... 155 5.4 Functional Analysis Formalism...... 156 5.4.1 Linear Operator Formalism of the Governing Equations...... 158 5.4.1.1 Steady-state Regime ...... 161 5.4.1.2 Time-dependent Regime...... 162 5.4.2 Properties of the Operators ...... 162 5.4.2.1 Norms of the Operators...... 163 5.5 Approximate Solution Methods for the Boundary Integral Equation ...... 171 5.5.1 Truncated Neumann Series and Iterated Kernels...... 171 5.5.2 Successive Approximations...... 173 5.5.3 The Nyström Method ...... 173 5.5.4 Finite Basis Methods ...... 174 5.5.4.1 The Point Collocation Method...... 175 5.5.4.2 The Least Squares Method...... 175 5.5.4.3 The Galerkin Method...... 176 5.6 The Diffuse Reflection Case...... 177 5.6.1 Kuttruff’s Integral Equation ...... 180 Chapter 6 – A New Combined Method for the Prediction of Room Impulse Responses ...... 183 6.1 Equation of Motion for the Acoustic Energy inside an Enclosure ...... 184 6.2 The Transport and Reflection Operator...... 185
viii 6.3 Solution of the Specular Operator Equation...... 192 6.4 Solution of the Diffuse Operator Equation ...... 193 6.5 The Combined Method...... 197 Chapter 7 – Implementation of the New Combined Method ...... 199 7.1 Introduction...... 199 7.2 Geometric Representation of an Enclosure...... 200 7.3 The Combined Prediction Method...... 202 7.3.1 The Extended Mirror Image Source Method...... 203 7.3.1.1 Geometric Calculation of the Mirror Image Sources...... 203 7.3.1.2 Accelerating Techniques...... 207 7.3.1.3 Extension to High Orders of Reflection...... 211 7.3.1.4 Specular Room Impulse Responses ...... 216 7.3.2 The Time-dependent Hierarchical Radiosity Method...... 217 7.3.2.1 Hierarchy Construction and Representation ...... 218 7.3.2.2 Solution of the Time-dependent Hierarchical Radiosity Method...... 224 7.3.2.3 Obtaining Diffuse Room Impulse Responses...... 229 7.3.3 Combination of the Results from Both Methods...... 231 Chapter 8 – Validation and Application of the Combined Method...... 235 8.1 Measurement Setup...... 235 8.2 Case Studies: Acoustic Measurements vs. Simulation Results ...... 238 8.2.1 Room 1: VA2 of IST...... 238 8.2.1.1 Acoustic Measurements in Room VA2 ...... 239 8.2.1.2 Simulation Results for Room VA2...... 243 8.2.2 Room 2: classroom V007 of IST ...... 250 8.2.2.1 Acoustic Measurements in Room V007...... 252 8.2.2.2 Simulation Results for Classroom V007...... 255 8.2.3 Room 3: Meeting Room 01.1 of IST...... 262 8.2.3.1 Acoustic Measurements in Room 01.1 ...... 263 8.2.3.2 Simulation Results for Room 01.1...... 266 8.2.4 Room 4: Room C9 of IST...... 273 8.2.4.1 Acoustic Measurements in Room C9...... 274 8.2.4.2 Simulation Results for Classroom C9 ...... 277 8.2.5 Room 5: Congress Centre Auditorium of IST ...... 284 8.2.5.1 Acoustic Measurements in Congress Centre Auditorium...... 285 8.2.5.2 Simulation Results for Congress Centre Auditorium...... 290 Chapter 9 – Conclusions...... 299 9.1 Summary and General Conclusions...... 299 9.2 Final Conclusions ...... 301 9.3 Future Work ...... 302 References ...... 304 Appendix ...... 312
ix LIST OF FIGURES
Number Page 1. Relative attenuation of spherical wave above impedance plane (grass) compared to free field (dependence on frequency) 34 2. Relative attenuation of spherical wave above impedance plane (grass) compared to free field (dependence on source-receiver distance) 34 3. Modulus of scattering factor for a 10 cm x 10 cm patch with specific impedance 10 over a plane boundary with specific impedance 1, plotted in function of incidence angle and frequency 36 4. Squared modulus of scattering factor for a 10 cm2 rough patch 38 5. Three-dimensional representation of the directional scattering characteristics of a wall 39 6. Notation used for the diffraction of a plane sound wave by the edge of a rigid half plane 40 7. Mean squared pressure of diffracted plane wave from a rigid half-plane. 40 8. Air absorption coefficient plotted in function of sound frequency 43 9. Example of absorption coefficient for a panel with 80 Kg/m2 density, plotted as function of frequency 45 10. Example of absorption coefficient for a porous absorber 46 11. Example of absorption coefficient for a resonant absorber 47 12. Sketch of the human ear and cross section of the cochlea duct 48 13. Equal Loudness Contours 50 14. Example of a reverberation decay curve of a room 70 15. Echogram with temporal structure of the room energy impulse response 72 16. Example of geometrical construction of mirror image sources of an arbitrary room 96 17. Finite element formulation with constant basis functions 194 18. Validity and visibility test for potential mirror image sources 204 19. Example of the validity and visibility test for potential mirror image sources 205 20. Back-face Culling accelerating technique 208 21. Example of two impossible polygon combinations 208
x 22. Example of a view frustum for discarding subsequent higher order images 209 23. Example of clustering of input polygons into one single “parent” polygon 210 24. Example of least squares fit of the number of visible images 213 25. Example of least squares fit of the distance of visible images to a receiving point 214 26. Example of specular echograms calculated with the EMISM with statistical extension to higher orders 217 27. Polygons are substructured and interact at an appropriate level. The corresponding quadtrees are shown 219 28. Hierarchical subdivision and links at various levels 221 29. Example of ray-casting for determining the occluded form factor between two polygons 222 30. Example of the subdivision of a convex polygon with six sides in six child polygons 223 31. Input polygons for a room 229 32. Mesh of the room, after the hierarchical refinement 229 33. Specular energy impulse response obtained with the implemented EMISM 232 34. Diffuse energy impulse response obtained by the implemented time-dependent hierarchical method 233 35. Total energy impulse response obtained by the combined method 233 36. Acoustic measurements setup 236 37. Anechoic chamber measurements 237 38. Auditorium VA2, looking at the backside of the room 238 39. Auditorium VA2, looking at the front side of the room 239 40. Wireframe drawing of the model of Auditorium VA2 240 41. Coloured legend for the materials used in the model of room VA2 244 42. 1000 Hz Specular energy impulse response for combination S1-A, room VA2 (5 ms integrated; linear scale) 245 43. 1000 Hz Diffuse energy impulse response for combination S1-A, room VA2 (5 ms integrated; linear scale) 245 44. 1000 Hz Total energy impulse response for combination S1-A, room VA2 (5 ms integrated; linear scale) 246
xi 45. 1000 Hz Total energy impulse response for combination S1-A, room VA2 (5 ms integrated; logarithmic scale 246 46. 1000 Hz Schröder backwards total energy impulse response for combination S1-A, room VA2 (logarithmic scale). 247 47. View of classroom V007 251 48. Another view of classroom V007 251 49. Wireframe drawing of the model of classroom V007 252 50. Coloured legend for the materials used in the model of room V007 256 51. 1000 Hz Specular energy impulse response for combination S1-C, room V007 (5 ms integrated; linear scale) 257 52. 1000 Hz Diffuse energy impulse response for combination S1-C, room V007 (5 ms integrated; linear scale) 257 53. 1000 Hz Total energy impulse response for combination S1-C, room V007 (5 ms integrated; linear scale) 258 54. 1000 Hz Total energy impulse response for combination S1-C, room V007 (5 ms integrated; logarithmic scale) 258 55. 1000 Hz Schröder backwards total energy impulse response for combination S1-C, room V007 (logarithmic scale) 259 56. View of meeting room 01.1 towards the front 262 57. View of meeting room 01.1 towards the back 263 58. Wireframe drawing of the model of meeting room 01.1 264 59. Coloured legend for the materials used in the model of meeting room 01.1 267 60. 1000 Hz Specular energy impulse response for combination S1-C, room 01.1 (1/225 ms integrated; linear scale) 268 61. 1000 Hz Diffuse energy impulse response for combination S1-C, room 01.1 (1/225 ms integrated; linear scale) 268 62. 1000 Hz Total energy impulse response for combination S1-C, room 01.1 (1/225 ms integrated; linear scale) 269 63. 1000 Hz Total energy impulse response for combination S1-C, room 01.1 (1/225 ms integrated; logarithmic scale) 269 64. 1000 Hz Schröder backwards total energy impulse response for combination S1-C, room 01.1 (logarithmic scale) 270
xii 65. Classroom C9, looking at the frontside of the room 273 66. Classroom C9, looking at the backside of the room 274 67. Wireframe drawing of the model of classroom C9 275 68. Coloured legend for the materials used in the model of classroom C9 277 69. 1000 Hz Specular energy impulse response for combination S1-A, classroom C9 (1/150 ms integrated; linear scale) 279 70. 1000 Hz Diffuse energy impulse response for combination S1-A, classroom C9 (1/150 ms integrated; linear scale) 279 71. 1000 Hz Total energy impulse response for combination S1-A, classroom C9 (1/150 ms integrated; linear scale) 280 72. 1000 Hz Total energy impulse response for combination S1-A, classroom C9 (1/150 ms integrated; logarithmic scale) 280 73. 1000 Hz Schröder backwards total energy impulse response for combination S1-A, classroom C9 (logarithmic scale) 281 74. Congress Centre Auditorium, looking at the frontside of the room 284 75. Congress Centre Auditorium, looking at the backside of the room 285 76. Wireframe drawing of the model of the Congress Centre Auditorium 286 77. Coloured legend for the materials used in the model of the Congress Centre Auditorium 291 78. 1000 Hz Specular energy impulse response for combination S1-E, Congress Centre Auditorium (5 ms integrated; linear scale) 292 79. 1000 Hz Diffuse energy impulse response for combination S1-E, Congress Centre Auditorium (5 ms integrated; linear scale) 292 80. 1000 Hz Total energy impulse response for combination S1-E, Congress Centre Auditorium (5 ms integrated; linear scale) 293 81. 1000 Hz Total energy impulse response for combination S1-E, Congress Centre Auditorium (5 ms integrated; logarithmic scale) 293 82. 1000 Hz Schröder backwards total energy impulse response for combination S1-E, Congress Centre Auditorium (logarithmic scale) 294
xiii LIST OF TABLES
Number Page 1. Audible ranges of the human ear 49 2. Standard centre, lower and upper frequencies for octave bands 52 3. Intensity levels measured in anechoic chamber for the Meyer Sound UPM-1 loudspeaker 237 4. Acoustic power of the sound source UPM-1 238
5. Measured T30 values - room VA2 241 6. Measured EDT values - room VA2 241
7. Measured Definition values D50 - room VA2 242
8. Measured Clarity values C80 - room VA2 242
9. Measured Lp values - room VA2 243
10. Reverberation times T30 predicted by the combined method - room VA2 247 11. Early decay times EDT predicted by the combined method - room VA2 248
12. Definition values D50 predicted by the combined method - room VA2 248
13. Clarity values C80 predicted by the combined method - room VA2 248
14. Steady-state sound pressure level Lp predicted by the combined method - room VA2 248
15. Difference between predicted and measured values of T30 - room VA2 249 16. Difference between predicted and measured values of EDT - room VA2 249
17. Difference between predicted and measured values of D50 - room VA2 249
18. Difference between predicted and measured values of C80 - room VA2 250
19. Difference between predicted and measured values of Lp - room VA2 250
20. Measured T30 values - room V007 253 21. Measured EDT values - room VA2 253
22. Measured Definition-D50 values - room V007 254
23. Measured Clarity-C80 values – room V007 254
24. Measured Lp values - room V007 255
25. Reverberation times T30 predicted by the combined method - room V007 259 26. Early decay times EDT predicted by the combined method - room V007 259
27. Definition values D50 predicted by the combined method - room V007 260
xi v 28. Clarity values C80 predicted by the combined method - room V007 260
29. Steady-state sound pressure level Lp predicted by the combined method - room V007 260
30. Difference between predicted and measured values of T30 - room V007 261 31. Difference between predicted and measured values of EDT - room V007 261
32. Difference between predicted and measured values of D50 - room V007 261
33. Difference between predicted and measured values of C80 - room V007 261
34. Difference between predicted and measured values of Lp - room V007 262
35. Measured T30 values - room 01.1 264 36. Measured EDT values - room 01.1 265
37. Measured Definition-D50 values - room 01.1 265
38. Measured Clarity-C80 values – room 01.1 265
39. Measured Lp values - room 01.1 266
40. Reverberation times T30 predicted by the combined method - room 01.1 270 41. Early decay times EDT predicted by the combined method – room 01.1 271
42. Definition values D50 predicted by the combined method – room 01.1 271
43. Clarity values C80 predicted by the combined method – room 01.1 271
44. Steady-state sound pressure level Lp predicted by the combined method – room 01.1 271
45. Difference between predicted and measured values of T30 – room 01.1 272 46. Difference between predicted and measured values of EDT – room 01.1 272
47. Difference between predicted and measured values of D50 – room 01.1 272
48. Difference between predicted and measured values of C80 – room 01.1 272
49. Difference between predicted and measured values of Lp – room 01.1 273
50. Measured T30 values - room C9 275 51. Measured EDT values – room C9 276
52. Measured Definition-D50 values – room C9 276
53. Measured Clarity-C80 values – room C9 276
54. Measured Lp values – room C9 277
55. Reverberation times T30 predicted by the combined method – room C9 281 56. Early decay times EDT predicted by the combined method – room C9 282
57. Definition values D50 predicted by the combined method – room C9 282
58. Clarity values C80 predicted by the combined method – room C9 282
59. Steady-state sound pressure level Lp predicted by the combined method – room C9 282
xv 60. Difference between predicted and measured values of T30 – room C9 283 61. Difference between predicted and measured values of EDT – room C9 283
62. Difference between predicted and measured values of D50 – room C9 283
63. Difference between predicted and measured values of C80 – room C9 283
64. Difference between predicted and measured values of Lp – room C9 284
65. Measured T30 values – Congress Centre Auditorium 287 66. Measured EDT values – Congress Centre Auditorium 288
67. Measured Definition-D50 values – Congress Centre Auditorium 288
68. Measured Clarity-C80 values – Congress Centre Auditorium 289
69. Measured Lp values – Congress Centre Auditorium 289
70. Reverberation times T30 predicted by the combined method – Congress Centre Auditorium 294 71. Early decay times EDT predicted by the combined method – Congress Centre Auditorium 295
72. Definition values D50 predicted by the combined method – Congress Centre Auditorium 295
73. Clarity values C80 predicted by the combined method – Congress Centre Auditorium 295
74. Steady-state sound pressure level Lp predicted by the combined method – Congress Centre Auditorium 296
75. Difference between predicted and measured values of T30 – Congress Centre Auditorium 296 76. Difference between predicted and measured values of EDT – Congress Centre Auditorium 296
77. Difference between predicted and measured values of D50 – Congress Centre Auditorium 297
78. Difference between predicted and measured values of C80 – Congress Centre Auditorium 297
79. Difference between predicted and measured values of Lp – Congress Centre Auditorium 297 80. Octave bands Sabine absorption coefficients 312
xvi LIST OF SYMBOLS
Symbol Page
p0 Ambient pressure 7 p Acoustic pressure 8
pinst. Instantaneous pressure 8
Lp Sound pressure level 8
pref Standard reference pressure 8 i Vector 8 v Particle velocity 9 ρ Fluid density 9
ρ0 Ambient fluid density 9 V Volume 9
SS Surface (area) 9 ∇i Divergence 10 c Speed of sound 11 ∇2 Laplacian 12 Φ Velocity potential 12 k Wave number 13 f Sound frequency 13 ω Angular frequency 13 λ Wavelength 13 T Period 13 ˆi Complex quantity 14 r Radial coordinate 15 I Acoustic intensity 16 w Acoustic energy density 16 n Unit normal vector 17
xvii i Expected value; time average 18
i Absolute value 18
∇ Gradient in direction of increasing r 18 r i* Conjugate 18 Re[i] Real part 19 e Unit vector in direction of increasing r 19 r Im[i] Imaginary part 21 S()ω Fourier spectrum 21 Ψ()ω Fourier phase 21 Π()ω Power; Power spectrum 21 θ,ϕ Angles 24 Γ(,,θ ϕω ) Directivity function 24 J (,,θ ϕω ) Radiation pattern function 25
Zˆ(,ω Ω ) Wall impedance at frequency ω for sound waves incident from direction Ω 27 R(,ω Ω ) Wall resistance at frequency ω for sound waves incident from direction Ω 27 X (,ω Ω ) Wall reactance at frequency ω for sound waves incident from direction Ω 27 ˆ ˆ ζω(,Ω ) Specific wall impedance Z(,ω Ω )ρ0c 28
Rˆ(,ω Ω ) Complex plane wave pressure reflection factor 28
θir,θ Incident direction, reflected direction (polar representation) 28 α(,ω Ω ) Absorption coefficient, at frequency ω , for waves incident from direction Ω 29 erfc Complementary error function 33 σ Effective flow resistivity 33 ˆ Ψ(,θiiϕθϕ ,,) Scattering factor 35
βˆ Specific admittance 35
22 Wξ ξ Correlation length of random fluctuation 36
Iir, I Absolute value of the incident intensity, of the reflected intensity 37
xviii i Norm 37
h Relative humidity 42 I Absolute value of the intensity 42
mT(,ω Celsius ,) h Air absorption coefficient (abbreviated: m()ω ) 42
att(,ω TCelsius ,) h Air attenuation (in dB Km ) 42
M p Panel mass density 44 b Plate thickness 44 Ξ Flow resistivity 45 d Width of enclosed space 46 ωˆ =+ωδi Complex angular frequency 58 δ Damping constant 59 κγˆ =+ki Eigenvalue (characteristic value) 60
Ls Length (coordinate s ) 60
Nfmodes () Number of eigenmodes up to frequency f 64 dN modes ()f Eigenmodes density at frequency f 65 df
δ NM Kronecker symbol 65 Q Volume velocity strength 66 δ ()rr− Dirac delta function 66 0 G(,rr ,ω ) Green’s function 66 0
fSch Schröder frequency 67
δ Mean of damping factors 67 gt(,rr ,) Impulse response (pressure) 67 0 Ψ(,rr ,)t Signal response (pressure) 68 0
T60 Reverberation time 69 D()e Directional energy density 72
T15 Reverberation time (measured from -5dB to -20dB) 73
xix T20 Reverberation time (measured from -5dB to -25dB) 73
T30 Reverberation time (measured from -5dB to -35dB) 73 Et() Decay curve 74 ht() Energy impulse response 74 EDT Early decay time 74
D50 Definition 74
C50 Speech clarity index 75
C80 Music clarity index 75 G Relative sound level 75
tI Initial-time-delay gap 75
tS Centre-of-gravity time 76 BR Bass ratio 76 LEF Lateral energy fraction 76 IACC Interaural cross correlation coefficient 76 STI Speech transmission index 77 ST Stage parameters 77 A()ω Absorption at frequency ω 91 α()ω Mean absorption coefficient at frequency ω 93
αi ()ω Area-averaged random-incidence energy absorption coefficient at frequency ω 93 M Number of walls 97 K Reflection order 97 ∆()ω Diffusivity coefficient 98 rr; Source’s position vector; Receiver’s position vector 98 SR NR Number of rays 100 B “Irradiation strength” (after [Kuttruff, 1979]) 106 (,Γ P ,)ρ Measure space 114 i Measure 114 M Two-dimensional manifold; Boundary of enclosure 115 σ Surface area measure (Lebesgue) 115
xx V Volume measure (Lebesgue) 115 Ω = (,θ ϕ ) Unit length vector in R3 , representing directions 116 S2 Unit sphere 116 χ Subset of the unit sphere 116
⊥ σs Projected solid angle measure (at point s ) 117 T ()s Tangent space (at point s ) 117 M HH22();ss () Positive; negative hemisphere (at point s ) 117 +− Γ Phase space 120 ρΓ Phase space measure 120 l Length measure (Lebesgue) 120 N Particle content measure 121 Absolute continuity 121 n Phase space density 121 Ψ Trajectory space 123 ρΨ Trajectory space measure 123 IP Particle event space 124 ν Trajectory space density 125 N (,)ω t Total number of particles having internal attribute ω at time t 126 ψ Trajectory space flux 126 q Trajectory space source function 129 Et(,)ω Particle emission function 129 St(,)ω Streaming function 130 σ (,r ω ) Particle absorption coefficient 131 A
CtA (,)ω Particle absorption rate 131 κ Volume scattering function 131
CtOUT (,)ω Out-scattering function 132
CtIN (,)ω In-scattering function 132 σ (,r ω ) Particle out-scattering coefficient 133 S
xxi σ (,r ω ) Particle total extinction coefficient 134
ϒϒIN; OUT Inwards; outwards-orientated rays 135
κ B Surface scattering coefficient 136 ρ (,s ω ) Surface transparency coefficient 136 B L[]()i p Laplace operator 137 d (,r Ω ) Boundary distance function 140 M L−1[]()i p Inverse Laplace operator 141
ξ(,rr' ,ω ) Total extinction function 142 E Energy content measure 143 ε Energy content density 144 ρIP Particle event space measure 144 β = β (,s Ω ) Acoustic angular power flux 145
E p Particle energy 146
B ()r Acoustic power flux (at r into direction Ω ) 147 Ω B()s Acoustic radiosity (at s ) 148
ϒS Wall scattering function (WSF) 148
ϒR Wall reflection function (WRF) 149
ϒT Wall transmission function (WRF) 149
RΩ Directional-spherical surface scattering 150
RH Directional-hemispherical surface reflectivity 150 σ (,r ω ) Energy distance attenuation factor 151 β
κ β Energy volume scattering function 151 Q Spectral angular power density function 151
()F, i Normed linear space 156
V Volume enclosed by M 156
2 Lp ()M ×S Lebesgue p-spaces 157
xxii i p p-norm 157
i ∞ Infinite-norm 157 S Scattering operator 159 P Propagation operator 159 ss(,Ω ) Ray-casting function 160 M T Transport operator 160
SL Laplace-conjugated scattering operator 160
PL Laplace-conjugated propagation operator 161
TL Laplace-conjugated transport operator 161 ρ (,s Ω ,ω ) Diffuse wall reflectivity function 177 D D RH Diffuse directional-hemispherical surface reflectivity 177 φ(,s ω ,)t Incident power 181 W dtn ⎡⎤ss→ ' Transition amplitude 184 dtn+1 ⎣⎦ T Transport and reflection operator 186 δ ()xx− Generalized Dirac function 188 µ 0 ∆(,s Ω ,ω ) Surface diffusivity function 188
T D Diffuse transport and reflection operator 189
T S Specular transport and reflection operator 189
vis(,ss' ) Visibility function 193
Pj Patch 194
dPP()ji, Area-averaged mean distance between Pj and Pi 195 T D Discrete diffuse transport and reflection operator 195
WPPτ ⎡⎤→ Area-averaged mean transition amplitude 196 τ +dtn ⎣⎦ j i
FP()ij→ P Form factor between Pj and Pi 196
Wi Polygon 201 v Polygon vertex 201 ij xxiii {,xijyz ij ,} ij Coordinates of vertex i of polygon j 201
dp() Wi Distance of polygon Wi to the origin 203 NK() Number of valid and visible images, per reflection order 214 DK() Mean distance to the receiver of visible images of order K 214
σ D ()K Standard deviation of the distances of visible images of order K 214 ρ(,)K ω Mean reflection coefficient for reflections of order K 214
σ R (,)K ω Standard Deviation of reflection coefficients of order K 214
1-∆ (K ,ω ) Mean specularity coefficient for reflections of order K 214
σ S (,)K ω Standard Deviation of the specularity coefficients of order K 214
A∆ Area threshold 220 FP()pq→ P Form factor estimate 221
F∆ Form factor threshold 222
xxi v
Chapter 1
INTRODUCTION
Architectural acoustics, or room acoustics, is the science of the acoustics of enclosed spaces. This science deals with the acoustical design of new rooms and with the study of the acoustic behaviour of existing rooms.
The acoustical signature of lecture rooms, concert and opera halls, drama performance theatres, and churches is quite different, since the sound propagation inside enclosures depends largely on the room shape and on the acoustical properties of its boundaries. Several complex physical mechanisms condition the sound propagation in rooms and are responsible for the resulting acoustic character. In a very simplified way, enclosed spaces can then be classified as having “good” or “poor” acoustics. This classification, being completely subjective, depends largely on the individual taste and education of the particular user of the enclosed space.
An important task of architectural acoustics consists on correlating the subjective acoustic impressions on the listeners to the objective knowledge of specific physical quantities involved in the characterisation of the sound fields that build up inside enclosures. The connection between the subjective acoustical impression of a room and well-established physical principles has been rather difficult to define until recent times. It is interesting to remember, for example, the difficult work of identifying in the past, the specific objective attributes that a room should meet in order to have “good” acoustics. The renowned architect Charles Garnier, author of the famous opera house in Paris, writes in [Garnier, 1878]: “I have conscientiously studied all this in the course of many months; I have read works in languages familiar to me and have had translated for me all that was published in tongues outside my range; I have sought other people’s views in conversation and discussion, and as a result I have made the following discovery: For a hall to have excellent acoustic properties it must either be long or wide, high or low, either of wood or stone, be round or square, have bare or covered walls, be floating or solidly based, involved or simple, hot or cold, empty or crowded, dark or illuminated. I have learned that some would have trees planted in the hall, some have it built entirely in crystal, while others contended that snow was the best sound conductor and
1 the wall, therefore, should be lined with artificial snow, and that some, finally, fell back on Vitruvius and insisted that urns be placed under the seats.”
It is commonly accepted that it was only with the systematic work of Wallace Clemens Sabine around the beginning of the twentieth century that architectural acoustics was established as a true science. The first objective physical parameters, which correlated well with the subjective acoustic impression, were defined, the most important being the reverberation time and the absorption of the rooms [Sabine, 1922]. However, systematic studies on the subjective effects of sound fields were only conducted after the second half of the twentieth century [Beranek, 1992]. These studies allowed the definition of additional objective acoustic parameters that more or less translate the subjective acoustic characteristics exhibited by different rooms.
The results of these studies showed that in order to be able to forecast the acoustics of a room or to completely understand the acoustics of existing ones, one necessarily needed better models of the sound fields inside enclosures compared to the simple Sabine model or its extensions (such as the Eyring model for example, see Chapter 3, section 3.2.1). The so-called wave theory of sound was by that time completely developed, but as mentioned in Chapter 2, its practical application is restricted to the simplest shapes of rooms. Moreover, even in this case, the mathematical complexity is so high that usually the computation is not feasible.
With the advent of digital computers and its widespread use after the 1960s, physicists and acousticians were offered a new powerful and flexible tool that could be applied for determining more rigorously the sound fields that build up inside enclosed spaces. New models were developed and implemented (and some “old” models could be finally implemented in practice), which simulated to a good degree of accuracy the sound propagation in these spaces. However, although much more exact than the simple “Sabine-based” models, the numerical methods still had their shortcomings. The introduction of the personal computer in the 1980s brought the access of computation power at a reduced cost almost to everyone. Computer based simulation models were allowed to develop more rapidly, and until today, this is a still fast growing research area.
The simulation of sound fields inside enclosures with walls of given acoustic properties can be accomplished by solving the acoustic wave equation in its linearised form and adopting the corresponding boundary conditions, as will be mentioned in Chapter 2. Solving the wave equation
2 analytically is only possible for highly idealised (and simple) situations. Therefore, several wave- based numerical methods were developed in order to obtain the solutions for sound fields inside enclosures (a review of the different modelling approaches of sound fields in rooms is presented in Chapter 4). However, the computation time taken by these methods is usually very high.
Other approaches made use of the so-called geometrical acoustics theory, where approximate solutions in the time or frequency domain can be obtained (see Chapter 4, section 4.2). In particular, the impulse responses of rooms can be easily computed, from which a variety of objective acoustic parameters that correlate well with the subjective acoustic impressions can be obtained (see Chapter 3, section 3.5.3).
For most practical purposes in room acoustics, it is only necessary to simulate how the sound energy propagates inside enclosed spaces. The resulting energy-based methods are usually sufficient to obtain accurate predictions of the acoustics of these spaces and the underlying calculations can be considerably simplified when compared to pressure-based methods. In fact, when the acoustic superposition of waves can be thought as being accomplished by the addition of incoherent components, then energy-based methods can be applied in a diverse range of problems. It is interesting to note that most of the objective acoustic parameters, which will be described in detail in Chapter 3, section 3.5.3, can be calculated from the energy impulse response of the rooms under analysis. Therefore, the justification for using energy-based methods for the simulation and prediction of sound fields inside enclosures gains an increased validity.
Energy-based models have been used in room acoustics for some time. Particularly, numerical energy-based methods have been implemented to some extent in the computer and the results show in general a good accordance with measured quantities. However, these energy-based models were introduced without a rigorous treatment of both its underlying physical assumptions and its mathematical framework.
This thesis presents a new model for the sound energy propagation inside enclosed spaces. The new model is developed starting from physical first principles and from the rigorous mathematical definition of the necessary quantities. The theoretical foundation allows the rigorous definition of the required energy-based quantities used for describing sound energy propagation inside enclosures and the definition of a general energy balance equation, which applies to any room. In
3 addition, new theoretical tools for the study of sound energy fields in rooms are presented, which are based on the mathematical language of functional analysis. The governing equations are thus expressed in terms of linear operators with very convenient properties, which will be mathematically detailed.
In addition, a new combined method that allows solving the derived equations is presented. This new combined method uses an extended mirror image source method (see Chapter 4, section 4.2.2) for solving the propagation of the specularly reflected energy components inside the enclosures, and a time-dependent hierarchical radiosity method for obtaining the solution for the propagation of the diffusely reflected energy components.
New improvements and algorithmic refinements are introduced in the computer implementation of this combined method.
Finally, it will be shown that this new proposed method is fast, flexible and accurate enough in order to be applied as an efficient method for room acoustics simulation.
This thesis is organised in nine chapters followed by a List of References:
Chapter 2 describes background theory concepts used in acoustics, with a particular emphasis on both the physical and the subjective descriptors.
Chapter 3 reviews the problem of sound fields in enclosures. This chapter starts with the correct wave-based treatment of the sound fields in rooms, portraying the main aspects of such approach. The important concept of reverberation is also dealt with and the most relevant quantitative and qualitative acoustic parameters for describing particular characteristics of the acoustics of enclosed spaces are mentioned and defined.
A comprehensive review of existing modelling approaches for room acoustics is presented in Chapter 4. This review results from an extensive bibliographic search on room simulation methods that were reportedly implemented on the computer.
4 In Chapter 5, the new model for sound energy propagation inside enclosures is presented. The underlying physical and mathematical framework is described and the resulting theoretical foundation is used to derive the equations, which govern this new model.
Chapter 6 introduces a new combined method that allows solving the particular equations of the new model. The combined method is derived rigorously and an analysis of the assumptions made in the derivation is presented.
The computer implementation of the new combined method is presented in detail in Chapter 7. Particularly, the algorithmic structure used in the implementation and the improvements and refinements adopted for the algorithms are portrayed.
Acoustic measurements carried out in a particular set of rooms are described together with results in Chapter 8. The same rooms were modelled in the computer and the corresponding impulse responses and acoustic parameters were predicted by the new combined method. Comparisons between measured and predicted values are presented.
Finally, Chapter 9 contains the conclusions of this work.
5
6 Chapter 2
BASIC THEORY CONCEPTS
2.1 Introduction Acoustics is the science of sound, including its production, transmission, and effects [Pierce, 1994]. In its broad scope, acoustics studies not only the phenomena responsible for the sensation of hearing, but it embraces also the study of related phenomena derived from analogous principles. Acoustic disturbances are different from other physical disturbances due to the mechanical characteristic of the sound.
In the case of phenomena occurring in gaseous mediums, sound is normally defined as the result of a disturbance in the ambient pressure, p0, derived from elastic vibrations of the particles in the medium, when the vibrations are of such magnitude and frequency that the disturbances can be heard by an average human ear or recorded by a measuring instrument. As the individual parts of the medium are coupled by elastic forces, it follows that a displacement of one part of this medium will alter the elastic values between one component and its neighbours, so that the later part will also undergo a displacement. This contiguous displacement will again produce displacement of fresh components, and in this way, the primary displacement will be transferred throughout the entire medium thus allowing the propagation of the acoustic disturbance up to considerable distances from the point of origin of the primary disturbance, the sound source.
The physical interpretation and related mathematical translation of sound as a mechanical disturbance of an elastic medium led to the development of the wave-based theory of sound. That sound could be interpreted as a wave phenomenon is an ancient belief, first documented in the works of the Greek philosopher Chrysippius, and later by Roman philosophers such as Aristotle. During the renaissance period, this assumption was further adopted, and, it was Newton, who began the mathematical theory of sound propagation as a wave phenomenon. After that, the work of Euler, Lagrange and d’Alembert developed a viable theory of sound propagation resting on firmer mathematical and physical concepts. Towards the final of the nineteenth century, the solid scientific work of Lord Rayleigh and von Helmholtz, among others, permitted to establish the
7 modern foundations of acoustic theory governing the propagation of sound, which has been further developed and refined during the twentieth century.
2.2 Physical Descriptors The adoption of sound being a manifestation of a mechanical disturbance in elastic media, such as air, introduces several physical descriptors originated in the fields of continuum fluid mechanics, field theory, and thermodynamics for the thorough characterisation and quantification of the underlying wave phenomenon.
Pressure, particle velocity, and fluid density appear as the basic physical quantities for describing an acoustic disturbance.
The acoustic pressure p is defined as the difference between the instantaneous pressure pinst. and
−2 the static ambient pressure of the medium, p0 . Its units are []Nm , which corresponds to one Pascal []Pa . The acoustic pressure is the most important quantity in describing acoustic disturbances since most of the sound receivers, particularly the human ear, are sensible to pressure. In general, the spatial temporal function corresponding to the variations in the acoustic pressure is called the sound field.
The dynamic range of the sound pressure to which the human ear is sensible is extremely vast, and therefore, it would be impractical to characterise the strength of a sound signal by its pressure. Due to this fact, it is a standard procedure to express the sound pressures using a logarithmic scale, the decibel scale, which defines the sound pressure level of an acoustic disturbance:
p2 LdBp =10log10 2 [] (1) pref
2 where p is the mean squared pressure of the disturbance and pref is a standard reference pressure, with a value of 210× −5 Pa for airborne sound, which correspondents roughly to the human hearing threshold for a signal with a frequency of 1000 Hz.
8 The particle velocity, v , given in units of []ms−1 , describes the instantaneous velocity of a fluid particle, which consists of all fluid contained within some moving volume. Each surface point of this volume is moving with the local fluid velocity. The particle velocity is thus defined as the mass- weighted local average particle velocity, or, as the local average momentum per unit mass.
The fluid density ρ at some point is defined as the local average of mass per unit volume calculated for a fixed volume inside the fluid, its units being []Kgm−3 .
Acoustic disturbances can be seen as disturbance of an ambient state of the propagation medium as mentioned above. For a fluid such as air, the ambient state is completely determined by the triple of values p , v and ρ , which quantify the pressure, fluid velocity and fluid density when there is 00 0 no acoustic disturbance present. The triple of values (,,)p v ρ satisfies the governing fluid- 000 dynamic state equations for the medium. When there are contributions from a sound wave, then these must be added to the ambient state values in order to obtain the new dynamic state of the fluid.
In the next sections, focus will be given to the underlying physical descriptors that allow the correct quantification and characterisation of a sound wave in air.
2.2.1 Characterisation of Acoustic Disturbances From the continuum physics point of view, acoustic disturbances in fluids such as air are described completely and univocally through the combined solution of three different physical laws: these are the law of conservation of mass, the law of conservation of momentum and the adiabatic pressure- density relationship.
The law of conservation of mass states that the time-rate-of-change of the mass inside some fixed volume located inside the fluid must equal the net mass per unit time entering and exiting the volume through the bounding surface. In integral form, this law can be written as:
d ρρ(,)rrvrnrtdV=− (,)(,) t ti () dS (2) dt ∫∫∫ ∫∫ S VSS
9 where nr()is the outward unit normal vector at some surface point. The law of conservation of mass can also be stated in differential form by applying Gauss’ Theorem (also know as Divergence Theorem) to equation (2), yielding:
∂ρ(,)r t + ∇=i()ρ(,)(,)rvrtt 0 (3) ∂t
The law of conservation of momentum is the translation of Newton’s second law for dynamic-fluid variables. This law, also known as Euler’s equation of motion for a fluid, is based on two physical assumptions. The first one asserts that the mass times acceleration of the centre of mass of a fluid particle equals the net force inflicted on it by the surrounding environment and by external forces. The second assumption is that the surface force inflicted by the fluid particle’s surrounding environment is directed towards the normal direction into the surface bounding the same fluid particle and that the external forces (for example gravity or the electromagnetic Lorentz force) are negligible. Taking both assumptions into account, the law of conservation of momentum can be stated as:
dtvr(,) ∇=−pt(,)rrρ (,) t (4) dt for an ideal fluid without consideration of viscosity effects. Note that since the gradient of the pressure points into the direction of increasing pressure, the resulting fluid particle acceleration is directed towards decreasing pressure. This mathematical statement translates the necessary physical principle that the restoring force acting on a disturbed fluid particle tends to bring it again to equilibrium. Without this principle, acoustic disturbances would grow indefinitely, and no sound waves would be possible to occur.
The third and final law, necessary for the description of an acoustic disturbance, is the adiabatic pressure-density relationship, which states that for a compressible fluid there exists a specific functional relation between fluid density and pressure. When the displacement of air by a disturbance is fast enough, such as in a sound wave, so fast that the heat resulting from the compression of the fluid particles does not have time to dissipate between wave cycles, the
10 phenomenon is called adiabatic compression. The resulting pressure-density relationship then takes the specific form:
p(,)rrt=× const .ρ (,) t γ (5) with γ equal to the ratio of the specific heat coefficients at constant pressure and at constant volume. The multiplicative factor const. stands for a numerical constant, which depends on the particular propagation medium and is equal to the ratio between ambient pressure and ambient density. The value of γ for air is 1.4.
2.2.2 The Wave Equation The three fundamental fluid-dynamic laws presented in the last section completely describe the behaviour of any acoustic disturbance in an inviscid fluid without the action of external forces. However, their application to solve any acoustic phenomenon of sound propagation is very difficult, mainly because the laws of conservation of mass and of conservation of momentum are non-linear equations. Therefore, some simplifications must be undertaken in order to obtain, from the mathematical point of view, more tractable equations. These simplifications are accomplished by linearising the above-mentioned laws and by adopting a Taylor-series development for the pressure-density relationship. The so-called linear acoustic equations then result:
∂ρ(,)r t +∇ρ ivr(,)t = 0 ∂t 0 ∂vr(,)t ∇=−pt(,)r ρ (6) 0 ∂t
22⎛⎞∂p pt(,)rr== cρ (,) t c ⎜⎟ ⎝⎠∂ρ 0 where now all quantities represent the acoustic contributions due to the disturbance to the total fluid quantities. These equations apply to a homogeneous and quiescent medium. Thermodynamic considerations require that the constant c2 introduced in equation (6) to be positive defined. This constant equals the value for the speed of sound in air. The speed of sound in air is temperature dependent. An approximate relationship between the two is given by the following expression, valid for dry air:
11 ⎡ −1 ⎤ cTms=+331 0.6 Celsius ⎣ ⎦ (7)
The increase of the relative humidity of air has only a negligible effect on the speed of sound.
The three linear equations can be combined into a single differential equation with one dependent variable yielding the so-called wave equation:
1(,)∂2 p r t ∇=2 pt(,)r (8) ct22∂ where the operator ∇2 equals the divergence of the gradient and is defined as the Laplacian operator.
The homogeneous wave equation (8) is of undeniable importance not only in acoustics but also in a variety of fields where analogous wave phenomena occur. It was first derived for the one- dimensional case by d’Alembert in 1747 for solving the problem of the vibrating string. Its three- dimensional derivation for the case of acoustic disturbances in air was subsequently derived by Euler and Lagrange in 1759 and 1760. In equation (8), the dependent variable is the acoustic pressure. However, the same equation is valid for the fluid density and for the divergence of the particle velocity.
In addition, one can define a velocity potential Φ through the following equations:
∂Φ(,)r t vr(,)tt=∇Φ (,) r ; pt (,) r =−ρ (9) 0 ∂t since the velocity field is irrotational. In this case it is also simple to see that the velocity potential satisfies the same wave equation (8). This velocity potential is a mathematical abstraction, but also a convenient quantity to describe an acoustic field in terms of a single variable that can be used to derive all physical quantities.
12 Plane waves
At a sufficiently large distance from a sound source, the wave fronts become almost plane wave fronts. In this case, all field quantities vary with time and with some Cartesian coordinate s, but are independent of position along planes normal to the s direction. The wave equation will reduce to an equation only for the coordinate s :
∂Φ22(,)st 1 ∂Φ (,) st = (10) ∂∂sct222
Progressive Wave Solution
The general solution of the one-dimensional wave equation, first obtained by d’Alembert, can be written as:
Φ=(,)()(st f12 ct −++ s f ct s ) (11)
where the functions f1 and f2 are arbitrary functions, not necessarily analytic. The s coordinate of the points for which the function f1()ct− s has the same value propagates in the positive s direction with the sound speed c. Similarly, the s coordinate of the points for which the function
f2 ()ct+ s has a constant value propagates into the negative s direction with velocity c. Therefore, the first function in equation (11) represents a progressive wave travelling in the positive coordinate direction, while the second function represents a regressive wave travelling in the negative s direction. Solution (11) is complete, since it contains two arbitrary functions, and it is particularly suited for studying wave propagation, i.e. for studying the evolution of an initial acoustic disturbance over the underlying medium.
A particular solution given by equation (11) is obtained by considering harmonic dependence of the velocity potential, with a specific frequency f. Introducing the wave number kfcc===22π ωπλ with λ being equal to the spatial wave length, meaning the distance the disturbance travels during a temporal period given by Tf= 1 , and since both functions in equation (11) are arbitrary, the harmonic progressive wave solution is necessarily of the form:
13 Φ=(,)s t A cos(ω t −++ ksφωφ00 ) B cos( t ++ ks ) (12) being the sum of a progressive and of a regressive harmonic plane wave. In equation (12) three numerical constants are used: φ0 represents an initial phase delay of the wave, and A and B are determined from the initial conditions. This expression can also be written in complex form as:
Φ=(,)st Aeˆ itiksωω−+ + Beˆ itiks (13) where i denotes the imaginary unit and where the standard convention consists of retaining only the real part of the complex expression in order to have the admissible physical solution.
Standing Wave Solution
Under the assumption that the system is linear and causal, then the solution of the wave equation for the velocity potential can be put into a separable form. Therefore, Φ(,)st=Σ () s Τ () t , and the one-dimensional wave equation (10) becomes:
1()1()∂22Τ∂Σts = (14) ct22Τ∂() t Σ () s ∂ s 2
The general solution of this equation can be easily shown to be:
Φ(,)st=Σ () s Τ () t = C cos( ks +φs )cos(ωφ t + t ) (15)
where C is the amplitude of the wave and φs ,φt are phase delays of the functions Σ and Τ , respectively.
The solution portrayed by this last equation is called a standing wave or natural vibration of the acoustic system, since now the temporal behaviour consists only in increasing or decreasing the spatial displacement determined by the harmonic function Σ()sC=+ cos( ksφs ).
Central Spherically Symmetric Waves
14 In addition to a plane wave, another common idealization of an acoustic disturbance is a spherically symmetric wave. In this case all the fluid-dynamic variables depend only on the radial distance to an origin point (as is the case of a point source). The governing wave equation can be rewritten in terms of a reduced set of spherical coordinates, which in this symmetrical case includes only the radial coordinate r:
11(,)∂∂Φ22rt rrtΦ=(,) (16) rr∂∂222 c t
Multiplication of this last equation by r produces exactly the same one-dimensional wave equation that governs plane-wave propagation, equation (10). Now, the only dependent variable is the quantity rΦ. Therefore, the solution of equation (16) for the determination of a spherically symmetric wave is:
f ()()ct− r f ct+ r Φ=(,)rt 12 + (17) rr
The first term on the right of the equal sign represents a “diverging” spherical wave that originates from a point source and travels into the positive r direction, whereas the second term represents a wave travelling in the negative r direction, and therefore converging to a point. This later “converging” wave is physically less significant. However, this wave is necessary for building up a standing spherical wave, which results from the superposition of a converging and diverging wave of equal amplitude.
For the harmonic wave case, the solution of the wave equation can be found in an analogous way as done for the harmonic plane-wave case, except that the dependent variable equals the product of rΦ :
Aeˆ itikrωω−+ Beˆ itikr Φ=(,)rt + (18) rr
The case of a harmonic spherically symmetric diverging wave is of the utmost importance in acoustics since, as it will be shown, it is a standard simplifying procedure to consider the generality
15 of sound sources as simple point sources, generating this type of wave. Note that, in this case, the wave amplitude decreases inversely proportional to the radial distance to the source.
2.2.3 Intensity and Energy Density A sound wave produces a mechanical disturbance in an elastic medium, such as air. Therefore, the classical notions of energy and power must also be present in any acoustic disturbance. In what follows, for the sake of simplicity of writing, the explicit dependence of all fluid-dynamic variables on the relevant spatial and temporal coordinates will be omitted, their inclusion being implicitly assumed.
A statement of energy conservation for acoustic fields, first obtained by Kirchhoff in 1876, can be derived from the three linear acoustic equations (6) as follows:
⎛⎞∂v p ∂ρ vvvvvii⎜⎟ρ0 =− ∇pppp =−∇() + ∇ ii =−∇ () − (19) ⎝⎠∂ttρ0 ∂
∂ ⎛⎞1 2 The left term of equation (19) can be written as ⎜⎟ρ0v and the last part of the right term as ∂t ⎝⎠2
∂ ⎛⎞1 p2 ⎜⎟2 yielding: ∂tc⎝⎠2 ρ0
∂w + ∇=iI 0 (20) ∂t where
2 1123p −− 2 wv=+ρ0 2 []; JmpWmIv = [ ] (21) 22ρ0c and where the scalar quantity w represents the acoustic energy density of an acoustical disturbance characterised by the squared fluid velocity v2 and by the squared acoustic pressure p2 , whereas, the vector quantity I translates the acoustic energy flux, also named as the acoustic intensity of the
16 disturbance. The quantity w is composed of the sum of the acoustic kinetic-energy density and of the acoustic potential-energy density.
Equation (20) is the law of energy conservation for acoustic fields in differential form. It can also be cast into integral form by integrating the expression over an arbitrary fixed volume inside the fluid and by transforming the volume integral of the divergence of the intensity by means of Gauss’ Theorem into a surface integral over the boundaries of the defined volume:
∂w dV=− ∇iiIIn dV =− dS ∫∫∫ ∫∫∫ ∫∫ S VVS∂t S (22) d wdV=− Ini dS dt ∫∫∫ ∫∫ S VSS
Therefore, for an acoustic field, the time-rate-of-change of the acoustic density inside some fixed region within the fluid equals the net amount of acoustic intensity entering (and exiting) through the boundary of this region.
Intensity and Acoustic Density in Plane Wave Fields
For the case of harmonic plane waves, the expressions for the acoustic density and intensity are simple to obtain. The general expression for a progressive harmonic plane wave in three dimensions can be written as:
itikω − nri Φ=(,)r tAeˆ (23) ∂Φ itikωω−−nrii itik nr pt(,)r =−ρρω =− iAeˆ = Peˆ 00∂t where the unit vector n points into the direction of propagation of the plane wave. The corresponding particle velocity of the plane wave can be obtained from the linearised momentum equation:
1(,)ik pr t vr(,)tptdtptdt=−∫∫ ∇ (,) r = n (,) r = n (24) ρρ00 ρ 0c
17 In a plane wave, therefore, pressure and velocity are directly proportional to each other. This proportionality is a constant, the characteristic impedance of the medium, given by the product
−21− ρ0c . Its value for air is typically 414 Kgm s , or 414 rayl. Therefore, acoustic plane waves are longitudinal waves.
Using this particle velocity into expressions (21) one obtains:
pp22 wcw===2 ; Inn ρρ00cc 2 p2 Pˆ 11 * wpp==22Re[ ] = 2 (25) ρρ00cc22 ρ 0 c 2 2 ˆ p 11 P In==Re[p* pcw ] nnn == ρρ00cc22 ρ 0 c for the instantaneous energy density and intensity and for the corresponding time-averaged quantities. The symbol in equation (25) means the time average and the symbol refers to the absolute value of the complex quantities. The factor 12 appearing in the time-averaged expressions derives from the time average of a squared sinusoidal function.
Intensity and Acoustic Density in Spherically Symmetric Wave Fields
The particle velocity in a diverging harmonic spherically symmetric wave can be obtained in a fashion similar to that of plane waves:
11⎡⎤+ ikr ve(,)rt=−∇ prtdt (,) = prtdt (,) ∫∫rr⎢⎥ ρρ00⎣⎦r (26) ⎡⎤1(,)(,)1+ ikr p r t p r t ⎡⎤ ==+ee1 ⎢⎥ rr⎢⎥ ⎣⎦ρω00ri ρ c⎣⎦ ikr where ∇ stands for the gradient in the radial direction r , and e means the unit vector pointing r r into the radial direction. Equation (26) indicates that the ratio of pressure and particle velocity in a spherically symmetric wave depends on the distance r and on the frequency of the disturbance. In
18 addition, the ratio is complex, indicating that a phase shift exists between the two quantities. For distances that are large when compared to the wavelength, the expression for the particle velocity approaches the expression for a plane wave.
The expressions for the instantaneous energy density and time-averaged energy density are therefore given by:
2 ⎛⎞ ρρ002 ⎡⎤p ⎛⎞1 v =+⎜⎟Re⎢⎥⎜⎟ 1 22⎝⎠⎣⎦ρ0cikr⎝⎠ 2 Pˆ 2 ⎡ 2 sin()ωωωtkr−−− cos()() tkr sin tkr⎤ =−++22⎢cos()ωtkr 22 2⎥ ; 2ρ0cr⎣ kr kr ⎦ 2 2 Pˆ 1 p 2 222=−cos()ωtkr ; (27) 22ρρ00ccr 2 Pˆ 2 ⎡ 2 sin()ωωωtkr−−− cos()() tkr sin tkr⎤ wtkr=−++22⎢2cos()ω 22 2 ⎥ 2ρ0cr⎣ kr kr ⎦ ˆ 2 P ⎡⎤1 w = 1+ 22⎢⎥ 22 2ρ0cr⎣⎦2 kr since the time average of the squared cosine and squared sine function is equal to 12, whereas the time average of the product of sine and cosine is equal to zero. Relatively to the instantaneous and time-averaged intensity for a spherically symmetric wave:
2 Pˆ ⎡⎤2 cos()()ωωtkr−− sin tkr Ie==pvr r 2 ⎢⎥cos()ω t −+ kr ; ρ0cr⎣⎦ kr (28) 2 Pˆ p2 Iee==2 rr 2ρρ00cr c
Thus, the expression for the average intensity in a spherical wave equals the expression obtained for a progressive plane wave. Note the dependence of the time-averaged intensity on the inverse square of the radial distance from the point of origin of the disturbance. This is the mathematical
19 statement of the so-called spherical spreading law. As the spherical wave front propagates and expands, its time-averaged intensity is distributed over a four-fold larger area.
2.2.4 Harmonic and Non-harmonic Sound Waves The particular solutions of the wave equation portrayed in section 2.2.2 are harmonic solutions, i.e. these are solutions where the temporal dependence is determined by a sinusoidal function. These are possible, but rather restricted, solutions of the wave equation. In this section, the issue regarding the relationship between these special solutions and the more general waveforms that are usually found in real acoustic disturbances will be discussed.
Two superposing progressive plane sound waves travelling in the same direction and possessing the same temporal frequency will yield another plane sound wave with the same frequency, but with resulting amplitude, that is a function of the relative spatial displacement between the two waves. This difference in the resulting amplitude of the wave represents the physical phenomenon of wave interference. Sometimes the relative spatial displacement allows for increased wave amplitude relative to both original amplitudes, yielding what is called constructive interference, but sometimes the relative spatial displacement between the two original sound waves results in decreased wave amplitude. In this later case, one speaks of destructive interference.
In the case of superposition of two progressive plane waves travelling in the same direction, but with different frequencies, the resulting wave is no longer harmonic, and it can no longer be represented by a single sine or cosine function. However, if the ratio of the larger to the smaller frequency equals a rational number, the resulting plane wave is a harmonic one with frequency given by the greatest common divisor between the larger and the smaller frequencies. If that is not the case, then the resulting wave is non-harmonic. Obviously, the superposition of three or more waves shows similar characteristics as for the case of the superposition of only two plane harmonic waves, with arbitrary frequencies.
In the general case of an arbitrary non-harmonic sound wave, one can show that it can be decomposed into elementary harmonic waves by means of Fourier Analysis. Then, any square integrable function, with at most a finite number of discontinuities, can be written as the integral of a continuous family of time harmonic functions [Bracewell, 1986]:
20 1 ∞ st()= ∫ Sˆ (ω ) eitω dω (29) 2π −∞ where Sˆ()ω is the Fourier Transform (with complex values) of the original non-harmonic function st(). Symmetrically, the Inverse Fourier Transform is defined in a similar manner as:
1 ∞ Sstedtˆ()ω = ∫ ()−itω (30) 2π −∞
Owing to the symmetry of equations (29) and (30) one can see that Sˆ()ω is not only the Fourier
Transform of st(), but st()is also the Fourier Transform of Sˆ()ω as well. Therefore, both representations describe completely the same physical phenomenon.
The interest of this dual representation stems from the assumption of the underlying linearity of acoustic disturbances. In this case, the solution of the wave equation for a non-harmonic wave disturbance can be obtained from the infinite summation of the harmonic solutions resulting in their Fourier synthesis.
The Fourier Transform of a given function st()comprises three component functions [Bracewell, 1986]:
- the Spectrum: SS()ω = ˆ ()ω describes the amplitude of the harmonic function owing
angular frequency ω , composing the original function.
⎡⎤Im⎡Sˆ (ω )⎤ - the Phase: Ψ=()ω tan−1 ⎢⎥⎣ ⎦ gives the phase delay of the harmonic function ⎢⎥Re⎡Sˆ (ω )⎤ ⎣⎦⎣ ⎦ owing angular frequency ω , composing the original function.
2 - the Power Spectrum: Π=()ω Sˆ ()ω gives the intensity of the harmonic function owing
angular frequency ω , composing the original function.
21 The Rayleigh-Plancherel Theorem [Bracewell, 1986] states that:
∞∞ 2 ∫∫stdt2 ()= S (ω ) dω (31) −∞ −∞ and therefore the Fourier Transform can be viewed as an unitary transform that preserves the total quantity of energy present on an arbitrary non-harmonic signal.
2.3 Sound Sources and Receivers A sound source is defined as a physical device capable of generating and radiating of acoustic waves. In the field of room acoustics, there are three basic types of sound sources: the human voice, musical instruments, and electro-acoustical devices such as loudspeakers.
Any of these sound sources has its own characteristic properties, which determines their ability to generate and radiate sound energy. Particular properties determine such physical attributes such as source acoustic power, frequency bandwidth, and directionality.
The acoustical power output of the human voice varies from 0.001 µW for whispering up to 1 mW for shouting, being approximately between 1 µW and 10 µW for normal conversation [Kinsler et al., 1982]. The power of a single musical instrument may lie in the range from 10 µW to 100 mW , but a large orchestra may easily produce an output power of 10W in fortissimo passages [Kuttruff, 1979]. For electro acoustic devices, the acoustical power output can be of any magnitude, from milli-watts to mega-watts.
Regarding the range of frequencies produced by the human voice, they lie approximately between 50 Hz and 10 kHz. For normal speech, the fundamental frequency is produced between 50 Hz and 350 Hz, being identical to the frequency of the vibrations of the vocal chords. However, in speech sounds, the overtones (i.e. multiples of the fundamental frequency) are much more characteristic than the fundamental tone, and they are especially strong in certain specific frequency ranges called the formants and that can extend up to 3.5 kHz. This is the case for spoken vowels and voiced consonants, and in the later case, these can have a continuous spectrum at frequencies up to 10 kHz and higher [Kuttruff, 1979].
22 The organ is the musical instrument that has the largest frequency bandwidth, being capable of producing tones whose fundamental frequency varies between 16 Hz and 10 kHz. Other musical instruments have fundamental frequency bandwidths not as large and lying somewhere within this range. Musical instruments generate sound tones there are rather complex from the frequency spectrum content owing to the production of higher harmonics. Therefore, the actual range of frequencies produced by a musical instrument can reach up to frequencies of about 15 kHz.
The frequency bandwidth of a characteristic electro-acoustical device lies approximately between 20 Hz and 20 kHz, this bandwidth being nowadays more or less standardised. Note that this bandwidth refers to a electro-acoustical device composed of two or more transducers, i.e. it can be composed of a bass-mid loudspeaker and of a tweeter. Nevertheless, specially designed and produced transducers also exist that are capable of generating and radiating sound waves with frequencies below the 20 Hz, infrasound waves, and above 20 kHz, ultrasound waves. These transducers are not relevant in the case of room acoustics.
The third important property exhibited by all sound sources is its inherent directionality. This is because the radiated acoustic waves have different intensities with regard to the different spatial radiation directions.
The directivity of a human speaker is based mainly on the sound shadow cast by its head. Sound waves are radiated in the front directions with increased intensity as related to the sound waves radiated in the back directions of his head. At low frequencies, the horizontal directivity pattern is more or less regular, approaching the circle as the generated frequencies decrease towards the lower end of the speech bandwidth. However, at mid and high frequencies, the characteristic horizontal directivity pattern becomes more and more irregular, showing various intensity lobes, whose shape varies from frequency to frequency. However, for most applications, the directivity patterns suffice to be determined for several contiguous frequency bandwidths, usually octave bands (see section 2.7)
Musical instruments normally have a pronounced directivity due to its geometrical construction, whereby the radiated sound waves obey some modal character of the generating material structure. However, the specific directivity pattern of a certain class of musical instruments is a rather complicated function of both frequency and direction. In addition, the same instrument can behave
23 very differently because of the different manufacturing processes. The violin is a particular example of this, whereby different manufactured violins have not only tonal characteristics that allow differentiating between them, but their directivities may also be disparate.
The case of electro-acoustical sound sources constitutes the case where the most varied directivities can be found. There are the so-called omni-directional loudspeaker devices, normally composed of several single loudspeakers arranged in a centrally symmetric lattice, and that can in principle radiate sound waves with equal intensities in all surrounding directions. In the other extreme, there are the so-called beam loudspeaker devices, usually composed of a single loudspeaker and specially designed sound focusing element that radiates most of its total acoustical power into a narrow spatial region, in the form of a beam.
2.4 Sound Radiation A usual simplification that is normally made for mathematically representing sound sources consists in assuming that its spatial extent is so reduced, when compared with the characteristic wavelength of the radiated waves, that it can be regarded as an infinitesimal point source. It is evident that such simplification is insufficient for the true description of real sound sources, but in the majority of practical cases, it proves to be rather satisfactory. The pressure field for harmonic waves radiated by an infinitesimal, omni-directional point source is derived from the first expression of equation (18), representing a symmetrically spherical diverging wave, and slightly modified in order to include the acoustical output power, Π()ω :
ρωcΠ()eitkr()ω − prt(,)= 0 (32) 2π r
If the source is not omni-directional, meaning that the radiated field is no longer symmetrically spherical, then the far-field pressure wave can be given approximately by:
ρωcΠ() eitkr()ω − prt(,)=Γ0 (,θϕω , ) (33) 2π r where the directivity function Γ [Kuttruff, 1979] is normalised for the direction of maximum wave amplitude. With far field, it is meant that the radial distance r where the acoustic pressure is
24 computed is considerably larger than the wavelength. The time-averaged intensity produced by such a sound source is according (28) equal to:
p2 2 Π()ω Ie==Γrr(,θϕω , ) 2 e (34) ρπ0cr 4 being now a frequency-dependent function of the polar angle θ and azimuth angle ϕ as defined by a local spherical coordinate system centred at the sound source. This last equation translates mathematically the usual statement that the time-averaged radiated intensity of a spherical point source falls of with the square of the radial distance: the spherical spreading law. In the case of a omni- directional simple point source the directivity function reduces to Γ(,θ ϕω , )= 1. Equation (34) can also be written in terms of the radiation pattern function J(,θ ϕω , )[Pierce, 1994], describing the acoustic power radiated by the sound source per unit solid angle:
Π()ω J (,,)θϕω Iee=Γ2 (,θϕω , ) = (35) 4π rr22 rr the relationship between Γ and J being simply:
Π()ω J(,θϕω , )=Γ2 (, θϕω , ) (36) 4π
2.5 Sound Receivers Receivers of sound waves are practically just of two types: the human ear and microphones. The human ear and its related perception characteristics will be discussed in section 2.7. Microphones are mechanical devices capable of transforming a sound wave impinging upon it into an electrical signal, which can be analysed or recorded in an analogical or digital manner for post-processing.
The above-mentioned receivers possess intrinsic unique properties. These properties are mainly characterised by their frequency bandwidth, sensitivity, and spatial resolution. The properties relating to the human ear are summarised in section 2.7.
Microphones can be built with different bandwidths, sensitivities, and spatial responses. A complete discussion of these specifications lies out of the scope of the current document. However, it can be
25 said that there are currently microphones with a huge bandwidth, with very high sensitivities and possessing almost any directivity.
It is obvious that these receivers are of finite spatial extent. However, and in analogy with what was written for sound sources, they can be approximated by infinitesimal point receivers. In this way, the same formalism adopted for point sources is also undertaken for modelling sound receivers, but in this case, the associated directivity functions will characterise the sensitivity in function of the direction of sound arrival.
When the sound sources or receivers are moving, then a special characteristic of sound must be referred, the Doppler Effect [Krane, 1983]. In this case, the sound wave suffers from a spatial compression in the direction of the source’s displacement, and respectively a rarefaction in the opposite direction, giving rise to a modified received sound frequency (this happens not only with sound waves but also with every wave motion in general). The current work was developed considering only stationary sound sources and receivers, and therefore the Doppler Effect is not considered.
In practical terms, the directivities of sound sources are normally given as gain functions that depend on the spatial directions and on the on-axis transfer function for different frequency bands. The sound level of a sound source is usually given by its acoustical power output (more usually it is given by the maximum admissible input electric power) or also by the sound pressure level generated at 1 meter from it.
2.6 Propagation of Sound Waves In the previous sections, only propagation of sound waves in free space was considered. By free space one means an unbounded homogeneous fluid and all the presented formalism and equations apply to this idealised propagation condition. However, in the real world, sound propagation is conditioned by further physical phenomena that will influence the propagation of a sound wave. Typical disturbances are due to reflections, diffractions, transmissions, refractions and attenuation in the propagation medium. These tend to attenuate the sound wave, but also to modify its direction of propagation. The sound fields that result from these disturbances are no longer simple to describe. Instead, their complexity generally disallows any formal mathematical procedure for solving an acoustic problem. Simplifications, approximations and, currently, numerical methods are
26 the only way to achieve the solution for sound propagation in the presence of obstacles such as walls, ceiling and floor, and in the case of the presence of two or more propagation media. In addition, real sound waves are neither plane waves nor symmetric spherical waves; nevertheless, to some extent they can be represented with small errors by such idealisations.
2.6.1 The Sound Field in Front of a Wall When a sound wave strikes a wall, part of its acoustic energy is generally reradiated in the form of another wave originated from the wall, while the rest of the energy is absorbed by it. By absorption, it is meant that the non-reflected energy fraction either is converted into heat or is further transmitted to the other side of the wall. The amplitude and phase of the reflected sound wave are usually different from the ones of the incident wave. In addition, the “shape” (waveform) of the reflected wave can be quite different from the original “shape” of the wave striking the wall.
A physical quantity that completely describes the acoustic behaviour of a wall is the acoustic impedance Zˆ =+RiX, a complex quantity dependent on the wave’s frequency ω and direction of incidence Ω . The real part of the acoustic impedance is denoted as the acoustic resistance and the imaginary part as the acoustic reactance. This wall impedance is defined as the complex ratio between the sound pressure p at the wall’s surface and the fluid particle’s velocity normal to it, vn , just outside the surface.
ˆ ⎛⎞p −1 Z(,ω Ω= )⎜⎟ ; [Pa m s ] (37) v ⎝⎠n surface
This normal velocity is due to a motion of the wall itself or else to a motion of air into pores in the wall. In either case, one considers a wall with low acoustic impedance as being “soft” and one with large impedance as being a “hard” wall. If the wall impedance has a real component, energy will be absorbed at its surface; purely reactive impedance implies only a change of phase upon reflection. In the case where the specific acoustic impedance of the wall is independent of the direction of wave incidence, which is approximately true for most acoustic materials, then one names it locally reacting surface.
27 The units for the wall impedance Zˆ can be defined more naturally from the values of the characteristic acoustic resistance of air for free plane waves, ρ0c . The resulting quantity is known as the specific acoustic impedance of the wall:
Zˆ(,ω Ω ) ζωˆ(,Ω= ) (38) ρ0c
The reciprocal of the specific acoustic impedance is called the specific acoustic admittance, its real part being known as the specific acoustic conductance, while its imaginary part is known as the specific acoustic susceptance of the wall.
• Reflection of Plane Waves from a Plane Wall
For the case of the reflection of a progressive plane wave at a plane wall with finite specific impedance, another quantity can be defined, which yields the relationship between the amplitude and phase of the reflected wave to that of the incident wave. This quantity is the complex plane wave pressure reflection factor [Kuttruff, 1979], which again depends on sound frequency and on the incident angle:
RReˆˆ(,ωωΩ= ) (, Ω )iγωR (,)Ω (39)
ˆ with γ R (,ω Ω )denoting the phase angle of R(,ω Ω ). The relationship between impedance and the plane wave pressure reflection factor is given by:
ζωˆ(,Ω )cos θ− 1 Rˆ(,ω Ω= ) i (40) ˆ ζω(,Ω )cos θi + 1 where θ represents the angle between the incident direction and the surface’s normal n . The i angle between the direction of the reflected wave and the surface’s normal can be shown to be equal to the incidence angle, θri= θ [Morse and Ingard, 1968]. In this case, specular wave reflection is assumed, which follows the so-called Snell-Descartes Law where the angle of incidence equals the angle of reflection. This reflection model is the most used in acoustics due to the wave
28 lengths of normal sound waves and to the magnitude of wall irregularities, which are generally small.
According to equation (25), the intensity of a progressive plane wave is proportional to the square of its pressure amplitude, and therefore, the intensity of the reflected plane wave is reduced by a
2 factor Rˆ(,ω Ω ). The fraction
2 1(,)(,)−Ω=ΩRˆ ωαω (41) of the incident intensity is therefore absorbed, during reflection, by the wall (this equation with the pressure reflection factor (40) inserted, is known as Paris Equation [Paris, 1927]). This fraction, which for real passive walls has values comprised between zero and one, defines the absorption coefficient of the wall. The relationship between absorption coefficient and specific impedance is obtained by combining (40) and (41), yielding:
⎡⎤ˆ 4cosθζω Re⎣⎦ ( ,Ω ) αω(,Ω= )2 ; ⎡⎤ˆˆ 2 (42) 12cosRe(,+Ω+Ωθ ⎣⎦ζω ) ζω (, )cos θ
θθ==ir θ
The distribution of sound pressure in front of the wall, due to a progressive harmonic plane wave can be obtained by considering the wall at the y-z plane of a Cartesian coordinate system. The incident wave is assumed to propagate in the direction given by nee=−sinθ cosθ and ixy according to the specular reflection law, the reflected wave propagates into the direction nee=+sinθ cosθ . The total field is then given by (using the notation of (23)): rx y
itikωω−−nrii itik nr pt(,)r =+ Peˆˆˆ ir R (ωθ , ) Pe ik y (43) ⇔=pxyt(, ,) Peeˆˆitω −ikx x ⎡ 2 R (ωθ , )cos( ky ) +− 1 R ˆ ( ωθ , ) ey ⎤ ⎣ y ()⎦
where kkxy==sinθ ; kk cosθ are the x, y components of the wave number k . Equation (43) defines a partly stationary wave along the distance y from the wall. The sum of the y -component
29 of the reflected wave with the fraction Rˆ of the y -component of the incident wave builds up a perfect standing wave. The remaining fraction (1− Rˆ ) of the y -component of the incident wave is still a progressive plane wave, given by:
pytPRefraction (,)=−ˆˆ⎡⎤ 1 (,)ωθ itky()ω + y i ⎣⎦ (44) ⇒=−++Re⎡⎤pytPRfraction ( , )ˆˆ⎡⎤ 1 (ω ,θω ) cos( tky γ ) ⎣⎦iy⎣⎦ which is responsible for the transfer of energy into the absorbing wall. Because of the energy losses at the wall, the pressure nodes at the partly standing wave are not pressure zeros, as it would be the case for a perfect rigid wall with Rˆ =1, but are pressure minima. Similarly, the pressure maxima are not as high as in the perfect reflector case. Obviously, the total pressure field of equation (43) exhibits a spatial harmonic character of the x -component, given by the factor e−ikx x , which stems from the oblique incidence of the plane wave.
The mean squared pressure field in front of the absorbing wall is calculated according to equation (25):
ik y− ik y 22ˆˆitω −ikx x ⎡⎤yy pxyz(, ,)== py () Pee⎣⎦ e + R (ωθ , ) e 2 (45) Pˆ 2 =+1RRˆˆ (,)ω θωθγωθ + 2 (,)cos(2 ky + (,) 2 ()yR and represents a perfect standing wave in front of the wall. This expression establishes the basic framework of a measuring method for the determination of the pressure reflection factors (and consequently of the specific wall impedance) of plane walls. It can be seen from equation (45) that the ratio of the mean squared pressure maxima and minima determines the absolute value of the reflection factor, while its phase being determined by the distance of the pressure minimum, or pressure maximum, that are located closest to the plane wall. The measurements are accurate only if the sound waves are truly plane waves. Accurate plane waves can easily be generated inside cylindrical tubes at low to mid frequencies, depending on the tube’s diameter. This measurement method is called the standing wave tube method.
30 • Reflection of Spherically Symmetric Waves from a Plane Wall
The case of the reflection of a spherically symmetric wave from a plane wall is a much more complex problem. If the wall impedance is very high, meaning that the wall is rigid, then the problem is straightforwardly solved by the image source method [Pierce, 1994]. This method is a conceptual technique that replaces the original boundary-value problem of a source and a wall by the problem of two sources, the original one and the mirror image of it, and no wall. Therefore, for the case of a single point source located at (0,0,zS ) , above a rigid plane wall located at the plane z = 0, the corresponding mirror image problem is that of two sources, the original point source, and a mirror image source located at (0,0,−zS ) . Thus, the total pressure field at some receiver point r = (,x yz ,) originated by a symmetrically spherical omni-directional point source at
(0,0,zS ) , with acoustic power Π()ω is given by:
−−ikR12 ikR TotalicΠ()ω ⎡⎤ e e iω t Φ=(,xyzt ,,) ⎢⎥ + e ωπρ2 01⎣⎦RR 2
−−ikR12 ikR totalρω0cΠ()⎡⎤ee iω t pxyzt(, ,,)=+⎢⎥ e (46) 2π ⎣⎦RR12 22 2 22 2 R12=++−xy(); zzSS R =+++ xy () zz and the corresponding mean squared pressure is given by:
2 111ρω0cΠ()⎡ 2cos[kR (12+ R )]⎤ pxyz(, ,)=++⎢ 22 ⎥ (47) 22π ⎣ RR12 RR12 ⎦
If the receiving point r is distant from the original source that is located at a small distance from the rigid reflecting plane, then:
Rrz12≈−SScosθ ; Rrz ≈+ cosθ (48) where (,r θ ) are the polar coordinates of the receiving point r , and obtaining the approximate equation:
31 2 12ρω0cΠ()⎡ 2cos[ 2kzS cosθ ]⎤ pxyz(, ,)≈+⎢ 22⎥ (49) 22π ⎣rr⎦
and for kzS 1 it can be seen that the intensity in the far field is increased by a factor of 4, or the intensity level increases by 10log 4≈ 6 dB , when compared to the situation of free field radiation.
In the case that the plane wall is not completely rigid, but it can be assigned a complex specific impedance ζˆ , therefore being of local reaction, then the exact solution for the total field is much more complex. The formal solution for this problem has been given by [Wenzel, 1974]:
−−ikR12 ikR 1 ⎡ee Rit⎤ ω Φ=(,x yzt ,,)⎢ ++Φ (, xyz ,)⎥ e (50) 4π ⎣ RR12 ⎦ where the amplitude factor has been omitted and where the third term inside the rectangular brackets defines the radiated field:
∞ ˆ ik2 λ e−+mz() zS Φ=R (,x yz ,) J (λ rd )λ ˆ ∫ o ζ 0 ⎛⎞ik mmˆˆ⎜⎟− (51) ⎝⎠ζˆ mkˆ =−λ 22; rxy =+ 22
No closed form solution of equation (51) has been found, and several papers have been published on different possibilities to achieve a more tractable equation and to obtain approximate formulas for special cases [Chien and Soroka, 1975; Thomasson, 1976].
Another way of solving the problem of a point source above an impedance plane can be pursued, if the original spherical wave is decomposed into elementary plane waves by using a spatial Fourier Transform, as done originally by Ingard [Ingard, 1951]. In this way, the plane wave pressure reflection factor, as given by equation (40), can be used for each of the elementary plane waves appearing in the Fourier decomposition in order to obtain the resultant reflected field from the plane wall. The interested reader can see Skudrzyk [1971], and Kawai [1982].
32 In either case, it can be shown that if the specific impedance of the wall is greater than one. An approximate expression is then valid:
−−ikR12 ikR ⎡ee⎤ itω Φ≈+(,x yzt ,,)⎢ Q (,φγ ) ⎥ e (52) ⎣ RR12⎦ where Q(,)φ γ is defined as the spherical wave pressure reflection factor, given by:
QRˆ(,)φ γφ=+−ˆˆˆ () (1 RE ())(); φγ
2 Eieerfciˆ()γπγγ=+ 1−γ ( − ); (53) 11+ i ⎛⎞ γφ=+kR2 ⎜⎟cos 2 ⎝⎠ζˆ and where Rˆ()φ is identical to the plane wave pressure reflection coefficient as given by equation (40) and erfc is the complementary error function. The angle φ appearing in the equations (52) and (53) is equal to the angle that the vector joining the image source and the receiving point makes with the normal of the wall n , i.e. it is equal to the incidence angle. If, in addition, the original source and receiver are not very close to the plane wall when compared to the wavelength of the radiated wave, the value of the function Eˆ()γ is approximately equal to zero, and one has that the total field is, to a very good degree of approximation, given by:
⎡⎤−−ikR12 ikR eeˆ ˆ itω Φ≈+(,xyzt ,,)⎢⎥ R (,φζ ) e ⎣⎦RR12 (54) ⎡ee−−ikR12ζφˆ cos− 1 ikR ⎤ =+ eitω ⎢ ˆ ⎥ ⎣ RR12ζφcos+ 1 ⎦
The spherical reflection factor is almost equal to the plane wave reflection factor, except at low frequencies or at the far field when source and receiver are located not too far from the impedance boundary. In Figures 1 and 2, both reflection factors are compared for the situation of an impedance boundary having values that correspond to a grass surface in the Delany and Bazley Model. This model is described in [Delany and Bazley, 1970; Ögren, 1997], and uses two parameters: the effective flow resistivityσ and the sound frequency f . The impedance is given by:
33 −−0.75 0.73 ⎡1000ff⎤⎡⎤ 1000 Zfˆ(σ , )=+ 1 9.08 + i 11.9 (55) ⎣⎢ σσ⎦⎣⎦⎥⎢⎥
6
4
l 2 e i f e r fd
v 0 i t a l e re B de -2
-4
0 1000 2000 3000 4000 Frequency
Figure 1: Relative attenuation of spherical wave above impedance plane (grass) compared to free field. Source and receiver height 1 meter; source-receiver distance 80 meters; Delany and Bazley Model with a flow resistivity of 200000. Green – plane wave reflection factor (54). Blue – spherical wave reflection factor (52).
4 l e i f e
r 2 fd v i t a l e re B de 0
-2
0 20 40 60 80 Source − receiver distance
34 Figure 2: Relative attenuation of spherical wave above impedance plane (grass) compared to free field. Source and receiver height 1 meter; frequency 100 Hz; Delany and Bazley Model with a flow resistivity of 200000. Green – plane wave reflection factor (54). Blue – spherical wave reflection factor (52).
• Scattering from Surface Irregularities
When a plane sound wave impinges on a real wall, large in extent with respect to a wavelength but irregular either in shape or in concerning its acoustic reaction, the wave reflected from the surface will be distorted because of the inhomogeneities. In the previous sections, it was shown that a plane ˆ wave of amplitude P , incident at an angle θi on a smooth, locally reacting, boundary plane of ˆ uniform point impedance ζ , is reflected as a plane wave at angle θri= θ to the wall’s normal, and ˆˆ ˆ with amplitude PR(,)θζi where the plane wave pressure reflection factor is given by equation (40).
Either if the specific impedance changes from point to point along the plane or if the boundary itself is not a perfect plane, the reflected wave will not be plane. If the surface irregularities are not large compared to the wavelength, the reflected wave may be represented as a plane wave plus a scattered wave, which measures the degree of distortion.
The simplest case where a scattered wave appears in the total reflected field is when a rectangular patch of dimensions lw× and of impedance ζˆ is placed over a locally reacting wall possessing ˆ specific impedance ζ 0 . In this case, an approximate formula for the scattered wave, at large distances from the scattering patch, is given by [Morse and Ingard, 1968]:
e−ikr pr(,θϕ , )=Ψ Pˆˆ ( θ , ϕ , θϕ , ) (56) sc i i r in spherical coordinates and where the scattering amplitude is given by:
ˆˆ iklw(ββ− )cos θ cos θ⎡ sin( µ l 2) sin(µ y w 2)⎤ Ψ=(,θϕθϕ ,,) 0 ix⎢ ⎥ ii ˆˆµµlw22 2(cosπθβθβi ++ )(cos0 )⎣⎢ xy⎦⎥ (57)
µxii=−kk(sinθϕ cos sin θϕ cos ); µ yii =− (sin θϕ sin sin θϕ sin )
35 where (,θiiϕ )are the spherical coordinates of the incident wave and (,θ ϕ )are the spherical ˆˆ ˆˆ coordinates of the scattered wave. β =1 ζ and β00=1 ζ are respectively the specific admittances for the patch and for the wall.
Figure 3 shows the modulus of the scattering factor for a 10cm× 10 cm patch as function of the scattering direction and frequency. The incident wave is at θ = π 4 . Note that for small frequencies the scattering is almost ideally diffuse. With increasing frequency, most of the scattered wave goes into the specular direction that equals the incident direction.
0.1 20000
0.05 15000
0 π − 10000 2 π − 4
0 5000
π 4
π 2
Figure 3: Modulus of scattering factor for a 10 x 10 cm patch with specific impedance 10 over a plane boundary with specific impedance 1, plotted in function of incidence angle and frequency.
Another situation where a scattering wave appears in the total reflected field is the case of random surface roughness. In this case, the squared modulus of the scattering factor equals:
2 1 2 22 2 A ⎡⎤cosθθ cos − kWγ ξ ˆ i 44 2 2 2 Ψ=(,θϕθϕii ,,) ⎢⎥kW γξ ξ e 2(cos)(cos)πθβθβ⎣⎦i ++00 (58) 22 2 γθθθθϕϕ=+−siniii sin 2sin sin cos( − )
36 where ξ is the normal displacement of the boundary surface over area A from its mean shape and
2 Wξ is defined as the correlation length of the random fluctuation.
In this case, it is seen that the scattering caused by the surface roughness avoids the direction of specular reflection, and, for very long wavelengths, the scattering amplitude becomes negligible. The maximum of the scattering factor occurs for negative incidence angles, meaning that the wave is scattered back to the source. This is shown in Figure 4 for the case of a 1cm2 rough patch.
It is obvious that these two examples do not represent the complex acoustic behaviour of a real wall, but they highlight some important characteristics of the sound field in front of real walls.
If the wall’s surface is not perfectly plane, suffering from shape irregularities that are of the same order of the considered sound wavelength, it is commonly idealised that the sound is reflected in a completely diffuse way. In this case, the incident wave is reflected in all possible directions with the same intensity Ir , for every possible incidence angle, and following the well-known Lambert Law [Kuttruff, 1979]:
II==()dni I d cos()θ (59) ri ii i i
where I is the incident intensity and d is the vector in the incident direction. This type of i i reflection is called diffuse reflection or lambertian reflection.
37 0.2 20000
0.1 15000
0 π − 10000 2 π − 4 0 5000
π 4
π 2
Figure 4: Squared modulus of scattering factor for a 10 cm2 rough patch with mean normal deviation ξ 2 =1 cm and correlation length
Wkξ = 12over a smooth plane boundary with specific impedance 1.
In reality, and as mentioned in both examples of this section regarding surface scattering, sound energy does not reflect from a real wall in either ideal specular or ideal lambertian way. Instead, the reflected wave shows a rather complex directional distribution of its intensity, as can be see in Figure 5. One underlines, however, that few work as been undertaken for obtaining experimental data concerning the directional distributions of real materials. This is therefore the main cause that the usual adopted reflection laws for modelling sound propagation inside enclosures are the ideal specular and ideal diffuse, with the ad-hoc introduction of a so-called diffusivity coefficient, which measures the amount of intensity being diffusely reflected. A focus on this diffusivity coefficient will be presented later in Chapter 4. A recent work on modelling surface roughness scattering is given in [Embrechts, 2000] with an application to a ray tracing formalism.
38
Figure 5: Three-dimensional representation of the directional scattering characteristics for a wall whose surface is “lined” with semi-cylinders of height 20 mm and width 60 mm . Angle of incidence 30º; Frequency 15 kHz. (After [Kleiner et al., 1997])
2.6.2 Diffraction of Sound Waves When the scattering object is large when compared with the wavelength of the scattered sound, one usually says that the sound is reflected and diffracted, rather than scattered. The effects are in reality the same, but the relative magnitudes differ enough so that there seems to be a qualitative difference.
Behind the occluding object there appears to be a shadow zone, where the pressure amplitude is rather small. In front of the object, and to its side, in the illuminated zone, there exists a combination of the incident wave and of the reflected and scattered wave from the surface of the wall. At the border of the shadow zone, the wave amplitude does not drop discontinuously from its value in the illuminated zone to zero. Instead, the amplitude oscillates about its illuminated value, with increasing amplitude, reaching its maximum just before the edge of the shadow, and then drops monotonically approaching zero well inside the shadow region.
The simplest problem where diffraction effects appear is that of a plane wave striking a rigid infinite half-plane, as displayed in Figure 6. However, even for this simplest case, the solution is far from being simple [Skudrzyk, 1971]:
−−+Ψikrcosθθ⎡⎤θ ikr cos( 2 ) ⎡ ⎛3πθ ⎞⎤ pr( ,θ )=+ Pe E⎢⎥ 2 kr cos Pe E⎢ 2 kr cos⎜⎟ −Ψ−⎥ ⎣⎦222⎣ ⎝ ⎠⎦ z (60) 1 2 Ez()= ∫ eit dt iπ −∞
39 where Ez() is (apart from a factor) the complex Fresnel integral.
Rigid half-plane r
Ψ θ Shadow Line O
Incident Plane Wave
Figure 6: Notation used in equation (60), giving the diffraction of a plane sound wave by the edge of a rigid half plane.
1.2
1
0.8 2 »L p P 0.6 » H
0.4
0.2
0 -4 -2 0 2 4 Distance from shadow line @mD
Figure 7: Mean squared pressure of diffracted plane wave from a rigid half-plane. Angle Ψ = 0 . Plotted as function of the distance from the plane perpendicular to the shadow line and passing across the origin O ; 1 meter behind the rigid half-plane. Frequency: 500 Hz.
In Figure 7 the diffraction pattern for the case when the angle Ψ = 0 is shown. For different angles the shape and magnitude of the curve remains practically unaltered.
40 General diffraction problems are usually treated by using the integral version of the wave equation, expressed by the Helmoltz-Huygens-Kirchhoff Integral Theorem [Skudrzyk, 1971], which in its time-independent form is:
−−ikrr −− ik rs −− ik rs 1 ⎡⎤ee S ⎡⎤ ∂Φ ∂ ⎛⎞ e Φ=()r + −Φ dS ⎢⎥∫∫ ⎢⎥⎜⎟S (61) 4π ⎢⎢rr−−∂∂−S S rsnn rs ⎥⎥ ⎣⎦ S ⎣⎦ ⎝⎠ where r is the position vector of the sound source, r is the position vector of some receiving S point, s is the position vector of some point over surface S and ∂ ∂n means the derivative along S the surface’s normal at point s . The surface integral represents the solution of the associated homogeneous equation that satisfies the boundary conditions of the problem.
A problem that appears when studying sound fields inside enclosures is the problem of sound diffraction by a quasi-rigid obstacle (for example a balcony, or the orchestra pit). This problem can be modelled by a sound wave propagating over a rigid, semi-infinite wedge. Although there is for this problem an exact analytic solution, an approximate solution is valid by the so-called Geometric Theory of Diffraction (GTD) [Keller, 1962]. In particular, the so-called Biot-Tolstoy-Medwin theory gives accurate results for this kind of problem albeit using rather simple mathematics. The interested reader can find more information in [Ouis, 2003].
2.6.3 The Attenuation of Sound in Air Up to now, no consideration has been given about the dissipation of acoustic energy in the propagation medium. In general, losses in the medium can be divided into three basic types: viscous losses, heat conduction losses and molecular exchanges losses.
Viscous losses are friction losses, which occur whenever there is relative motion between adjacent layers of the fluid, such as during the compression and rarefaction cycles that occur during the passage of a sound wave. Heat conduction losses result from the conduction of thermal energy between higher temperature elements and lower temperature elements. Therefore, in reality the acoustic process is not entirely adiabatic as assumed in section 2.2.1. Finally, molecular exchange losses result from the additional energy that is “stored” in the medium’s molecules when a compression takes place. This absorption of the medium can principally lead to the conversion of
41 kinetic energy of the molecules into stored potential energy and into increased internal rotational and vibrational energies (in the case of polyatomic molecules).
The magnitude of the air attenuation depends on the sound frequency, air temperature, and relative humidity in practical terms. A simple model that considers the effect of sound absorption on sound intensity results if one considers that the attenuation of a plane wave over distance is constant:
dI (,ωωThmTh ,)=− (, ,) (62) ds Celsius Celsius
where ω = 2π f is the angular frequency, TCelsius is the air temperature, h is the relative humidity of the air and s is the path variable (a measure of the distance travelled along the path since the origin). The quantity mT(,ω Celsius ,) his called the air absorption coefficient. Solving for the intensity yields:
−mT(,ω Celsius ,) hs Is()= Ie0 (63)
where I0 equals the initial intensity.
The international standard ISO 9613-2 [1996] establishes the basic framework in order to obtain the attenuation of sound in air for a broad range of frequencies, temperatures and relative humidities. The values given by this norm are given in units of dB/ Km . In order to convert these values into the air absorption coefficient the following expression can be used:
⎡⎤att(,ω TCelsius ,) h ln⎢⎥ 10 10 mT(,ω ,) h= ⎣⎦ [ m−1 ] (64) Celsius 1000
where att(,ω TCelsius ,) h is the value of the attenuation in dB/Km given in ISO 9613-2 [1996].
42 0.04
0.03 D 1 − m @
m 0.02
0.01
0
100 200 500 1000 2000 5000 Frequency @Hz D
Figure 8: Air absorption coefficient plotted in function of sound frequency. Red - TChCelsius ==15º , 10% . Green -
TChCelsius ==15º , 20% . Blue - TChCelsius ==25º , 50%
2.6.4 Sound Absorbers When a sound wave strikes a real wall, with some specific mechanical properties, which for a locally reacting wall can be described by its specific impedance, some portion of its energy is removed from the reflected wave as seen under section 2.6.1. This absorbed energy is transferred to the wall either by setting the surface into motion, which in turn may initiate new waves on the other side of the surface, accounting for transmission phenomena, or by setting the air inside the wall’s pores into a vibratory motion. In either case, the extent to which absorption takes place at the wall depends upon many factors, including the materials that comprise the wall, the frequency of the impinging wave front, and the angle of incidence of the wave front. Comprehensive data and practical use of sound absorbers can be found in [Brüel, 1951].
Sound absorbing materials may be divided into four basic types: porous absorbers, membrane absorbers, resonance absorbers, and people and furniture.
2.6.4.1 Membrane Absorbers This type of absorber is usually a nonporous panel mounted away from a solid backing that vibrates under the influence of an incident sound wave. The dissipative mechanisms in the panel convert
43 some of the incident acoustic energy into heat. Such absorbers are very efficient at low frequencies, whereby the addition of a porous absorber in the space between the panel and the backing wall will further increase the efficiency of the low-frequency absorption.
For normal incident sound waves, the specific impedance of the panel is given by [Kuttruff, 1979]:
ωM ζωˆ()=+ 1i p (65) ρ0c
where M p equals the surface density of the panel. The corresponding absorption coefficient is obtained from equations (40) and (41):
−1 ⎡⎤2 2 ⎛⎞ωM p ⎛⎞2ρ c αω()=+⎢⎥ 1⎜⎟ ≈⎜⎟0 (66) 2ρωcM⎜⎟ ⎣⎦⎢⎥⎝⎠0 ⎝⎠p
It is noteworthy that a perforated panel exhibits essentially the same properties as a non-perforated panel, except that now an equivalent mass of the perforated panel, per unit area must be used in the above equations. This equivalent mass M eq is approximately given by:
S2 −2 M eq =+ρδ0bKgm(1 2 ) [ ] (67) S1 where b is the plate thickness in meter and δ accounts for the “end correction” of the effect of the panel holes. S1 and S2 are respectively the cross section area of the holes and the panel area per hole. The “end correction” factor is approximately given by:
S δ ≈ 0.45 1 (68) b
An example of the absorption coefficient for a membrane absorber, calculated from expression (66) is given in Figure 9.
44 0.8
0.6
α 0.4
0.2
0 50 100 500 1000 5000 10000 Frequency @Hz D
Figure 9: Example of absorption coefficient for a panel with 80 Kg/m2 density, plotted as function of frequency.
2.6.4.2 Porous Absorbers Porous Absorbers such as mineral wool, fabrics, fibreboard, plastic foams, acoustic plasters, and carpets have an open pore structure. Conversion to heat is produced by friction when the vibrating air molecules are forced to move through the pores and interact with the pore walls.
This type of sound absorber is effective primarily for high frequencies with short wavelengths. In a sound wave which is incident on a rigid wall, the maximum particle velocities occur at 1/4λ and 3/4λ . If the thickness of the absorber is less than one quarter of the wavelength, they have little effect.
A highly idealised model for a porous absorber is the so-called Rayleigh model [Kuttruff, 1979], where the porous material consists of a great number of similar equally spaced semi-infinite wave- guides. This model gives for the specific impedance of the porous material, characterised by its flow resistance Ξ (in units of Kgm−−31 s ):
SSΞ ζωˆ()=−21 1i (69) SS120ρ ω
A porous sheet placed some distance away from a solid wall will have almost the same effect as a thicker absorber. The maximum effect is achieved when the distance to the wall surface from the
45 centre of the absorber equals 1/4λ and is restricted to a comparatively narrow frequency band. This is because the maximum amplitude of both the incident and reflected waves will occur within the porous material, as can be seen in Figure 10.
Figure 10: Example of absorption coefficient for a porous absorber. In the case of lower frequencies, the porous absorber must be located more distant to the wall as for higher frequencies in order that the same magnitude of absorption is achieved. (After Andrew Marsh – The Fridge, Architectural Science Lab).
2.6.4.3 Resonant Absorbers These may be flexible sheets stretched over supports or rigid panels mounted at some distance from the front of a solid wall. Conversion to heat takes place through the resistance of the membrane to rapid flexing and due to the resistance of the enclosed air to compression. They will be most effective at their resonant frequency, which depends upon their surface density M p and the width of the enclosed space d [Brüel, 1951]:
2 160ρ0c fresonance =≈[]Hz (70) 2π Mdp Mdp
In practice, the method of fixing and the stiffness of the panels will also have some effect, as the panel itself will tend to vibrate. This means that it will act as a sound radiator so it is rare to find such a system with an effective absorption coefficient greater than 0.5. Membrane resonant absorbers are most effective at low frequencies, which is really why they are normally used. An example can be seen in Figure 11.
46
Figure 11: Example of absorption coefficient for a resonant absorber. (After Andrew Marsh – The Fridge, Architectural Science Lab)
2.6.4.4 People and Furniture The sound absorption by people is due mainly to clothing and its porosity. Since clothing is usually not very thick, the absorption will be considerable mainly at the mid and high frequency ranges. It is obvious that there are only mean values for people absorption, and the associated error of theses measurements are usually very high. The absorption by people can be given in terms of absorption areas, but when people are located near to each other, it is common to use an absorption coefficient, as just used for sound absorbing materials.
In the case of furniture, various absorption mechanisms are present, depending on the materials they are made of. Most furniture is made of wood, including chairs, school desks and tables for which there exists considerable data concerning their average absorption coefficients, but for several other pieces of furniture, only a vague guess can be made regarding their absorption.
A good data pool with absorption coefficients for people and furniture can be found in [Beranek, 1962, Kuttruff, 1979].
2.7 Sound Perception For the purposes of practical acoustics, it is important to understand the subjective perception by the human ear in its relation to the sound intensity and spectral content of a sound signal.
The physical characteristics of sound can be measured with considerable precision by means of standard acoustic instrumentation, and the results of such measurements can be expressed in terms of precise physical parameters, as described in section 2.2. By contrast, the interpretative
47 characteristics of human hearing are expressible in terms of a set of subjective parameters, determined by experiments that lead to statistical predictions of the subjective appreciation of sound by an average human person.
2.7.1 Fundamental Properties of the Human Ear The human ear is an exceptional receiver, superior to any artificial receivers such as microphones. In fact, the ear allows a very fine analysis of the registered sound waves, be it a frequency analysis or a directional analysis, which can take place for a much-extended dynamic range.
The small pressure fluctuations that a human can perceive as sound waves are imposed on a relatively stable atmospheric pressure (average value: 101.325 kPa ). Given the way the ear is constructed, it is not sensitive to this constant pressure only to the smaller acoustic fluctuations.
The human ear is one of the most intricate and delicate mechanical structures in the human body, and consists of three main parts: the outer ear, the middle ear and the inner ear. Sound waves approaching the ear enter either directly or are reflected by the pinnae down the meatus (auditory canal, or ear canal) and are conducted to the cochlea by the three auditory ossicles, the malleus (hammer), the incus (anvil) and the stapes (stirrup). Their vibrations are conducted up the cochlea by the basilar fluid, which excites about 30000 small hair cells on the surface of the Organ of Corti, which is attached to the top of the basilar membrane. The vibrations of these hair cells (dendrites) stimulate neurons to produce electrical impulses, which are sent along the auditory nerve to the brain for processing.
Figure 12: Left: Sketch of the human ear. Right: Cross section of the cochlea duct. (After [Kinsler et al., 1982])
48 Sound waves travel through the meatus (auditory canal) down to the tympanic membrane (eardrum). The auditory canal can resonate and amplify sounds within a frequency range of about 2000 Hz to 5500 Hz by up to a factor of 10. Successive compressions and rarefactions of air reaching the eardrum result in a change in pressure between the outer ear and the middle ear. The Eustachian tube helps to keep the middle ear at atmospheric pressure. The difference in pressure between the sound wave striking the outer surface of the eardrum and normal atmospheric pressure on the inside of the eardrum causes the eardrum to vibrate. Within the middle ear, an air- filled cavity of about 2 cm3 , vibrations travel through the three ossicles to the cochlea. These ossicles act as interlocking levers, which amplify the force of the eardrum striking the hammer. Since the oval window of the cochlea is smaller than the eardrum, this causes a further amplification of the sound vibration, up to 20 times at some frequencies.
2.7.1.1 Intensity Perception of the Auditory System. Thresholds. Through extensive empirical testing it has been clearly shown that the ear's response to a sound is proportionate, not to the absolute value of a stimulus, but to the ratio of the actual intensity of the sound to the threshold of audibility intensity. The threshold of audibility is the minimum perceptible free-field intensity of a tone that can be detected at each frequency over the entire audible range of the ear. Further to this fact, Fechner's law states that the relationship is a logarithmic one.
⎡ measured intensity ⎤ response ∝ log ⎢ ⎥ (71) ⎣ threshold of audibility⎦
Sound level measurements are generally referenced to a standard threshold of audibility at 1000 Hz for the human ear which can be stated in terms of a reference sound intensity of 10−−12Wm 2 , which corresponds approximately to a reference pressure of 210× −6 Pa .
Frequency: 20Hz− 20000 Hz
Intensity: 10−12Wm−− 2− 10 Wm 2
Pressure: 2×− 10−6 Pa 200 Pa
Table 1: Audible ranges of the human ear.
49 Table 1 details the frequency, intensity and pressure ranges of audible sound. The upper limit actually represents the threshold of pain, when the sound is so loud that it actually hurts the ear and may cause physiological damages.
2.7.1.2 Equal Loudness Contours Sounds that seem to the ear to be equally loud are said to have the same subjective strength or loudness. In determining the loudness of a sound, one compares it with a normal tone of 1000 Hz, whose physical energy is adjusted so that the sound and the normal tone appear to the ear to be equally forcible. The loudness level, expressed in phons, of the sound in question is then equal to the physical intensity of the normal tone. At 1000 Hz, therefore, phon scale and decibel scale become identical, but for all other frequencies, they will be different.
By comparing various tones of differing loudness levels with the normal tone one obtains the equal loudness level contours. These contours, which are shown in Figure 13, are based on data collected by Robinson and Dadson [1956] and are available as an international standard [ISO 226, 1987]. The equal loudness curves represent averages derived from testing a large number of human subjects.
Figure 13: Equal Loudness Contours.
50 2.7.1.3 Frequency Perception of the Auditory System The sensibility of the human ear to frequency varies considerably between individuals. For infants, the highest audible sound frequency is situated near 20 kHz, whereby for adults the value is located near 16 kHz or less. This value decreases considerably with growing age of people. The lowest audible sound frequency is generally located near the 16 Hz, and more or less independent of age. The bandwidth of an average human ear is therefore approximately equal to 20 kHz, which justifies, among other factors, that the usual sampling rates in digital audio are twice this bandwidth (44.1 kHz for CD, 48 kHz for DVD or DAT) according to the well-known Nyquist sampling frequency.
The human ear is particularly sensitive to sound frequencies comprised between 500 Hz and 5 kHz. This particular sensitive zone, which corresponds roughly to the frequency content of speech, is not based on any particular property of the inner ear, but is simply due to the resonance boosting in the auditory canal, whose 3 cm length corresponds to a quarter wavelength at 3 kHz. In this zone, the human ear can perceive differences in frequency of the order of the 3 Hz. For frequencies lower than 70 Hz, this value grows up to about 10 Hz, and for high frequencies, it is around 30 Hz.
An average human ear can distinguish around 2000 changes in frequency content across the audible range. These changes in frequency content are more exactly described by another subjective characteristic of sound, the pitch. Like loudness, it is a complex characteristic and is dependent on various physical quantities. While determined primarily by frequency, intensity and waveform also modify its perception. The most pronounced decrease in pitch with increasing loudness occurs for tones with frequencies below 300 Hz. For frequencies between 500 Hz and 3 kHz, the pitch of a tone is relatively independent of its loudness, while for tones with frequencies above 4 kHz, the pitch increases with loudness, the increase being larger for higher frequencies.
For any particular loudness, it is possible to attribute numbers to perceived pitches describing how high they sound, thereby establishing a pitch scale for pure tones. The reference frequency is 1 kHz, and the tone corresponding to this frequency is said to have a pitch of 1000 mel. A tone whose pitch is 500 mel sounds as half as high, and a tone with 2000 mel sounds twice as high, as the 1000 mel tone.
51 2.7.1.4 Critical Bandwidths and Masking When sounds with rich frequency content are perceived, a phenomenon known as masking can take place, whereby a part of the signal can become inaudible. Studies about this masking phenomenon have shown that one sound could mask another one only if it possesses frequencies comprised in a band centred on the frequency of the sound being masked. This result shows that the auditory system (at the level of the basilar membrane) analyses the sound signals in different critical bandwidths. These bandwidths overlap themselves in a continuous manner, but they are commonly divided into 24 adjacent bands. The critical bandwidth is nearly 1/3 octave for frequencies above about 400 Hz. One considers therefore, that it is unnecessary to use larger frequency resolutions, justifying the current practice of doing acoustic measurements in 1/3 octave steps. This behaviour of the auditory system, acting like a collection of parallel filters, each with its own bandwidth, is equally the origin of the methods of acoustic simulations in octave bands.
Centre Frequency [Hz] Lower Frequency [Hz] Upper Frequency [Hz] 16.0 11.3 22.4 31.5 22.4 45.0 63.0 45.0 90.0 125.0 90.0 180.0 250.0 180.0 355.0 500.0 355.0 710.0 1000.0 710.0 1400.0 2000.0 1400.0 2800.0 4000.0 2800.0 5600.0 8000.0 5600.0 11200.0 16000.0 11200.0 22400.0 Table 2: Standard centre, lower and upper frequencies for octave bands [ISO 266, 1997]
The frequency scale is normally divided into contiguous bands. The b th band begins at the lower frequency f1 ()b and ends at the upper frequency f2 ()b . The partitioning is said to be into proportional frequency bands if the ratio f12()bfb () is the same for each band b . The centre frequency f0 ()b of any such band is defined as the geometric mean f12()bf () b, which is always lower than the corresponding average mean. An octave band is a band for which f21= 2 f , while a
52 13 1/3 octave band is one for which f21= 2 f . The standard centre frequencies and related octave band intervals are given in Table 2 [ISO 266, 1997].
2.7.1.5 Time Perception The auditory system possesses equally a certain time resolution. Different studies have shown that the auditory system realizes a kind of time integration of the received sound signals.
A phenomenon linked with the temporal resolution of the auditory system is the masking intervening on the arriving of two coherent sounds at neighbouring instants. In this case, the first arrived sound masks the second one, and this phenomenon is named Anteriority Effect, or “The Law of the First Wave Front”, first discovered in 1849. With the work of Haas after 1951 [Haas, 1951] this phenomenon is currently called the Haas Effect, which has equally consequences in the perception of sound reflections in enclosures. It is obvious that this masking effect depends on the relative sound level of both signals.
If it is clear that the auditory system is particularly sensitive to the intensity and frequency variations of a sound signal, the effect of phase variations on the perceived sound are not so clear. Some studies indicate that there are some perceptual differences dependent on the phase of the signals [Plomp and Steeneken, 1969], while there are others that claim that phase variations are almost inaudible [Kuttruff, 1991]. Nevertheless, it seems that any existing perceptual differences are more pronounced with low frequency stimulus, being more or less independent of the subjective intensity of the signal.
2.7.1.6 Spatial Sound Perception The human auditory system perceives subjective characteristics of sound signals, but, in addition, it also provides spatial information about the location and size of a particular sound source. This means that the reconstructed spatial information is three-dimensional, although the angular discriminative capability is much finer in the so-called horizontal plane than in the median plane (the plane that crosses a human head, splitting it in the left and right brain hemispheres).
Perception of the Distance to a Sound Source
53 The distance to a sound source perception is mainly determined by two factors, the first one being the inverse square law attenuation of a free-field spherical sound wave, and the second being the changes to the spectral content of a sound signal due to atmospheric attenuation and diffraction around obstacles.
Further factors can also account for the distance perception. For example, inside enclosures the ratio between the direct sound and reflected sound energies permits to gain the perception of how far the sound source is.
Perception of Direction
This is one of the most important characteristics of the auditory system. Sound source localisation is achieved in a static case primarily with three cues [Blauert, 1983]: the interaural time difference ITD, the interaural level difference ILD and the frequency-dependent filtering due to the pinnae, the head and the torso of the listener.
The combined frequency (or time) domain representation of these static localisation cues is often referred to as the head-related transfer function, or in short the HRTF. A HRTF represents a free- field transfer function from a fixed point in space to a point in the listener’s ear canal, and can be measured either for an open or for a blocked ear canal. The knowledge of the HRTF of a listener allows the reproduction of the three-dimensional spatial localisation of a virtual sound source.
Dynamic and non-free-field cues such as head movements or room reflections are also important factors in the directional perception of the auditory system, mainly for sound localisation in the median plane.
The directional angular resolution in the horizontal plane is around two degrees. Studies have shown that the auditory system is able to detect ITDs as small as 30µs [Bauer, 1984]], which allows extraordinary angular resolution, as mentioned above.
Perception of Multiple Sources
54 The phenomena of time integration and of masking have consequences in the localisation of sound in the case of multiple sources. If two sources emitting coherent signals are considered, then three distinct perceptions can take place, as summarised in the following points:
• If the delays and sound levels of both signals are almost identical, then the auditory system perceives a single event, whose localisation depends on the position of both sources.
• If the delay of one sound signal relative to the other is comprised between 630µs and 1ms , then the Haas Effect takes place. A single event is perceived, its localisation depending on the position of the source whose signal arrives first.
• If the relative delays (larger than about 2 ms) and relative levels become more pronounced, then two events are perceived, each localised at the sources’ positions.
55
56 Chapter 3
SOUND FIELDS IN ENCLOSURES
3.1 Introduction Sound fields inside enclosures are a superposition of the sound coming directly from the source and of the sound reflected and scattered by the walls and by the different objects present in the room. Sound components that have undergone one or more reflections constitute the so-called reverberant field. This reverberant field corresponds to a series of echoes for an impulsive sound source (a source emitting an infinitesimally vanishing short signal). In some rooms, the direct wave radiated by a sound source can predominate relatively to the reverberant field. Examples of this type of rooms, which are called “dead” rooms, are recording studios and small, well-furnished rooms. On the contrary, in other enclosures the reverberant field can be orders of magnitude higher that the direct field alone. In this later case, one speaks of “live” rooms. Churches, industrial buildings (with hard walls and few fittings) and underground stations are good examples of this type of enclosures. In the limiting case, when the direct wave largely predominates everywhere, the room is said to be anechoic, and rooms designed in this manner are called anechoic chambers. On the contrary, if the reverberant field predominates considerably, then the designed rooms are called reverberation chambers.
Sound fields that built up inside enclosures exhibit a very complex spatial and time structure, or, in completely corresponding terms, the frequency response at a receiver is a very complicated function. Because of the wave nature of sound, this frequency response shows a high degree of variation with source and receiver position; just a small shift in the positions of either source or receiver can considerably change the corresponding time structure of the reverberation.
Theoretically, the response of the room can be solved exactly by using the wave equation together with the boundary conditions that represent the acoustical behaviour of the walls of the enclosure. In practice, however, the complexities introduced by any but the simplest conditions make this kind of calculation often intractable. In addition, even if practical, such a calculation will yield far more information than often necessary to characterise the sound field inside the room to the level of
57 detail that is useful. Nevertheless, only a wave-based approach to the problem of room acoustics can shed light into the complete complex transient and steady-state response of a room to an acoustical disturbance.
3.2 The Wave Equation and Boundary Conditions The sound field inside an enclosure can be obtained by solving a set of differential equations, which are the wave equation (8), and the boundary conditions that translate the acoustic reaction of the walls. Only locally reacting walls are normally considered, whereby the relation (37) can be used in order to state the boundary conditions that the sound pressure field must obey over the surfaces of the enclosure.
If a harmonic time law for the pressure variations is considered, that is Φ=Φ(,)rrte ()itωˆ , then the governing equations are:
1(,)∂Φ22r t ωˆ ∇Φ22(,)rrrt = ⇒ ∇Φ () + Φ () = 0 ct22∂ c 2 (72) ωωˆ =+if δ;2;Im[]0; ω = π δ = inside the volume of the room, and
ωˆ ζωˆ(,ssnss )∇Φ ()i () +i Φ () = 0 (73) out c over the enclosure’s boundary that is described by a spatially varying specific acoustic impedance ζˆ(,s ω ). ns() refers to the outward normal to the wall. ωˆ is a complex quantity called the out complex angular frequency of the solution, whereby its real part is just the normal angular frequency for a harmonic time varying quantity and its imaginary part is called the damping constant, allowing the acoustical disturbance to be damped in time.
The time independent version of the wave equation, as given by (72), is known as the homogeneous Helmoltz Equation and the boundary conditions in (73) are a type of Robin boundary conditions.
58 Most analytic solutions to the Helmoltz Equation together with the boundary conditions (73) are obtained by power series development of an initial guess solution, the “Ansatz”. It should be noted that the usual approach for solving the Helmoltz Equation (and other partial differential equations) is based on separation of variables, whereby the multi-dimensional problem is converted into a set of coupled one-dimensional differential equations. Theoretical analysis is thus restricted to room shapes corresponding to the eleven coordinate systems, which allow separation of the wave equation. It is fortunate that the most usual room shape, the rectangular enclosure, is also the easiest to obtain an analytic solution. However, even for this case, the analysis is complex enough so that the physical picture is often lost among the obtruding mathematical details.
3.3 Natural Modes for the Rectangular Room The concept of a room mode, first introduced by Lord Rayleigh [1877], leads to the modal theory of room acoustics. This theory asserts that every sound field inside an enclosure can be thought of the superposition of several natural room modes, which are excited by a sound source. According to this interpretation, room acoustics equal the acoustics of a resonating cavity.
The case of a rectangular enclosure with locally reacting walls, with each of them having homogeneous specific impedance, is rather simple to solve by separation of the three Cartesian variables, and has been considered for example in [Morse and Bolt, 1944]. Therefore, one assumes a solution of the form:
Φ(,)r txyzeexyze =ΦΦΦ ()()()itω −δω t =ΦΦΦ ()()() itˆ xyz xyz (74) ωωˆ =+i δ where a complex angular frequency ωˆ is introduced in order to cope with the inherent fact that the normal modes decay with time because of the energy losses at the boundaries. δ is thus called the damping constant. This “Ansatz” (74) when inserted into the homogeneous Wave Equation yields:
2 d 2Φ 2 2 111d Φ x y d Φ z ωˆ 2222+++=0 (75) ΦΦΦxyzdx dy dz c
59 Since each differential term only depends on its own coordinate, these terms must be equal to some separation constant in order that when summed to the ωˆ 22c term the result is equal to zero. Therefore, one has that:
ωωˆˆ22 −−κκκˆˆˆ222 −+ =0 ⇔ == κκκκ ˆˆˆˆ 2 222 + + xyzcc22 xyz (76) ωδ+ i κγˆ ==+ki c where κˆ is the characteristic value or eigenvalue. Its real part is the wave number parameter and its imaginary part is the attenuation parameter.
For the spatial functions, the following conditions apply:
111d 2Φ d 2Φ d 2Φ x ˆˆˆ222y z 222=−κκκxyz;; =− =− (77) ΦΦΦxyzdx dy dz which can be immediately solved for:
ˆˆ ˆˆ+−isκκss− is ˆ Φ=s ()sAeAeAss + = s cosh[ isκˆˆ ss +ϕ ] (78) 11ˆˆ AAe+−==ˆˆϕϕss; AAe− ss22 ss where the variable s means either of the three Cartesian coordinates, x,,.yz
The room is considered to lie parallel to the three coordinate axis, spanning the space between
x = 0 until x = Lx , from y = 0 until yL= y and from z = 0 to zL= z . The specific acoustic impedances of the six walls, which are considered as function of the angular frequency ω , are
L ˆˆ000Lx ˆˆy ˆˆLz denoted respectively by ζ xx,,,,,ζζζζζ yy zz.
The trial solution (74) , together with the solution functions given in (78), when inserted into the room’s boundary conditions (73) yield the set of simultaneous equations from where the unknowns
κˆs and ϕs are determined:
60 κˆ coth[]ϕζˆ = s ˆ 0 ; ssκˆ (79) κˆ coth[]iLκˆˆ+=−ϕζs ˆ Ls s ssκˆ s which can be combined into one transcendental eigenequation:
⎡⎤κκˆˆ ⎡ ⎤ ˆ −−10ssˆˆ 1Ls iLκζζss++coth⎢⎥ s coth ⎢ s ⎥ = 0 (80) ⎣⎦κκˆˆ ⎣ ⎦
It should be underlined that the set of equations (76) and (79), which involve 12 real valued unknowns, is consistent and that the acoustic specific impedances (i.e. the value of the angular frequency ω ) must be fixed for each solution of the eigenvalue problem in order that the resulting eigenfunctions form a complete orthogonal set of functions in Hilbert space L2. Further details can be found in [Morse and Feshbach, 1953]
As discussed in [Morse and Bolt, 1944], equation (80) has an infinite number of complex roots
κˆs,n . There is at least one root (and no more than two) with ks,n between zero and 1, another root with ks,n between 2 and 3, and so on. The different roots are distinguished by assigning to them different values of the subscript n, with n=0 for the value of κˆs with the smallest value of ks , n=1 for the next smallest value of ks , and so on. When both parallel walls have arbitrary, however,
ˆˆ0 Ls ˆ equal impedances ζ s ==ζζss then the transcendental eigenequation is simplified to:
ˆˆˆ ˆ ⎡⎤−iLxsκκ sκ ˆ ⎡⎤ L ss κ coth⎢⎥=⇔=ζζssi tan ⎢⎥ (81) ⎣⎦22κκˆˆs ⎣⎦
Plots of the real and imaginary parts of the roots of equation (81) as functions of the magnitude and phase angle of the specific acoustic impedance were given in [Morse and Bolt, 1944] and are known as Morse charts (conformal transformations) in the literature.
A numerical procedure can be established, in principle, for finding the complex roots of the eigenequations, but it should be noted that these equations are not well suited because of the infinitely many branch cuts existing in the conformal transformations. In addition, the fact that the
61 eigenequations involve inverse hyperbolic functions is prone to numerical difficulties. This means that the numerical method can jump without control from one solution to the other. An alternate form of the eigenequation (80) that is better suited for a numerical solver, which involves an entire complex function, is given in [Bistafa and Morrissey, 2003].
In cases where the acoustical specific impedances of the walls are considerably large (ζˆ >15 for frequencies above 500 Hz for a medium sized room, as indicative values), then the following approximate solution for the roots of the eigenequation are valid:
1 ⎡⎤iκˆ ⎛⎞112 κˆs,0 =+⎢⎥⎜⎟; ⎜⎟ˆˆ0 Ls ⎢⎥Ls ζζss ⎣⎦⎝⎠ (82) 2 πκi ˆ ⎛⎞⎛⎞11κˆ 2 L 11 ˆ s κ sn, =+nn s⎜⎟⎜⎟ ++33 +; s = 1,2,3... Lnππ⎜⎟⎜⎟ˆˆ00LLss n ˆˆ ss⎝⎠⎝⎠ζζss s ζζ ss
The solution of equations (79) for the three pairs of walls makes it possible to compute the eigenvalues and the associated eigenfunctions of the boundary value problem. The eigenfunctions given by the “Ansatz” (74) together with the spatial functions (78) satisfy the wave equation. Therefore, for each room mode, a direct substitution results in:
ωˆ 2 ∇Φ2 (,rrωωˆˆ ) +N Φ (, ) = 0 (83) NN c2 NN
where N stands for the trio of numbers nnnxyz,,. The corresponding eigenvalues
ωˆ NNN=+ωδi are obtained from the following expression:
2 ωωδˆˆˆˆ22222=+()ic =⎡ κκκ + +⎤ (84) N NN⎣ xnynzn,,,xyz⎦
The real part ωN of the complex angular frequency ωˆ N is the natural angular frequency of the damped standing wave Φ (,r ωˆ ) and the imaginary part δ is the temporal damping constant, as NN N mentioned before. The damping constant is usually much smaller than the natural angular frequency of each mode and therefore an approximate expression for these is given by:
62
1 ωγγγ≅−+−+−ck⎡⎤2222 k k 222 ; N⎣⎦ xnxnynynznzn,,,,,,xxy yzz (85) ⎡⎤k γ k γ k γ xn,,xx xn yn,,yy yn zn,,zz zn δ N ≅++c ⎢⎥ ⎣⎦⎢⎥κκκˆˆˆ
In summary, each damped natural room mode, portrayed by a standing wave inside the enclosure, exhibits spatial attenuation as given by the attenuation parameter γ . The spatial behaviour of s,ns each damped natural mode is described by the wave number parameter k and each of the s,ns eigenmodes possesses a natural frequency of oscillation as given by (85) and an associated temporal damping, determined by the imaginary part of the complex angular frequency, that is approximately given by the δ N in equation (85).
The particular case of a rectangular enclosure with rigid walls having infinite specific acoustic
ˆˆ0 Ls impedance is immediately obtained from the above equations. Letting ζζss= →∞ in equation (82) one sees that the roots for the enclosure with rigid walls are given by:
π κ sn, ==nn s; s 0,1,2,3... Ls (86) γ = 0 sn, s and the eigenvalues, which are real in this case, are therefore:
2 2 2 ⎡⎤⎛⎞n π ⎛⎞n π ⎛⎞n π 2 ⎢⎥x y z κ nnn=++⎜⎟⎜⎟⎜⎟;nnn x , y , z = 0,1,2,3... (87) xyz ⎢⎥LLL⎜⎟ ⎣⎦⎝⎠xyz⎝⎠⎝⎠
Any combination of integers (,nnnxyz ,), each assumed as nonnegative to avoid redundancy, gives a specific room mode. The associated natural mode frequencies are given, accordingly to (85) as:
1 2 2 2 2 c ⎡⎤⎛⎞n π ⎛⎞n π ⎛⎞n π ⎢⎥x y z fnnnnnn=++⎜⎟⎜⎟⎜⎟; x , y , z = 0,1,2,3... (88) xyz 2π ⎢⎥LLL⎜⎟ ⎣⎦⎝⎠xyz⎝⎠⎝⎠
63 and the associated eigenfunctions are simply products of harmonic functions of the own Cartesian coordinates:
⎡⎤n π ⎡⎤nyπ ⎡n π ⎤ Φ=ΦΦΦ(r ,txyzeAx ) ( ) ( ) ( )itω = cosx cos⎢⎥ y cos z ze itω (89) xyz ⎢⎥ ⎢ ⎥ ⎣⎦LLLxyz⎣⎦⎢⎥⎣ ⎦
The standing waves described by this equation (and by the corresponding formulas for the damped natural modes case) can be divided into three classes: those for which none of the n’s are zero will be called oblique waves, those for which one n is zero will be called tangential waves and those for which two n’s are zero will be called the axial waves. Each of these waves has different properties, for example in their damping constants for the case of non-rigid walls [see for example Brüel 1951 for a simplified discussion on the damping of the different waves]. In addition, there is also a difference in energy content. In general, the axial wave has four times the energy of an oblique wave and twice the energy of a tangential wave. However, the axial waves group has the feeblest damping and thus govern, in general, the final slope of the sound decay in rooms [Brüel, 1951].
The normal modes and their natural frequencies depend primarily on the shape and size of the room, whereas their rates of damping depend mainly on the values of the specific acoustic impedances of the walls.
The frequency distribution of the normal room modes, first given in [Bolt, 1939, Maa, 1939]:
32 4ππ⎛⎞ffLf ⎛⎞ Nfmodes()=+++ V⎜⎟ S S ⎜⎟ Of () 348⎝⎠ccc ⎝⎠
V== volume;; SS total surface area (90)
LLLL=++4(xyz ) and it can be shown that for rooms of arbitrary shape the first and second terms of equation (90) are still applicable, provided that SS is replaced by an effective surface area (often smaller than the actual total surface area) being determined by the area of a smoothed-out average surface around the room such that the total volume contained is equal to V [Morse and Bolt, 1944]. The first term, being always the leading one, accounts for the number of oblique waves, while the second and third ones account respectively for the tangential and axial waves.
64 The number of modes in a frequency band of width ∆f and centred at frequency f (modal density) can be estimated by:
dN ffL2 π modes ()f =+++ 4πVS Of () (91) df c3228S c c
As an example of the number of natural room modes, consider a medium sized room with a volume of about 168 m3 (for example a rectangular living room with 8 m x 7 m x 3 m). In this case the number of allowed eigenfrequencies up to 1 kHz is equal to 19208 and the average spacing between successive natural frequencies at 1 kHz is around 0.018 Hz. In the case of upper frequency 16 kHz (to cover the audible range) the allowed eigenfrequencies are 7.36× 107 and the average spacing becomes only 7.25× 10−5 Hz. Therefore, one can see that these figures underline the enormous volume of numerical calculations, which are required to determine the sound field accurately inside a very simple room. If the room becomes larger, then the numbers grow in proportion to the volume. One can thus conclude that the modal theory of room acoustics is only applicable for small-sized rooms and for low frequencies.
3.4 Steady-state and Transient Sound Fields inside Enclosures The set of eigenfunctions given in equation (89) has the property that they all are mutually orthogonal. In the case of modal eigenfunctions for general shapes of rooms, it can also be shown that they are mutually orthogonal [Pierce, 1994]. Furthermore, since any eigenfunction multiplied by a constant is still an eigenfunction, one chooses a determined factor such that they are normalized to have a mean squared volume average of one. The set of orthonormal eigenfunctions therefore satisfy:
ΦΦ()rr () =δ V ∫∫∫ NM NM (92) V
where δ NM is the Kronecker symbol and V is the enclosure’s volume.
The set of orthonormal eigenfunctions can be shown to form a complete set, whereby any well- behaved spatial function within the room can be approximated as a linear combination of the
65 Φ ()r , since they are eigenfunctions of a self-adjoint (Hermitian) differential system of the second N order [for example Apostol, 1969]. Therefore:
∞ fa()rr=Φ (); ∑ NN N (93) 1 afdV=Φ()rr () NN∫∫∫ V V
The analysis shown in the previous section was valid for the homogeneous wave equation, i.e. the equation without any source terms. In the case of having a simple point source with strength Q of volume velocity (units []ms−1 ) inside the room under analysis, the governing equation is now the Inhomogeneous Helmoltz Equation:
ωˆ 2 ∇Φ2 ()rrrr + Φ () =−Qδ ( − ) (94) c2 0
By using the modal expansion for the source term and for the solution that one seeks, the steady- state velocity potential in a room excited by a point source can be written as:
Q ∞ ΦΦ()rr ( ) ˆ 2 NN0 Gc(,rr0 ,ω )=− ∑ 22 (95) ViN ωωδˆ −+()NN where G(,rr ,ωˆ ) is the Green’s function of the room under consideration and represents the 0 spatial distribution of the radiation from a point source of frequency ω /2π located at the point r . 0 It can be shown [Mores and Bolt, 1944] that very close to the source this series approaches in value the free-space Green’s function as given by equation (18), or more specifically:
ω −−i rr 1 e c 0 G(,rr ,ω )= (96) 0 4π rr− 0
This part of the velocity potential can be called the coherent part. Farther away from the source, the series (95) differs from the free-space Green’s function because of the waves reflected by the
66 room’s walls. Therefore, it is made of the sum of the coherent part and of an incoherent part, which has no definitely directed flow of energy or ordered wave motion.
The steady-state velocity potential has a resonance whenever the driving frequency ω is equal to c
th th times the real part of one of the eigenvalues κˆN . At this n resonance, the corresponding n standing wave Φ N predominates considerably, having an amplitude inversely proportional to the imaginary part of κˆN for that frequency. The resonance peak for each standing wave is fairly narrow, the half-widths being given by γ N π . For usual rooms, the half-widths are of the order of 1 Hz. One can thus see that one single half-width always covers many eigenfrequencies. With a sound source emitting a pure sinusoidal wave, it is thus quite impossible to excite a single room resonance separately. Instead, the resulting steady-state sound field is made up of many simultaneously excited room modes. Exceptions can be observed only in small rooms at low excitation frequencies, where the eigenfrequencies are well separated. The limiting frequency, below which this occurs, is approximately given by the so-called Schröder cut-off frequency [Pierce, 1994]:
31c3 f = (97) Sch 4 Vδ where δ is the mean value of the damping factors, which are associated with neighbouring eigenfrequencies. The Schröder’s rule then asserts that above the cut-off frequency a sum over mode indices can be reasonably approximated by an integral. In this case, the series that yields the steady-state velocity potential is converted into an integral over frequencies.
The transient characteristics of the sound field inside the room is obtained from the steady-state response by using the operational calculus, or more specifically by applying a Fourier transform, as given by (29), to equation (95). This yields:
1 Q ∞ ⎡ ∞ ΦΦ()rr ( ) ⎤ gt(,rr ,)=− c2 NN 0 editω ω (98) 0 ∑ ⎢ ∫ ˆ 22⎥ 2π ViN ⎣−∞ ωωδ−+()NN ⎦
If the damping factors are assumed much smaller than the wave frequencies, then one has:
67 ∞ ⎡ ∞ ˆ itω ⎤ AeN gt(,rr0 ,)= ⎢ dω⎥ ∑ ∫ ωω22−−2i δω N ⎣−∞ NN⎦ (99) 1 Q iϕ Acˆ =−2 Φ()rr Φ ( ) = AeAN NNNN2π V 0
whose denominator possesses simple poles at approximately ω = ±+ωδNNi . The integral in (99) can be readily evaluated by using the Residue Theorem, yielding:
⎧0:0t ≤ ⎪ ∞ gt(,rr0 ,)= ⎨ A −δ t (100) 8sin()πωϕN te+ N :0 t> ⎪∑ NAN ⎩ N ωN
The resulting sound wave in the room, after t = 0 , consists of a series of individual standing waves oscillating with its own natural frequency ωN and damping out in time with its own damping factor
δ N . This kind of sound wave, started at t = 0 and subsequently dying out exponentially, is called reverberation. Equation (100) defines what is called the impulse response of the room under consideration, since it gives the room response to an infinitely short sound impulse (a Dirac delta stimulus). Therefore, impulse response and steady-state room response are related to each other by the Fourier Transform.
In the more general case when a room is excited not by an ideal impulse but by a preliminary unspecified signal st()which is zero for all time t > 0 , then the resulting room’s response is given by the convolution integral:
0 Ψ=(,rr ,)tsgtd ()(,τττ rr , − ) : t ≥ 0 00∫ −∞ (101) ∞ A Ψ=(,rr ,)te 8παωϕβωϕN −δN t sin( t +− ) cos( t + ) 0 ∑ ()NNANNN NA N ωN and where the coefficients of the trigonometric functions are given by:
00 α ==sedsed()cos(τωττ )δτN ; β ()sin( τωττ ) δτN NN∫∫ NN (102) −∞ −∞
68 It is obvious that only the modes whose eigenfrequencies are enlaced in the spectrum of the excitation signal contribute overwhelmingly to the reverberation process.
The mean squared pressure of the reverberant field is given simply by:
∞ 22−2δN t pe= ∑ηN ; N ⎛⎞2 (103) 2222⎛⎞δ N 22 ηπραβNN=++64A 0 ⎜⎟ 1 ⎜⎟⎡ NN⎤ ⎜⎟ω ⎣ ⎦ ⎝⎠⎝⎠N and thus it is equal to a combination of exponentials, and not just a single constant times a single exponential.
3.5 Reverberation Time The concept of reverberation time is one of the most important in room acoustics. The reverberation time is defined as the length of time it takes the mean energy of the wave to reduce to a millionth part of its initial mean value, or in completely equivalent terms, it is the time needed to the sound pressure level to decay 60 dB relatively to its initial mean level.
It can be shown [Kuttruff, 1979] that if the damping factors are distributed more or less uniformly around some mean value, then the reverberation time is given by the following expression:
3ln10 Ts= [] (104) 60 δ where δ is the weighted average of the nearly uniform damping factors of all the natural room modes that participate in the reverberation process.
In more or less complex shaped rooms, the reverberation curve is more or less a straight line (when plotted on a logarithmic scale), from whose negative slope one computes the reverberation time. Superimposed on this straight line there are certain random or quasi-random fluctuations, as can be seen in Figure 14.
69 Decay Curve HLogarithmic Scale L 0
-10
-20
dB -30
-40
-50
0.1 0.2 0.3 0.4 0.5 Time @sD
Figure 14: Example of a reverberation decay curve of a room
Typical reverberation times go from about 0.3s for small living rooms up to 10 s in the case of large churches and reverberation chambers, while most multi-purpose large rooms present values between 0.7s and 2 s . Therefore, the average damping factors found in practice have values comprised between 1 s−1 and 20 s−1 .
The solution to obtain the sound field inside an enclosure was described in the previous sections. However, due to the mathematical details, the actual sound propagation mechanism inside the enclosure can be somewhat obscured.
It was shown that the sound from an omni-directional point source in free space spreads over a sphere of continually increasing radius. In time, the sound will pass a listener, and thereafter is lost as far as the particular position is concerned. The situation in an enclosed room is quite different. The sound begins to spread out on the surface of the sphere of continually increasing radius but very soon some part of this spherical wave front reaches the walls, floor and ceiling, where it is reflected. Other parts of the original spherical wave are reflected by bounding surfaces until the original wave front is broken up and reflected into many different directions. At each reflection, some sound energy is absorbed by the reflecting surfaces, with the result that after many reflections the sound energy is ultimately reduced to nothing. During this process of reverberation, or sound
70 decay, the listener will have received some of the sound directly from the source, the amount depending on the relative positions of the source and receiver, and some indirectly by the various acoustic paths, which reach him from different directions and at different times. The sound is therefore recorded only once by the direct path but it will be recorded many times via the reflected paths, successive echoes becoming weaker until they become inaudible.
As the reflected paths are longer than the direct path from source to receiver, the time taken for sound to reach the listener is not a single value but is spread out in time for about one second in a medium room and several seconds in a large hall. As the listener receives sound by a number of different acoustic paths and the brain integrates these, the loudness will be greater in a room than for a corresponding source and receiver relation in free space. The actual increase in loudness depends on whether much or little sound energy is lost at each reflection. The acoustic sensation resulting from the “slow” decay of sound is the reverberation of the enclosure.
3.5.1 Time Distribution of Reflections It is interesting to examine the time structure of the impulse response of a room by analysing the so-called room energy impulse response, or echogram. This echogram represents the instantaneous power of the impulse response in function of time. The time structure of the reverberation inside a room can usually be subdivided into three distinct parts, which are more or less clearly visible in the echogram of Figure 15:
• The direct sound from the sound source arriving at the receiver
• The Early Reflections; these are the reverberant components of the sound wave that have suffered some reflections before they arrive at the receiving point. Normally, the reflection order of these early reflections is between 1 and 5
• The Late Reverberation; numerous reverberant wave components that suffered a high number of reflections of high orders arrive at the receiver. These numerous components merge into a more or less continuum and therefore the reflections lose their individuality
71 -25
-50
-75
-100
-125
-150
-175
0.05 0.1 0.15 0.2 0.25
Figure 15: Echogram with temporal structure of the room energy impulse response
3.5.2 Spatial-Directional Characteristics of Reverberation. Diffuse Sound Fields. The late reverberation is composed of numerous high order reflected components that stem from all possible directions surrounding a receiver. In this case, the concept of a diffuse sound field becomes apparent. By definition, a diffuse sound field is characterised by both following conditions [Pierce, 1994]:
1. Spatial Uniformity: the local spatial average of the acoustic energy density is the same throughout the room. Therefore, the energy density (for a given frequency
band) is given by a constant value wb .
2. Directional Uniformity: in any position inside the room, incoherent sound waves are incident from every possible direction, with equal intensity and with random phases. By defining a directional energy density D()e as the energy per unit volume and per unit solid angle of propagation direction, the case of a diffuse sound field is represented by Dw()e = 4π . b
72 These are rather strong conditions for a sound field and they constitute the basis of the so-called statistical acoustics, which allow the immediate determination of the reverberation sound field in enclosures. Conditions 1 and 2 are seldom met in practice and depend among many factors, the main ones being the volume of the room, the absorption of the walls and, also, the degree in which they are able to “diffuse” the incident sound waves. A general discussion on the validity of the diffuse field assumptions can be found in [Hodgson, 1996].
It should also be underlined that the concept of diffuse sound field is sometimes misunderstood with the concept of diffusely reflecting surfaces. Diffusely reflecting surfaces are surfaces, which do not reflect a sound wave in a pure specular way, but scatter some part of the wave into other directions (see also section 2.6.1). It can be shown that, even for rooms possessing diffusely reflecting walls, the resulting sound field may not be completely diffuse.
The diffuse field hypothesis allows the establishment of energy balance equations from which the general laws of reverberation can be derived. In particular, under this hypothesis, it is readily shown that the sound decay inside a room follows a pure exponentially decreasing law, meaning, in connection with the wave theory of room acoustics, that there is a single damping factor for the reverberation process.
3.5.3 Quantitative Measures for Sound Fields inside Enclosures Traditionally, reverberation time and other sound decay measures were considered the primary evaluation parameters for the acoustic quality of rooms. However, researchers recognized the inadequacies of using only these measures alone and introduced a variety of additional measures aimed at characterising more correctly the sound fields inside rooms. A big effort has been made in order to correlate the proposed measures with the subjective acoustic perception of human listeners. Among the great panoply of proposed quantitative measures, the most used in current studies, the majority of which is defined in [ISO3382, 1997], are the following:
1. Reverberation Time (TTT15;; 20 30 ). As defined under section 3.5, it is the time needed so that the sound pressure level decays 60 dB relatively to its initial mean level. In general, the reverberation time is calculated after having obtained the integration of the energy inside the room impulse response, which gives the sound
73 decay curve. This procedure is known as Schröder’s backward integration [Schröder, 1965]. Thus, one has that the sound decay curve E()t is given by:
1 ∞ Et()= ∫ h2 (τ ) dτ E(0) t (105) hg()ττ∝ [] ()2
where ht()is the energy impulse response of the room under consideration. A linear regression is then fit to the logarithm of the sound decay curve between two given values of the linear decay. These values can be (-5 dB, -35 dB) yielding then
the corresponding T30 reverberation time, (-5 dB, -20 dB) yielding the T15
reverberation time or (-5 dB, -25 dB) yielding the T20 reverberation time. All these times are translated to 60 dB decay, therefore the determined times that sound takes to drop x dB is multiplied by a factor equal to 60 / x . The adopted unit is the second.
2. Early Decay Time (EDT) [Jordan, 1970](after work from [Atal, Sessler and Schröder, 1965]). The procedure is the same as with the calculation of the reverberation time, but with the linear regression fit between the values (0 dB, -10 dB). The EDT is then equal to the time taken for the decay multiplied by a factor of 6. The adopted unit is also the second.
3. Definition ( D50 ) [Thiele, 1953]. Definition is the percentage of sound energy that arrives at the receiver in the first 50 ms (early energy) in relation to the total energy. It is correlated with speech intelligibility.
50ms ⎡⎤2 ⎢⎥∫ htdt() ⎢⎥0 D50 =100∞ [%] (106) ⎢⎥2 ⎢⎥∫ htdt() ⎣⎦0
74 4. Speech Clarity Index (C50 ) [Beranek and Schultz, 1965]. It is given by the logarithm of the ratio between sound energy arriving in the first 50 ms and the sound energy arriving after this instant. It is therefore a measure of the early to late sound energy ratio. This measure correlates also with speech intelligibility.
50ms ⎡⎤2 ⎢⎥∫ htdt() ⎢⎥0 CdB50 =10log∞ [ ] (107) ⎢⎥2 ⎢⎥∫ htdt() ⎣⎦50ms
5. Music Clarity Index (C80 ) [Reichardt et al., 1975]. It is given by the ratio between sound energy arriving in the first 80 ms and sound energy arriving after. This measure correlates well with the clarity at which music is heard in a concert hall.
80ms ⎡⎤2 ⎢⎥∫ htdt() ⎢⎥0 CdB80 =10log∞ [ ] (108) ⎢⎥2 ⎢⎥∫ htdt() ⎣⎦80ms
6. Relative Sound Level (G ) [Lehmann – 1976]. The logarithmic ratio of the total sound energy ratio in the measured impulse response to the total sound energy in
2 the measured impulse response htA ()of the same source at a distance of 10 m in an anechoic chamber.
∞ ⎡⎤2 ⎢⎥∫ htdt() GdB=10log⎢⎥0 [ ] (109) ⎢⎥∞ htdt2 () ⎢⎥∫ A ⎣⎦0
7. Initial-time-delay Gap (ITDG) (tI ) [Beranek, 1962]. The separation time, in milliseconds, between the arrival of the direct sound and the arrival of the first reflection.
75 8. Centre-of-Gravity Time (tS ) [Kürer, 1969]. Gives a measure of the temporal localisation of the decay process and is given by:
∞ ∫th2 () t dt 0 tsS = ∞ [] (110) ∫ htdt2 () 0
Several studies report that the central time correlates highly with both clarity
indexes C50 and C80 .
9. Bass Ratio (BR) [Beranek, 1962]. Equals the ratio between the reverberation times determined for the low frequencies and the reverberation times determined for the high frequencies:
THzTHz(125 )+ (250 ) BR = (111) THzTHz(500 )+ (1000 )
10. Lateral Energy Fraction (LEF) [Barron, 1971]. The fraction of the sound energy that arrives from lateral directions within the first 80 ms of the impulse response:
80ms htdt2 () ∫ 8 5ms LEF = 80ms (112) ∫ htdt() 0
2 where ht8 ()is the energy impulse response as measured by a “figure-of-eight” directivity microphone. The null of the microphone is intended to be directed towards the source so that the recorded impulse response is not affected by the direct sound from the source. The lower limit of 5 ms in the integration makes sure that the direct sound is eliminated in the recorded impulse response.
11. Interaural Cross Correlation Coefficient (IACC) [Keet, 1963] The IACC is a measure of the difference between the sounds at the ears of a listener. If both
76 sounds are alike then IACC = 1. On the contrary, if the sounds at the two ears are independently random signals, then IACC = 0. For impulsive sources, the IACC is determined from:
t2 htht() (−τ ) dt ∫ LR IACC()τ = t1 (113) tt22 h22() t dt h () t dt ∫∫LR tt11
where L and R designate the entrances to the left and right eardrums, with the listener facing the source. The parameter τ can vary from -1 ms to +1 ms, which is roughly the time difference between the arrival at the two ears of a wave from the left side or left side of the head. If the integration limits are chosen as
t1 = 5 ms;t2 = 80 ms, then one obtains the IACCE (early), while by setting the
limits as t1 = 80 ms; t2 = ∞ one obtains the IACCL (late). The IACCE is correlated
with a subjective sense of sound spaciousness (see next section), while the IACCL is highly correlated with the subjective sense of diffuseness of the late reverberant sound field. The maximum value of the IACC versus parameter τ for a listener facing the source is called the Maximum Interaural Cross-correlation Coefficient,
IACCMAX [Gottlob, 1973]. The smaller this value, the greater the number of reflections that come from outside the median plane, meaning that either a substantial number of reflections are lateral, or else that the sound field is diffuse.
12. Speech Transmission Index (STI); Rapid Speech Transmission Index (RASTI) [Houtgast et al., 1980] The STI and RASTI are determined from the Modulation Transfer Function of the room as is described in the norm [IEC 60268-16, 1998], and has a direct relation with the perceived speech intelligibility.
13. Stage Parameters (ST) [Gade, 1989] There are three stage parameters as given by:
77 100ms ⎡⎤2 ⎢⎥∫ htdt() ⎢⎥20ms STearly =10log10ms [ dB ] ⎢⎥2 ⎢⎥∫ htdt() ⎣⎦0 1000ms ⎡⎤2 ⎢⎥∫ htdt() ⎢⎥100ms STlate =10log10ms [ dB ] (114) ⎢⎥2 ⎢⎥∫ htdt() ⎣⎦0 1000ms ⎡⎤2 ⎢⎥∫ htdt() ⎢⎥20ms STtotal =10log10ms [ dB ] ⎢⎥2 ⎢⎥∫ htdt() ⎣⎦0
STearly is used as a descriptor of ensemble conditions, i.e. the ease of hearing other
members in an orchestra, STlate describes the impression of reverberance and
STtotal describes the support from the room to the musicians own instrument.
3.5.4 Subjective Evaluation of Room Acoustics The listeners’ impression of the acoustics of a particular room is of a complete subjective nature. Nevertheless, there are some common attributes of concert halls as subjectively judged by listeners. Beranek [Beranek, 1992] devised a rating system for audience-observed sound quality in rooms by using seven independent subjective attributes. These are subdivided into five independent room attributes and into two independent stage attributes, which apply exclusively to performance spaces. Although this rating has been derived mainly for the quality assessment of concert halls, its underlying five room attributes can be equally applied to every enclosure.
The room related attributes are:
1. Intimacy: the term intimacy characterises the listening attribute of closeness of communication between the sound source and the listener. In terms of music, this attribute is responsible for the contact between musicians and listeners. Intimacy correlates well with the initial-time-delay gap (ITDG), with the lateral energy
fraction (LEF) present in the room and also with the IACCE.
78 2. Liveness or Reverberance: this attribute is a translation of the subjective feeling of a live, resonant and reverberant room, as opposed to a highly damped sounding room, usually termed a “dead” room. This attribute correlates well with the
objective parameter reverberation time T30 , but sometimes it is even better correlated with the EDT. In this context, the concept of optimum reverberation time becomes important, meaning that according to the purpose of the room, different amounts of liveness are requested. For example, one wishes that a normal living room should not be “very live”, since people are usually accustomed to some damping in the sound inside such a room. On the contrary, for a concert hall, and depending on the musical programme, one wishes that the overall reverberance be high enough so that the perceived sound decays more or less in a slow way.
3. Warmth (also known as room timbre): this attribute characterises the sense of sound’s richness in low frequency contents, as opposed to a “thin” sound. Other terms that translate this sensation are “rich in bass” or “mellow”. This attribute correlates well with the objective parameter bass ratio (BR), defined as the ratio of the average reverberation time at 250 Hz and 500 Hz to that at 500 Hz and 1000 Hz.
4. Loudness: Every person is aware of the loudness of sound inside a room. This attribute is connected with the reverberation time and with the size of the room, being obviously a function of the acoustic power output of the sound source. Thus, two kinds of loudness can be defined: one for the direct sound and the other for the reverberant sound, a distinction that becomes quite important in large rooms. Loudness correlates well with the objective parameter relative sound level G . Beranek states that it correlates well also with the combined measures of distance from the sound source and ratio of reverberation time to the cubic volume of the room, but these measures correlate also with the relative sound level.
79 5. Diffusion and Blend: the reverberant sound field instigated from any sound source should quickly become diffuse so that there is a good blend of the overall sound throughout the enclosure. This attribute translates the subjective sense of envelopment created in a sound field, as opposed to the sense of a harsh or overly brilliant sound that originates if the room only possesses large smooth walls. Diffusion is created by all sorts of irregularities in the way of the sound waves, be it from the walls of the room or from obstacles that scatter the sound into
different directions inside the room. Diffusion correlates well with the IACCL.
The stage related attributes are:
1. Balance and Blend: these are related to the physical arrangements immediately surrounding the performers on stage, affecting the performance of the music or, to a lesser extent, the performance of drama. The loudness balance between sections of an orchestra can be affected by details of the design of the stage and furnishing. Balance and blend also depend on acoustical conditions not related to the room, as for example the music programme and the placement of the performers. There must be a return of reverberant sound from the rest of the enclosure to the stage area in order to achieve a good blend, because otherwise the performers will feel that they are operating into a sonic void space. This can cause a lecturer to speak too rapidly for the audience, project his voice too strongly, or strain his voice. In the case of musical performances, it can cause an orchestra and conductor to misjudge the behaviour of the room.
2. Ensemble: describes the ability of musicians to hear themselves and each other, affecting their ability to play in ensemble. It can be determined by the stage’s enclosure, the presence of overhead reflectors or downward-reflecting sidewalls, and by the feeling of vibrations of the stage floor.
80 Chapter 4
ACOUSTIC MODELLING OF THE SOUND FIELD IN ENCLOSURES
In order to be able to evaluate the acoustics of a room at the design stage the acoustician must be able to predict the behaviour of the sound fields inside this virtual environment. From the knowledge of the temporal and spatial behaviour of the sound field, the acoustician can then extract several important features that will lead to the determination of various acoustic measures. As it was seen in Chapter 3, there exists a definite correlation between objective acoustic parameters and the subjective assessment of the sound quality inside enclosed spaces. Therefore, the correct knowledge of these parameters permits, in principle, to determine the acoustic sensations that a certain designed room will impress on the listeners using it.
Apart from design purposes, the correct modelling and simulation of sound fields inside enclosures is also needed in order to obtain “real feeling” simulation tools and applications, with uses in entertainment, simulation, and training products. The degree of reliability of the modelled sound field will depend, of course, on the final usage of the simulation, but there is a number of minimum requirements that must be taken into account. The first very important requirement is the accurate modelling of the intervening sound sources since they are the ones that are responsible for the direct signal that reaches the listener. Other important requirements are the ability to model as correctly as possible the behaviour of the sound reflections inside the room, since they are responsible for the sense of reverberance of the perceived sound field, apart from other attributes as pointed out in section 3.5.4.
Classic approaches for the modelling of sound fields inside enclosures resorted mainly to wave based solutions. However, these techniques resulted in practice only for a class of simple geometries. Another widely used technique consisted in employing the image source method for predicting the reverberant sound field. The image source method has been applied to either approximately determine the sound field arising from the first order reflections inside an enclosure, or to obtain the statistical descriptors for the reverberant sound field inside rooms [Kuttruff, 1979].
81 With the advent of the digital computer, physicists and acousticians realized that numerical algorithms could be applied to model the sound fields inside rooms and to obtain simulation results. They could then derive many objective acoustical parameters for describing the major characteristics of the sound fields. The first documented essay on the possible use of digital computers for the prediction of sound inside rooms is due to Schröder, Atal and Bird [1962] and Schröder and Atal [1963]. These authors envisage the use of a ray tracing technique to obtain the energetic temporal response inside a reverberant room. They even point out a way of obtaining synthetic digital sound signals from anechoic recordings that simulate the reverberation of enclosures and presenting them to listeners through transaural techniques. These methods were later put into practice [Schröder, 1970].
Since then, a variety of modelling and simulation approaches has been derived and employed for predicting sound fields inside rooms of arbitrary shapes. All of the developed methods can be mainly subdivided into two categories: wave-based modelling approaches and geometrically based modelling approaches. The first one tries to solve the problem of sound propagation inside rooms by using the wave formalism of sound, which, in general, produces computationally expensive methods of moderately restricted practical application. The second modelling category simplifies underlying physical principles of sound propagation, which in general terms consists in the substitution of a sound wave by the description of the propagation of sound particles inside rooms (that usually mimic the propagation of a spherical wave front), whose trajectories are simply described by sound rays. These geometrical based techniques, which constitute the acoustic analogue of the geometrical optics approach to wave based optical problems, have to date been more or less extensively employed with satisfactory results.
In the next sections, a summarised portray of these modelling and simulation approaches is drawn, with advantages and shortcomings of every method being highlighted
4.1 Wave-based Methods The formal way of solving acoustic propagation problems inside enclosures consists in translating the underlying physical phenomena into a corresponding mathematical model. In its simplified form, this mathematical model reduces to the well-known linearised wave equation (8) together with a set of boundary conditions translating the acoustical behaviour of the walls of the room.
82 However, even in this simplified form, the mathematical complexity restricts obtaining the solutions of many acoustic problems. Moreover, even if analytic solutions can be found for a certain class of problems (normally simple), they are tremendously cumbersome to apply in practice with the result that the physical phenomena become hidden behind all the formulas (as for the case of sound fields inside rectangular enclosures, see sections 3.3 and 3.4).
Wave-based models are usually given in terms of partial differential equations. In some cases, these equations may be solved using analytical methods, which range from separation of variables methods to integral formulations in terms of variational or integral equations. For separation of variables, the solution is generally found in terms of special functions, such as Bessel, Hankel, Legendre, Hermite, Laguerre, and Whitaker functions. Quite often, the solution may be found in terms of an eigenmode expansion or a weighted sum involving special functions. The weights are found by using transforms, such as Fourier, Laplace, Hankel and Mellin. For other cases, the solution may be found in terms of sums involving orthogonal polynomials, such as the Legendre, Laguerre, or Chebychev polynomials.
Variational methods include a variety of techniques, which are used to derive a domain integral form of the governing equations. The objective of this variational approach is typically to minimize an integral of the error of the approximation over the domain or to minimize an energy or Lagrangian integral. These methods constitute the basis of the numerical Finite Element techniques, described in section 4.1.3.
Boundary Integral Equations, on the other hand, are found by using Green’s Functions. The Boundary Integral Equations may be viewed in some contexts as the superposition of a large number of Green’s Function solutions multiplied by source strengths where these strengths are adjusted to satisfy the boundary conditions. When feasible, this method usually involves an expansion in terms of eigenfunctions of the integral equation, which often converges rather slowly, particularly near the boundaries.
Wave-based models are characterised by creating very accurate results at single sound frequencies, in fact too accurate to be useful in relation to architectural environments, where results in octave bands are usually preferred. Nevertheless, there are currently actually several wave-based numerical
83 methods that are being developed and applied for successfully solving some restricted acoustic problems.
4.1.1 Analytic Methods 4.1.1.1 Modal Decomposition The theory of modal acoustics (the example of the modal approach for a rectangular enclosure was given in section 3.3) permits to decompose any solution of the wave equation in terms of the natural eigenmodes of the related homogeneous boundary conditions problem, which comprises a set of linearly independent orthonormal basis functions, the so-called eigenfunctions of the enclosure. As it was indicated for the case of the rectangular room, these eigenfunctions correspond to certain quantified eigenvalues of the wave number and of the related eigenfrequencies of the acoustical system.
It should be underlined that the determining eigenfunctions, being solutions of the associated homogeneous equations, can always be obtained as linear combinations of the basis vectors that generate the supporting Hilbert Space [Kolmogorov and Fomin, 1999]. Thus, every specific eigenmode can be obtained as a functional series of any complete basis of functions spanning the corresponding dimensional Hilbert Space (possible basis are the trigonometric functions, Legendre or Laguerre Polynomials, Bessel Functions, etc.). In this case, the steady-state response of the room must be given in terms of a double series (in contrast with equation (95) where the single series enlaces only the different room modes) over all the basis functions and over all the eigenfunctions. Another shortcoming of the modal approach resides in the fact that for obtaining the steady-state room response, from which the corresponding room impulse response is derived, many terms in the series given by equation (95) must be calculated. As pointed out in the last part of section 3.3, the number of eigenmodes grows with the third power of the driving frequency, which means an enormous calculation effort to obtain the correct steady-state response.
Usually, for the sake of simplicity, one uses for the set of basis functions the eigenfunctions that correspond to the same problem but with rigid boundaries. However, this approach shows a slow convergence rate in the resulting series, even if modal coupling is taken into account, as pointed out by Pan [1993; 1995]. A possible practical solution for increasing the convergence rate consists in using the so-called extended mode shape functions, instead of the rigid boundary eigenfunctions
84 [Pan, 1999]. An example of the modal expansion approach to interior sound field modelling is given by Jayachandran [1998].
4.1.1.2 Integral-based Methods. The Helmoltz-Huygens-Kirchhoff Theorem. The Helmoltz-Huygens-Kirchhoff Integral Theorem was already written in section 2.6.2. This Theorem results from a direct application of Green’s Formula [Apostol, 1969] to the Helmoltz Equation. Formally, integral-based methods for solving the wave equation consist in projecting the three-dimensional problem into an equivalent two-dimensional problem.
The Helmoltz-Huygens-Kirchhoff Integral Theorem states that the sound field inside a given volume is completely determined by the direct fields of the sound sources inside this volume and by the diffracted sound field originated by the boundary conditions, which represent whatever is reflected at the boundaries or, also, enters through the boundaries from outside. In terms of the acoustic pressure, this Theorem states that it suffices to know the distribution of the sound pressure and of the normal particle velocity over the surface that delimitates this volume in order to obtain the sound field at some receiving point located inside this volume.
The Helmoltz-Huygens-Kirchhoff Integral Theorem has been extensively applied in problems of sound diffraction and sound radiation in general acoustics problems, but, it has also been applied to some extent to model sound fields inside rooms [Johnson et al., 1998, Berkhout et al., 1999, Too and Wang, 2002] and even to design high-quality electro acoustic systems [Berkhout et al., 1993].
As with every wave-based technique, analytic solutions of the Helmoltz-Huygens-Kirchhoff Integral Theorem exist only for very simple problems. For practical problems in acoustics, only numerical techniques for solving the equations can be applied. However, the computational requirements are also very high, and the solutions are only achieved for rather low frequencies and simple geometries. In some cases, the numerical techniques are only even applied to solve two- dimensional problems [Berkhout, 1999, Too and Wang, 2002] and always for low frequencies. Finally, another problem of the integral based method is that one obtains frequency responses of the rooms. Further computation is therefore required if one wants to obtain room impulse responses in the time domain.
85 4.1.2 Finite Element Formulations Finite Element formulations solve the wave equation and associated boundary conditions by subdividing space and time into elements, which contrary to continuum infinitesimal quantities are of finite size and shape. The wave equation is then expressed as a discrete set of linear equations for these elements.
4.1.2.1 The Finite Element Method (FEM) The Finite Element Method (FEM) has been traditionally applied to solve structural vibration problems in mechanics. The first application of FEM to acoustics was done by Gladwell [Gladwell, 1965], who also established the general formalism in [Gladwell, 1966].
FEM has also been rarely applied to predict the characteristics of sound fields in enclosures, mainly inside passenger and luggage compartments in cars [Craggs, 1972; Shuku and Ishihara, 1973; Kopuz and Lalor, 1995], but it has also been applied to study sound fields inside small rooms [Craggs, 1986] possessing non-local reacting surfaces.
FEM is usually used in order to determine the first eigenmodes of the room under study, which then allows the calculation of the frequency transfer function of the room. However, due to the high memory and computational requirements of FEM, the calculated eigenfrequencies seldom extend above 500 Hz for rather small rooms. Therefore, FEM’s application is only viable for low frequency sound fields inside small sized rooms.
Finally, FEM has also been applied for determining the decay rates of individual room modes and the associated transient response of damped rooms, possessing planar walls but also some curved walls. In these studies, the absorptive behaviour of the boundaries was given by the acoustic specific impedance over each wall, but always considered as frequency independent. Also in this case, only the decay rates for the first hundred, or sometimes, thousand, eigenmodes have been calculated, which translates itself in the fact that the impulse response only gives the first early reflections (maximum length of the impulse response about 200 ms). These studies are reported in [Easwaran and Craggs, 1995, 1996a, 1996b]. In addition, here, the inherent restrictions of FEM become also apparent.
86 4.1.2.2 The Boundary Element Method (BEM) The Boundary Element Method (BEM) distinguishes itself from FEM in that only the boundaries of the space under consideration are discretised into finite elements. On the contrary, in FEM the whole space has to be discretised into finite elements. In practice, this means that matrices used by FEM solvers are very large but sparsely filled, whereas BEM matrices are smaller but densely filled. BEM calculations are very time consuming. A comparison between FEM and BEM is given for example in [Becker and Waller, 1986; Kopuz and Lalor, 1995].
BEM is a numerical technique for solving the boundary integral form of the wave equation, which is mathematically given by the Helmholtz-Huygens-Kirchhoff Integral Theorem. Only the boundaries of the environment are subdivided and the pressure (or particle velocity) is assumed a linear combination of a finite number of basis functions defined over the elements. One can impose either that the wave equation is satisfied at a set of discrete points (Point Collocation Method) or ensure a global convergence criterion (Galerkin and Least Squares Method).
One may distinguish between three main classes of BEM. The Direct Boundary Element Method DBEM is based on the Helmholtz Equation. The Indirect Boundary Element Method IBEM is derived from potential theory, in which the field is written in terms of source layers at the surface and may be interpreted as an integral form of Huygens’ Principle. The Variational Boundary Element Method, VBEM, is formulated either in terms of the direct or in terms of the indirect boundary problem using a variational formulation for the error of the solution on the boundary.
The main advantage of BEM is the reduction of the set-up time (meshing, data preparation) that results from reducing the dimensionality of the problem by one. However, BEM suffers from a great number of difficulties, some of which are of complex resolution [see for example Atalla and Bernhard, 1994].
The application of BEM for the determination of sound fields inside cavities and rooms has been realised, but to a much lesser extent than FEM, and application examples, like FEM, are normally within the determination of sound fields inside passenger and luggage compartments [Suzuki et al., 1989]. Application to room acoustics has been described for example in [Kludszuweit, 1991]. Again, due to the high computational requirements, BEM solvers are only applicable to small-sized rooms and to low frequencies.
87 4.1.2.3 Finite Difference Methods (FDM) Among the Finite Difference Methods (FDM) one can distinguish between Finite Difference Frequency Domain (FDFD) methods and Finite Difference Time Domain (FDTD) methods. FDFD converts the partial derivatives of the Helmholtz Equation into finite differences. This method was applied for the prediction of steady-state sound fields inside ducts [Alfredson, 1973].
FDTD has been more extensively applied to solve transient acoustic problems. [Botteldooren, 1995] applies the FDTD method for simulating the low frequency (63 Hz octave and 125 Hz octave) response of a simple auditorium. The walls possessed complex frequency-dependent impedances and the excitation consisted in a narrow Gaussian pulse. The results were computed using a 25 cm grid, whereby the 232560 cells were Voronoi cells. The author states a computation time of 1 hour to simulate 1 second of sound propagation in a modern workstation (1995). Sakamoto demonstrated a FDTD simulation of a small concert hall (5000 m3) up to a maximum frequency of 1.4 kHz. The model of the hall possessed more than 100 million elements and ran parallel on eight computers with total RAM memory of 11 GB for approximately 34 hours long [Sakamoto et al., 2004]. Therefore, one can see that FDTD methods are also very costly from both memory and computational requirements.
In [Savioja et al., 1994] a variant of FDTD, known as the digital wave-guide mesh, has been applied for the transient simulation of sound propagation inside rooms. But also in this case, the room used for the simulations is rather small (8x8x2 metres) and the upper allowed sound frequency for a 10 cm mesh unit is approximately 3 kHz, needing 12000 time steps to calculate the impulse response for 2 seconds using special hardware (Silicon Graphics).
4.2 Geometrical Acoustics Based Methods The last section showed that wave-based methods for modelling sound fields inside enclosures are useful and conceptually fruitful only in cases where the wavelength of the sound is large enough. At these low frequencies only a few eigenmodes are excited, so that the series expansion converge rather rapidly, the resonances are distinct, and the waveforms are simple. As described in section 3.4, above the Schröder cut-off frequency the resonances merge, the series expansions contain several hundreds of nonnegligible terms, and the wave-based approach can no longer portray the sound fields inside rooms in a simple form. In addition, even if the computation of the sound field
88 by the wave-based approaches could be successfully undertaken, the results yield too much detailed information, more than would be necessary and meaningful for the judgement of the acoustical properties of the room.
In order to capture a meaningful description for the higher frequencies of sound one turns to a description of the sound field in terms of acoustic rays and average intensities, analogous to geometric optics. The concept of an acoustic sound ray is an idealisation for high frequencies of sound. It can be identified with a small portion of a spherical wave with vanishing aperture, which originates from a certain point, having a well-defined direction of propagation. This sound ray is subject to the same laws of propagation as a light ray. In room acoustics, it is considered that the sound ray travels in straight lines, thereby neglecting any diffraction effects (however, some extensions of geometrical acoustics consider also some kind of diffraction [Torres et al., 2001]). The greatest difference between geometrical acoustics and its equivalent optical approach resides in the fact that the sound velocity is much smaller than the light velocity (that can be considered in practical cases as being infinite) and must be considered in all circumstances, since it is responsible for many important effects such as reverberation.
It must be also underlined, that some authors prefer to consider the sound rays as trajectories of point particles of sound, having a definite space position and momentum [Joyce, 1975]. However, the results of geometrical acoustics are the same regardless of the conceptual picture.
The traditional methods based on geometrical acoustics consider that the sound rays are reflected in a pure specular manner by the walls of the room. In this case, as stated in section 2.6.1, the angle of the reflected sound ray equals the incident angle on the wall. However, as pointed out in that section, in the case of surface irregularities, either in the form of surface roughness or in the form of small variations in the specific acoustic impedance, then the reflected sound is scattered over different directions. To extend the validity of the geometrical acoustics approach some authors (especially Kuttruff and Hodgson) introduced the so-called diffusivity coefficient, ∆()ω , which equals the ratio of the amount of sound energy that is reflected into non-specular directions to the amount of total reflected energy. This coefficient, which varies with the wavelength, has values between zero and one. A coefficient equal to zero means that the sound is totally reflected into the specular direction, whereby a coefficient equal to unity means that the sound is totally reflected into
89 non-specular directions. In this latter case, usually one assumes that the sound is reflected in a completely diffuse manner according to Lambert’s Law (see section 2.6.1). The international standard [ISO/DIS 17497-1, 2002] defines the procedure for measuring random-incidence diffusivity coefficients of surfaces in a diffuse field.
4.2.1 The Diffuse Field Theory The first quantitative experiments in room acoustics were those started in 1896 by Wallace Clemens Sabine. His immediate task consisted in determining why the lecture hall of the Fogg Art Museum at Harvard had such poor acoustics, and in suggesting modifications to solve the problem. He made of this particular problem a start of a general study of the behaviour of sound in rooms.
Sabine observed that, while the lecture hall was modelled after its neighbour, the acoustically successful Sanders Theatre, the materials of the surfaces were very different, with Sanders faced with wood and the lecture hall with tile on plaster. By ingenious experiments and inductive reasoning, Sabine arrived empirically at the now well-known reverberation theory and to the reverberation time formula:
KV TsSabine ()ω = ; [] (115) ∑αωii()S i where K is a constant depending on the particular system of units used (MKS or CGS), V is the room’s volume, and Si is the exposed area of each type of material possessing an absorption coefficient αi , which Sabine defined as the average ratio of absorbed to incident sound intensity for the material. Just as the absorption coefficient is frequency dependent, so too is the reverberation time. While this formula does not account for the effects of such factors as the proximity of absorptive materials to the sound source, for example, Sabine himself noted that “it would be a mistake to suppose that…[the position of absorption within the room] is of no consequence” [Sabine, 1922].
Sabine’s reverberation theory resides on a certain set of essential assumptions for the sound field inside the room:
90 1. Uniform, diffuse distribution of sound energy throughout the room at any instant
2. Equal probability of propagation of sound in all directions
3. Continuous absorption of sound by the boundaries
Clearly, this simplified picture (diffuse-field theory, see section 3.5.2) is strictly geometrical, whereby sound energy is considered to travel in rays, and all wave phenomena are neglected. When these assumptions are true, then one can show that there exists a very simple relationship between the mean acoustic energy density inside the room, w , and the power flux I incident on any unit area in the room, independent of its position and orientation [Morse and Ingard, 1968]:
1 I()ωω= cw (); [ Wm−2 ] (116) 4
Under these assumptions, a simple differential equation derived from the conservation of energy inside the room (energy variation = source energy – energy absorption by the walls) can be formulated as:
dw c VAwt+=Π()ω (,)ω (117) dt 4 where w is the mean acoustic energy density inside the room, assumed to be uniform, and Π(,t ω ) equals the net acoustic power supplied by the sources in the room at instant t and for exciting frequency ω . The quantity AS()ωαω= ∑ ii () is the total absorption of the room (also known as i absorbing power) and has the dimensions of area; if the area is given in square feet, A()ω is said to be given in units of sabins, whereas if it is given in square meters, it is said to be given in units of metric sabins.
The solution of equation (117) is given by [Morse and Ingard, 1968]:
ActAc()ωωτt () 1 − wt(,ω )=Π e44VV∫ e (τω , ) d τ (118) V −∞
91 indicating that the energy density at a given instant depends on the power output for the previous 4()VAω c seconds, but depends very little on the power output before that time.
Two cases are of interest: the first one consists in assuming that the source is turned off at instant t = 0 (and assuming that prior to this instant, it operated continuously since infinite time with acoustic power output Π()ω ). The second consists in assuming that the source is only turned on at t = 0 (and assuming that the source’s power becomes instantly equal to some steady-state output Π()ω ). These initial value problems when substituted into the solution (118) give respectively:
Ac()ω 4()Π ω − t wt(,ω )= e4V ; [ Jm−3 ] (119) Ac()ω and
Ac()ω − t 4()Π ω ⎡⎤ −3 wt(,ω )=−⎢⎥ 1 e4V ; [ Jm ] (120) Ac()ω ⎣⎦
The rate of the exponential decay from equation (119) can be used to derive the corresponding reverberation time (see section 3.5), which corresponds to the Sabine reverberation time:
24ln10 V Ts()ω = ; [] (121) S cA()ω
This equation is the same as the one obtained experimentally by Sabine, which emphasizes the fact that Sabine’s picture was essentially a geometrical and statistical one.
The second initial value problem with the solution given by equation (120) allows the calculation of the steady-state reverberant energy density inside the room, which is therefore given by:
4()Π ω wJm(,)∞=ω ; [−3 ] (122) Ac()ω that indicate that a small value of the room’s absorption is associated with a large value of the ultimate reverberant energy density.
92 The first important modification of Sabine’s theory was the replacement of the assumption of continuous absorption by the boundaries by that of a process of discontinuous drops in intensity during the decay. The resulting formula, first derived by Schuster and Wätzmann [1929], and later independently by Eyring [1930] and Norris [1929] is
24ln10 V Ts()ω = ; [] SW;; N E cS−−ln(1αω ( )) S (123) 1 αω()= ∑ αii () ωS SS i
where SSSi= ∑ is the total area of the room, composed by the sum of the individual exposed i areas Si . The coefficients αi denote the area-averaged random-incidence energy absorption coefficients of the walls, which correspond to the Sabine definition. The diffuse field assumptions are retained, but sound energy is assumed to travel one mean free path and then to be reduced abruptly by an amount depending on the mean absorption coefficient of the walls [Pierce, 1994]. Equation (123) is normally known as the Eyring reverberation time formula. The Sabine reverberation time can be obtained from the Eyring formula by developing the natural logarithm into a power series and retaining only the first term of the expansion, which is valid for small values of the mean absorption coefficient. Therefore, the Sabine theory appears as a simplification of the Eyring theory. The distinction between both formulas becomes appreciable when α ()ω is of the order of 13 or greater. Since TSW;; N E ()ω is less than the TS ()ω , it implies a more rapid decay of sound inside enclosures. Measurements in relatively dead rooms show that equation (123) is more accurate than the Sabine equation (115) for these cases.
Other reverberation formulas were afterwards derived, which differ from that of Sabine and Eyring in how they determine the absorption inside the room for cases where the absorption is not uniformly distributed over the walls, or, they use different mean free paths. A discussion on the mean free path problem in room acoustics is given by Joyce [1975]. A well-known reverberation formula is due to Millington [1932] and Sette [1932]:
93 24ln10 V TsMS; ()ω = [] (124) cS−−∑ iiln(1αω ( )) i
This, however, has the fundamental deficiency that if any single absorption coefficient is unity, regardless of the size of the corresponding material, then automatically the reverberation time becomes equal to zero.
Eyring’s theory assumes that the energy in the room resumes uniform distribution after each set of reflections, during the discontinuous decay process. Millington-Sette theory follows the course of a bundle of sound rays through many reflections and assumes that, on average, a particular ray will strike a given surface a number of times proportional to its area. Both forms assume the geometrical acoustics conditions, but the averaging is obtained differently. Eyring’s theory takes an arithmetic mean over the absorbing surfaces while Millington-Sette’s theory takes a geometric mean. Since geometric means are always less than arithmetic means, it follows that:
TTM ;;;;S>< SWNE; α MSα SWNE ;; (125)
A review of other reverberation time formulas is done by Kuttruff [1979] and by Kinsler et al. [1982].
Whenever a continuous sound source is present inside a reverberant room, two sound fields are produced. One is the direct sound field from the source and the other is the reverberant sound field that is produced by the reflections from the boundaries of the enclosure. In the diffuse field theory of Sabine and Eyring, the sound energy density is obtained from the following expressions:
Π()ω ⎡ 1 4 ⎤ wr(, ) ; ω =+⎢ 2 ⎥ crR⎣4()πω⎦
RSSSiiS()ωαα==∑ ; (126) i
−−SS ln(1α ) RSW;; N E ()ω = 1−αSS where R()ω is the room constant and where r is the distance from the sound source. Air attenuation can be included in all of these models by the adoption of the air absorption coefficient
94 defined in section 2.6.3. In this case, the term 4(,mTω Celsius ,) hVmust be added to the total absorption of the walls of the room in all of the above-mentioned expressions.
While these statistical acoustics theories provide some indication on the character of the sound field created by a source within an enclosure, they are limited by the simplifying assumptions implicit in their derivations. Energy density is not uniform throughout the enclosure, due in part to the irregularity of the geometry and the non-uniform distribution of the absorptive material. Further, as already said in section 3.5.3, during the last decades a number of acoustic measures have been developed, which require more detailed information about the sound field than these calculations provide. The values for many of these measures vary for different positions throughout the rooms, and require directional and temporal data along with the intensity of sound for each passing wave front in terms of impulse responses. For a review on the practical applicability of the diffuse-field theories in different rooms (classrooms, industrial buildings, gymnasiums, etc.), see [Hodgson, 1996].
4.2.2 The Mirror Image Source Method (MISM) The concept of a mirror image source was already pointed out in section 2.6.1 for describing the total sound field produced by a spherically symmetric point source above a rigid plane. This concept can be generalised, under the scope of geometrical acoustics, to multiple reflecting walls and to higher order reflections, by creating images of the images of the original source (see Figure 16). Thus, the Mirror Image Source Method (MISM) replaces walls and other reflecting surfaces with a set of virtual sources radiating into “free field”. The amplitudes of the mirror image sources are adjusted so that the boundary conditions, i.e., the wall impedances at the positions of the walls, are fulfilled. This can be done exactly for just a few idealised cases, for example a rectangular room with rigid walls [Allen and Berkley, 1979]. For the case of nonrigid walls, even the effect of a single locally reflecting surface, as pointed out in section 2.6.1, is quite complicated to portray. Therefore, a precise statement of the effects of a room with finite impedance walls is impossible. However, the MISM yields solutions that are asymptotically correct at high frequencies, even for walls with specific acoustic impedances of locally reacting surfaces.
95
Figure 16: Example of geometrical construction of mirror image sources of an arbitrary room. S denotes the sound source and P the receiving point. MIS3 is obtained by mirroring the first order MIS2 on the plane where R3 is located.
The concept of a mirror image source dates back to Euler [Pierce, 1994], and was extensively used in room acoustics throughout the twentieth century, but only to obtain statistical descriptors of sound fields, or, in order to obtain some ray trajectories inside rooms. The first implementation of the MISM on the computer is due to Gibbs and Jones [1972], who applied it to calculate the distribution of the sound pressure levels inside a rectangular enclosure. These authors calculated the sound pressure levels for an enclosure with five rigid walls and one absorbing wall and for a rigid rectangular enclosure with six absorbing patches placed centrally over the walls. Air attenuation was also included in their study.
The extension of the MISM to rooms of arbitrary geometries, although composed of plane walls, was first described by Juricic and Santon [1973]. These authors applied the MISM on the computer in order to obtain echograms and cumulative energy charts for a rectangular room and for a small reverberation chamber without parallel walls. However, the complete algorithmic representation of the extended MISM for arbitrary polyhedra was given by Borish [1984], who also applied it on the digital computer for obtaining the visible images in a simple, quasi-rectangular room.
Traditionally, the MISM has been almost exclusively applied to rectangular enclosures, since for this idealised case, one can show that all the valid mirror images are located on a regular rectangular lattice, each cell representing a mirror image of the original room. Therefore, in this case, the coordinates of the mirror image sources for every reflection order can be analytically determined, which highly facilitates the application of the method [Allen and Berkley, 1979; Hammad, 1988].
96 For arbitrary polyhedra, the application of the MISM poses some problems, both from the algorithmic point of view and from the calculation demands, although the deterministic construction of the geometrical image sources is simple. The main problem lies in the fact that a certain constructed image source influences a receiver only sometimes, due to the finiteness of the walls of the room. Since the MISM is based on geometrical acoustics, this means that a validity test, in terms of a ray connecting the receiver and the image source, must be conducted in order to determine if the potential image source can directly radiate to the receiver or not. In addition, since the rooms can have any shape, there exists the possibility that occluding walls be present along the propagation path from potential image source to the receiver. These validity and visibility tests are computationally expensive to conduct, although there have been some improvements in the basic algorithm so that these tests are reduced to a minimum [Lee and Lee, 1988; Kristiansen et al., 1993]. This reduction is achieved by the preclusion of invalid potential image sources right before they are in fact geometrically constructed, therefore avoiding later tests.
Another problem with the MISM resides in the fact that the number of potential image sources grows exponentially with the reflection order. More specifically, for a room with M walls the number of potential image sources up to some reflection order K is given by [Vorländer, 1989]:
MM(1)1−−K MMM+−+−++−=(1)(1)...(1) MM21 MMK − ≈− (1) M K (127) M − 2
It becomes apparent that the MISM scales badly both with the number of walls, but even worse, with the reflection order considered. Moreover, an even more important fact is that, from the huge number of potential image sources only a very small fraction of it constitutes the set of valid and visible image sources that radiate some receiver inside the room. To cope with this problem, some authors tried to extrapolate the valid and visible images to higher orders of reflections based on data collected from the geometric construction phase of the MISM for low orders of reflection [Kristiansen et al., 1993; Martin et al., 1993].
Once the valid and visible image sources are found by the MISM, then the problem of calculating the response at some receiver becomes trivial. It should be underlined that the response can be calculated either in the time domain or in the frequency domain. If applied directly in the time domain, then usually the MISM produces an energy impulse response of the room, since all the
97 contributions from the image sources are simply added, without taking into account any phase differences between them. This energy impulse response can be written as:
⎛−⎞rrSR δ ⎜⎟t − c 2 Π(,t ω ) ⎝⎠−−m rrSR gt()=+ e 4π rr− 2 SR ⎡⎤⎛−⎞rr (128) δ t − iR Nvisible ⎢⎥nrefl, i ⎜⎟ Π(,t ω ) ⎡⎤c −−m rr ⎢⎥⎝⎠ iR ⎢⎥(1−−∆αωik,, ( ))(1 ik ( ω )) e 4π ∑ ⎢⎥∏ 2 i ⎣⎦k rriR− ⎢⎥ ⎣⎦⎢⎥ where the first term accounts for the direct field from the source and the second term is due to all the valid and visible image sources located at r . The sound source is located at r and the receiver i S at r . The exponential factors account for the air attenuation of sound. From this energy impulse R response, several objective acoustic parameters can be calculated [Kirszenstein, 1984] and, in addition, one can derive, in principle, a broadband pressure impulse response [Vorländer, 1989].
In the frequency domain, phase effects can be included in the simulation. These phase effects are due to the propagation delays and to the phase shifts that occur when the sound is reflected from a wall. In this case, the total complex sound pressure is given by:
−+((ik m )rr − ) ρωcΠ()e SR pˆ()ω =+0 2π rrSR− (129) Nvisible nrefl, i −+((ik m )rri −R ) ⎡⎤⎡⎤ ρω0cΠ() e −−m rriR ˆ ⎢⎥⎢⎥Reik,,(ωω ,Ω−∆ )(1 ik ( )) 2π ∑ ∏ rr− i ⎣⎦⎢⎥⎣⎦k iR where Rˆ denotes the complex pressure spherical reflection factor, as given by (53). Gensane and Santon [1979] were the first to use this complex pressure MISM using the plane wave approximation for the reflection factors as given by (54). They applied this model to show the general validity of the complex MISM in enclosures of special geometry, namely in a layer case, in a wave-guide and in a rectangular enclosure, when compared to the modal theory of room acoustics. Later, other authors used the same complex MISM in the frequency domain, but used the more
98 exact spherical reflection factors as given by (53) [Lemire and Nicolas, 1989; Suh and Nelson, 1999]. Others still simplified the problem further and used just the plane wave pressure reflection factor (41) for predicting the steady-state effective sound pressure decay in function of the distance to the source in industrial buildings [Dance et al., 1995]. In either case, the computation times are substantially higher than the MISM in the time domain using energy summation, since complex quantities must be processed. Usually, the results given by the complex MISM are in good accordance with measurements made in the frequency domain.
The MISM only applies to pure specularly reflecting surfaces, and its possible extension in order to simulate diffusely or partial diffusely reflecting surfaces is apparently impossible. In addition, curved surfaces are not handled by this method, so piecewise planar approximations by polygons must be employed. Such flat approximations can not represent the spreading effect of convex surfaces correctly without applying some artificial scattering effect.
Nevertheless, the MISM is the only method based on geometrical acoustics capable of simulating correctly the early specular reflections for a given room, thereby generating the corresponding impulse response.
4.2.3 The Ray Tracing Method Ray tracing was first introduced in room acoustics by Schröder and Atal [1963] and later by Krokstad et al. [1969] to calculate the acoustic response of rooms. In the original method, the sound source emits sound rays in all spatial directions, some directions being privileged according to possible directivity characteristics of the source. Each ray is specularly reflected when it hits a wall of the room until it strikes the audience area where it is assumed to be absorbed totally. The authors could thereby calculate the relative delay of each sound ray relative to the direct sound and draw impact maps relative to certain time intervals.
Juricic and Santon [1973] applied ray tracing in order to obtain echograms of rooms and Santon [1977] applied ray tracing for computing the sound pressure in a room (it is interesting to note that this paper of Santon suggests a way of reconstructing image sources from ray tracing, an idea much later taken by Vorländer [1989]).
99 Kuttruff [1979] introduced the idea that sound rays could be reflected either specularly or according to some predefined reflection law, including Lambert’s Law. When a ray hits some surface, owing a diffusivity coefficient ∆()ω , a random number is generated. If this number is less than the diffusivity coefficient, then the ray is reflected in a pure specular manner; on the contrary, the sound ray is reflected in a completely random way according to a statistical distribution that mimics the Lambert’s cosine distribution of possible directions (this constitutes an example of a Monte- Carlo Method). It could be viewed as more realistic and accurate if each ray generated a number of secondary rays upon reflection on a partly diffuse wall, but such an approach would lead to an exponential growth of the number of sound rays for high orders of reflection [Dalenbäck, 1996].
One can distinguish between two types of ray tracing: energy ray tracing, and hybrid ray tracing [Lehnert and Blauert, 1992]. In the first type of ray tracing, each ray that is emitted from a sound source carries a certain amount of energy, which is accordingly reduced by the air attenuation and by the different absorptions of the different walls that the ray encounters since it is detected at some receiving volume or receiving area. In the second type of ray tracing, conceptually introduced by Santon [1977] and later more formally formulated by Vorländer [1989], the rays just serve to determine meaningful paths of wave propagation from which the virtual mirror image sources can be reconstructed. Energetic ray tracing allows the determination of echograms, from which an integrated energy impulse response can be obtained. From this energy impulse response, many of the most important objective measures for room acoustics can be obtained, as described in section 3.5.3.
For simulating an omni-directional sound source, the rays are emitted in all possible directions with a uniform spatial distribution. This spatial distribution can follow a deterministic function [Krokstad et al., 1968], or, else, can use a probabilistic approach using Monte Carlo methods [Juricic and Santon, 1973]. For an omni-directional source, each emitted ray will have an initial energy equal to Π()ω NR , where NR is the number of total rays emitted. For the simulation of a directive source, two approaches can be followed. The first one consists in sending more rays into directions where the source has a greater directivity, each ray having the same initial energy. The second approach consists in sending the rays uniformly in all directions, but now each ray has an initial energy proportional to the directivity function of the source.
100 The main problem when using ray tracing consists on the fact that the detector (receiver) can not be a point, due to the infinitesimal cross-section of a ray. The only way of being able to detect some sound ray consists in introducing a finite-sized receiver, which can be considered as a surface or a volume [Krokstad et al., 1968, Vorländer, 1989]. In any case, there is a risk of registering false reflected rays, multiple detections, and that some possible reflection paths are not found [Lehnert, 1993]. In the case of energy ray tracing, the problem of invalid paths and multiple detections is not of great concern, but this is not the case for hybrid ray tracing, where multiple detections offer the possibility of constructing repeated valid image sources. This problem can be solved by back- tracing rays from the receiver to the source in order to check if the detected path is valid or not and by keeping a list of already constructed image sources. Details can be found in [Lehnert, 1993; Lehnert and Blauert, 1992] along with possible correction schemes for these kinds of errors.
The second problem with ray tracing resides in the resulting limited spatial resolution due to the finite number of emitted rays. It can be shown that for achieving a good spatial resolution one must consider a minimum number of rays NRmin [Rindel, 2000]:
2 8πc 2 NRmin ≥ tmax (130) Smin
where tmax is the maximum simulation time and Smin is the area of the smallest surface that one wishes to contemplate in the ray tracing. According to this equation, a very large number of rays is necessary for a typical room. As an example, a minimum surface of 10 m2 and propagation time up to only 600 ms leads to around 100.000 rays as a minimum value for the ray tracing. Obviously, this means a high calculation time for complex rooms with high reverberation.
Finally, the ray tracing method scales rather well with the complexity of the room and with the maximum reflection order considered. In fact, the relationship between computation time and both number of walls and maximum propagation time is almost a linear one [Vorländer, 1989]. Nevertheless, for rooms with many walls the method becomes also computationally inefficient, since each wall must be tested in turn against the sound ray in order to find the ray’s impact point. However, a variety of spatial subdivision schemes (octrees, binary space partition BSP, Seads, etc.) exists that allow reducing considerably the computation time for ray tracing. These accelerating
101 techniques stem almost all from the field of computer graphics and a good review can be found in [Watt and Watt, 1992].
A method that leads to a reduction of the computation times in relation to conventional ray tracing has been proposed in the field of computer graphics by Shinya, Takahashi and Naito [Shinya et al., 1987] and is called pencil tracing. In this method, one considers two types of rays: the axial rays and the paraxial rays. A pencil is a bundle of rays consisting of a central axial ray, surrounded by a set of nearby paraxial rays. Each paraxial ray is represented by a four-dimensional vector that represents its relationship to the axial ray. Two dimensions express the paraxial ray’s direction. In many cases, only an axial ray and solid angle suffice to represent a pencil. If pencils of sufficiently small solid angle are used, then reflection can be approximated well by a linear transformation expressed by a 4x4-system matrix. Thus, pencil tracing accelerates conventional ray tracing, in that only a more or less small number of axial rays must be completely traced within the environment until maximum propagation time. The rest of the other rays are considered as paraxial rays and their propagation, including reflections, are calculated through matrix-vector manipulations. To the author’s knowledge, no implementation of pencil tracing in room acoustics has been done to date.
Finally, it is noteworthy making a small observation regarding the so-called particle tracing method, which is a method completely analogous to the ray tracing method. The only difference lies in the detection mechanism: with energy ray tracing, one computes intensities at some receiving point, while with the particle tracing method the quantity directly computed is the energy density in a detector volume. Therefore, with particle tracing the energies of the crossing sound particles have to be weighted with the inner crossing distances. The idea is that the longer a particle stays in the detector volume when crossing it, the higher is its contribution to the energy density.
4.2.4 Beam, Cone and Pyramid Tracing As pointed out in the last section, the ray tracing method poses two main problems. The first one consists on the detection condition of valid rays on a finite-sized receiver. The second one, the limited spatial resolution achieved by the method. In order to solve partially both problems, some authors introduced, instead of simple rays, refined geometrical elements.
Cone tracing [Amantides, 1984; Maercke, 1986] is similar to ray tracing, but uses the concept of space coherence portrayed by a bundle of almost parallel sound rays. Infinitesimally thin rays are
102 therefore substituted by a cone, whose apex is located at the sound source. In approximate cone tracing, only the axis of the cone is traced just like in conventional ray tracing. After each reflection of the axis ray, the apex of the cone coincides with an image of the source taken on the same sequence of walls as the ones involved in the ray’s history. If a receiving point is found inside a truncated cone between two successive reflections, the corresponding image is potentially visible and an acoustic response has to be taken into account. The cone method greatly improves the performance of classical ray tracing because it eliminates most of the uncertainties due to the statistical nature of the rays, especially for the higher orders of reflection. As it describes the sound field in terms of image sources, not only energy responses, but also pressure impulse responses and complex transfer functions can be obtained.
Cone tracing has the problem that it is impossible to cover a whole sphere with them, unless some overlapping of cones is introduced. In this case, a single source-receiver path will be detected several times within different adjacent cones. This error can be corrected by maintaining a list of already detected image sources, just like in hybrid ray tracing.
Another possibility is to assign to each overlapping cone a special weighting function, with values that go to zero with increasing distance from the axis of the cone. In this case, one speaks of beam tracing [Maerke and Martin, 1993]. Therefore each ray is considered as the axis of a highly directive elementary source, called a beam. The superposition of the directivities of adjacent beams leads to an omni-directional radiation pattern.
Finally, another possibility for solving the problem of cone overlapping is to consider that the sphere is subdivided into a number of pyramids, with apex at the source [Lewers, 1993]. These pyramids cover completely, and without any overlap, the sphere surrounding a sound source. In approximate pyramid tracing, just like approximate cone tracing, only the axis ray is traced within the environment. Therefore, approximate pyramid tracing correctly samples the space for the direct sound from the source, but suffers from the same problems as cone and ray tracing regarding detection of invalid acoustic paths and multiple detections.
As for the case of ray tracing, scattering from the walls can also be taken into account in cone, beam, and pyramid tracing, by a Monte Carlo procedure.
103 In order to solve the problems of invalid and multiple detections, one should trace the faces of the cones or of the pyramids throughout the room. Therefore, a true cone or pyramid tracing procedure should actually trace the faces, which would then be split up at reflections across edges, subsequently requiring separate tracing of each part of the newly created cones or pyramids. This is the basis of the so-called “beam tracing” in computer graphics [Heckbert and Hanrahan, 1984] and of the so-called “adaptive beam tracing” in room acoustics [Drumm and Lam, 2000], although it has been earlier introduced with a different perspective by Stephenson [1996]. The main difference in these two approaches resides in how the splitting of the traced pyramids is done at the edges of walls. Drumm and Lam [2000] use the same approach as originally defined by Heckbert and Hanrahan [1984] for finding the different exact areas of illumination of the incident pyramid on the edges, while Stephenson [1996] uses direct polygon-polygon intersections in order to construct the different reflected portions of the original pyramid.
The adaptive beam tracing is a correct geometrical acoustics based algorithm for finding all the valid acoustic propagation paths between a sound source and a point receiver. It does not suffer from the errors of approximate tracing techniques (in which only the axis ray is traced) at the expense of much more complex algorithms for clipping the incident beams along the edges of the reflecting surfaces. In addition, with every beam split-up, the number of existing new beams to trace grows enormously with the considered reflection order. Thus, the scalability of this method is bad. Due to this fact, Drumm and Lam [2000] use the adaptive beam tracing method only up to low orders of reflection, typically six or seven. Finally, since the exact pyramid faces are traced, there is no simple way of introducing scattering from the walls directly in the algorithm. Stephenson suggests a random shift of the constructed image source if the intervening surface possesses some scattering characteristics, but this seems not to be correct because then all energy continues to be propagated along the so defined new beam. In [Drumm and Lam, 2000] diffusely reflected sound components are therefore treated separately by another simulation method (the radiosity method, see section 4.2.6).
A way of limiting the growth of new beams produced after split-up is adopted in [Campo et al., 2000], where the authors limit the generation of new beams to a maximum of three per split-up. In this way, the calculation does not grow indefinitely.
104 Finally, unlike approximate tracing techniques, the adaptive tracing methods are only applicable to rooms constituted by planar walls.
4.2.5 Finite Sound Ray Integration Method (FSRIM) In the Finite Sound Ray Integration Method (FSRIM) [Sekiguchi and Kimura, 1991], sound rays are traced throughout the environment just as in conventional ray tracing. The rays are specularly reflected at each wall of the enclosure, which might be planes or also curved surfaces, until some maximum propagation time is reached. The impact points over the walls are stored for later processing, by considering that every impact point constitutes a secondary radiating sound source. All contributions from the secondary sound sources are then added, with corresponding time delays, at some receiving point.
The processing is done in the frequency domain, with the absorption properties of the walls given in terms of a complex transfer function. The secondary waves radiated from every secondary sound source are considered as differential waves, meaning that they are obtained from the time derivative of the incident waveforms.
In the original work of Sekiguchi and Kimura, a half triangular pulse was used as the waveform emitted from the sound source, but any waveform can be used in the simulation. This elegant formulation allows one to calculate the pressure response of a room to any excitation signal, including the pressure impulse response if a Dirac impulse is used for excitation. However, the authors do not consider the further propagation of the diffracted waves after they pass the receiving point, which means that the error in the waveform increases gradually. The main problem about the FSRIM is again the high computation times that are required. For this reason, the authors choose to end the simulation soon, the maximum propagation time being less than 700 ms, which is insufficient in order to obtain the complete reverberation tail of the room’s response.
4.2.6 The Radiant Exchange Method. The Radiosity Method Radiant Exchange methods are geometrically based methods that were developed during the 1950s in order to simulate thermal transfers between surfaces of different solids. These methods were introduced in room acoustics in the early 1970s by Kuttruff [1971] and later in the field of computer graphics in 1984 [1984].
105 Radiant exchange methods consider energy balances in the process of thermal, acoustic, or light radiation. Therefore, the underlying mathematical formalism that translates the governing equations consists of integral equations for the relevant energy quantities. Traditionally, radiant exchange methods were developed entirely for solving problems in which the boundaries reflected energy in a purely diffuse way. This has also been the case in the application of radiant exchange methods in acoustics and in the field of computer graphics. A short review on the background to the different contributions of radiant exchange methods in room acoustics for the case of purely lambertian reflections is given in [Alarcão and Bento Coelho, 2003].
The integral equation for acoustic radiation inside enclosures with diffusely reflecting walls was first derived by Kuttruff for the steady-state case [Kuttruff, 1971]:
εϑϑcos cos ' B()rrr=−⎡⎤ 1α ('' )BdS ( )' ; π ∫∫ ⎣⎦ R2 S SS (131) R =−rr' where B()r is defined as the “irradiation strength” at the boundary point r , i.e. the sound energy incident per unit area on the surface element dS ' , whose position vector is given by r . The factor S ε ≥1 was introduced to cope with the case B = constant .
Kuttruff later extended his integral equation [Kuttruff, 1976] in order to include the time dependence of B through the inclusion of the finite time of sound propagation between surface elements:
' 1coscos''⎛⎞R ϑϑ ' B(,)rrrtBtdS=−⎡⎤ 1α ( ) , − ⎣⎦⎜⎟2 S (132) π ∫∫ ⎝⎠cR SS where c is the sound velocity and R is the distance between r and r' . Equation (132), usually referred simply to as time-dependent Kuttruff’s Integral Equation, does not consider the direct field from a sound source, and in this form, its application is restricted to convex rooms.
106 In the case of image synthesis for computer graphics, the governing integral equation is due to Kajiya [Kajiya, 1975] and differs from Kuttruff’s Integral Equation because it is stated in terms of three-point transport quantities, therefore being more general than equation (132).
Analytic solutions for Kuttruff’s Integral Equation (in its steady state or time-dependent form) exist only for a small number of very simple geometrical arrangements, like spherical enclosures [Carrol and Chien, 1977; Carrol and Chien, 1978; Joyce, 1978; Gilbert, 1988] and flat and long enclosures [Kuttruff, 1985, Kuttruff, 1989].
There is a number of many mathematical techniques for solving integral equations, as will be described in more detail in Chapter 5. One of them is based on the finite element approach, where the integral equation is discretised. This results in a set of simultaneous linear equations that can be solved by standard procedures. This was the technique applied in [Kuttruff, 1976] for determining reverberation times, and in [Goral et al., 1984] for calculating diffuse light reflections. This technique is usually known in the current literature as the Radiosity Method. Since its introduction, the Radiosity Method has suffered many important and crucial improvements, mainly due to extensive work in the field of image synthesis [Sillion and Puech, 1994].
In acoustical radiosity, instead of letting sound rays, which are emitted from the sound source, sample the boundaries of a room, larger wall elements (known as patches) predetermine the wall interreflections. The contributions between all patches, and from the source to the patches, and finally from all the patches to the receiver are determined via the so-called form factors (also known in the earlier literature as configuration factors or shape factors). In the case where the walls reflect sound (or light, or heat) in a pure diffuse way, as given by Lambert’s Law, then the form factors are solely geometrical relationships between pairs of patches, and are more or less easy to obtain.
Several papers, due to Kuttruff mainly, outline the acoustical radiosity method and use numerical procedures for obtaining the sound field inside enclosures with diffusely reflecting walls. Miles presented the discretised version of equation (132) and calculated both the steady state and the transient sound field in a rectangular enclosure with non-uniform absorbing surfaces by solving the equations numerically [Miles, 1984]. The transient solution showed the interesting fact that, although the beginning of the decay is rather complicated, the sound field ultimately exhibits the expected exponential behaviour [Miles, 1994]. Shi et al. [1993] gave an explicit numerical algorithm
107 based on full matrix radiosity [Sillion and Puech, 1994] in connection with image synthesis. In the same year, Lewers [1993] used a radiant exchange approach for deriving the diffuse reverberation in a hybrid method, where the specular components were calculated by approximate pyramid tracing. The prediction of steady-state levels inside a rectangular duct through acoustical radiosity is reported in [Cianfrini et al., 1998] and, more recently, Kang [2002] used acoustical radiosity for determining decay curves and reverberation times in long enclosures with rectangular cross sections. The most recent work to date on the implementation of acoustical radiosity is due to Nosal et al. [2004], who outline algorithms for sound predictions in arbitrary polyhedral rooms, although validation results are only presented for the case of spherical enclosures. Nosal and Hodgson [2002] have earlier presented results on the validation of acoustical radiosity by comparing simulations with experimental measurements of reverberation and steady-state sound levels for three rectangular rooms. Finally, a practical comparison between radiosity and energy ray tracing has been reported in [Le Bot and Bocquillet, 2000], where the authors simulate steady-state sound fields inside an L-shaped room and inside a factory with a simple geometry. Results between both simulation methods are very well correlated. This is of no surprise if one recalls that ray tracing is a Monte Carlo solution technique for integral equations, which is sometimes not immediately recognized by some authors.
It should be noted that in most of the works reported above, no specific data on the computation times required for each prediction is given. Usually, the sole statement is that the computation is rather short, but this is sometimes a misleading and incomplete statement. An exception is the work of Nosal et al. [2004], where the authors state that the computation times for predicting 600 ms of the impulse response for a spherical enclosure with mean absorption coefficient of 0.2 takes around 420 s on a 2 GHz class Pentium machine. This refers to a meshing of the sphere in 288 patches, which on the course of simulation yields 408 receiving elements.
The computation times in time independent classical radiosity are highly dependent on the mesh of patches (meaning the number of patches into which the enclosure’s boundary is subdivided) considered, because there exists approximately a squared relationship between computation time (and memory) and the number of patches [Sillion and Puech, 1994]. In acoustical radiosity, the computation effort is even more aggravated due to the time dependence of the equations. Therefore, when some authors state that the computation times are short [Lewers, 1993; Kang,
108 2002], normally a very coarse mesh is used, whereby the computation time grows considerably when one implements a finer model of the room.
Some extensions to classical time independent radiosity in order to include non-diffuse reflections in the models have been reported in the field of image synthesis [Sillion and Puech, 1994], but the resulting models are often very time and memory consuming. In the case of time dependent radiosity, extensions in order to include perfect specular reflections are reported only in [Tsingos, 1998; Rougeron et al., 2002]. The first author uses some advanced algorithms for radiosity, namely the so-called Hierarchical Radiosity (see Chapter 7), in conjunction with an extension based on the MISM to handle perfect specular reflections. The complete extended algorithm is elegantly formulated, but the necessary computation times are enormous. It is reported, for example, that in order to obtain the impulse response of a simple rectangular room up to forty orders of reflection the computation time was 48 hours! Rougeron et al. [2002] use an accelerating technique based on an averaging algorithm for speeding up the calculation of late reverberation. Nevertheless, computation times are of the order of a quarter of an hour for more or less simple geometries with 400 patches on a 500 MHz class Pentium machine. In addition, these two extended methods only allow walls that are either pure diffuse reflectors or pure specular reflectors. Mixed pure and specular walls with a given diffusivity coefficient are not considered.
4.2.7 Hybrid Methods Many hybrid approaches have emerged that incorporate the best features of the above-referred methods. The MISM is best used for modelling early reflections where directional and temporal accuracy is critical. The method may be paired with ray, cone, pyramid, and beam tracing for establishing valid image sources. The MISM is rarely used for later reflections due to the exponential increase in the number of potential images that must be considered. For example, the room acoustics prediction softwares ODEON and CATT Acoustics use approximate cone tracing to enumerate the valid propagation paths for the early reflections (normally up to maximum reflection order three) and then use the same ray tracing algorithm for a Monte Carlo sampling of the reflection characteristics of the walls [Naylor, 1993; Dalenbäck, 1996]. This Monte Carlo sampling is based on the method exposed in the third paragraph of section 4.2.3, whereby at each ray’s impact point a secondary source is generated, which radiates directly the receiver according to a diffuse distribution.
109 Lewers models late sound with the radiant exchange method [Lewers, 1993]. Other authors use a statistical approach based on the results of an earlier, or ongoing cone-tracing phase [Heinz, 1993; Martin et al., 1993]. Heinz presents an approach in which surfaces in the enclosure are assigned a wavelength dependent diffusion coefficient, which is used to transfer energy from the incoming ray to the diffuse reflected field. The diffusely reflected components of the sound field are simulated by imposing a certain time-variable power spectrum on a simulated Poisson process. The time-variable power spectrum for each given time interval of the used sampling rate is obtained from a low- resolution ray tracing, by smoothing out and filtering over the octave band echograms. Martin et al. [1993] use the data obtained for the low orders of reflection for extrapolating statistically valid acoustic paths for higher orders of reflection, and the reverberation tail is modelled as Gaussian filtered white noise. Various approaches are used to combine the early and late response simulations.
Finally, some hybrid methods have been suggested in order to expand the overall accuracy along the complete audible bandwidth. These hybrid methods include therefore the natural combination of a more rigorous, wave-based simulation method for low sound frequencies and a more efficient, geometrical acoustics based method for mid and high frequencies, e.g. the MISM or some ray tracing based method. One such hybrid method was used in [Savioja et al., 1999] where the authors used a MISM for calculating the high frequency components and a digital wave guide mesh method for modelling the low frequency components of the sound fields inside enclosures.
110 Chapter 5
A NEW MODEL FOR SOUND ENERGY PROPAGATION INSIDE ENCLOSURES
5.1 Introduction In the last chapter, an overview of the different approaches for modelling the sound fields that build up inside enclosures when an acoustic excitation is produced by sound sources was presented. Some of the approaches are of rather simple and immediate application, such as the statistical diffuse-field theories, while some others, namely those that use wave-based formalisms, are much too difficult to apply in practical cases, or the computation effort required is so high that the methods become inadequate.
As highlighted in Chapter 4, only wave-based approaches can accurately model most of the physical phenomena occurring in the propagation of sound inside enclosures. However, it became apparent that the degree of detail achieved by these methods is generally too high, whereby the underlying phenomena are often obscured by the complexity of the mathematics, or even of the numerical schemes used. In addition, it can be argued that such a high degree of detail is incompatible with the human hearing perception, which acts as an integrating filter for the sound waves incident on the ear [Kuttruff, 1979].
It is interesting to note that all the objective parameters for describing the most important attributes of sound fields inside enclosures, as given in Chapter 3, section 3.5.3, use the concept of sound energy instead of sound pressure. The only exception is the IACC, which uses the pressure at both ears in order to derive an objective correlation between them.
The simplification of using energy quantities for describing sound fields, thereby neglecting all phase effects, is justified by the fact that the dimensions of most rooms and halls are large when compared to the intervening sound wavelengths. In this case, as already mentioned in Chapter 4, section 4.2, in this large room limit, the Schröder cut-off frequency (97) is usually low enough and the density of eigenfrequencies so high that a strong overlap of eigenmodes will take place. Therefore, any audible sound signal excites many room modes simultaneously, and since their
111 phases are nearly randomly distributed, all phase effects are cancelled when they are superimposed [Kuttruff, 1979].
When only energy is considered as the main quantity for describing sound fields in enclosures, some approximate methods based on the geometrical acoustics assumptions can be applied. Because of the underlying concept of a sound ray in geometrical acoustics, most energy-based methods resort to tracing some geometrical tokens within the room under consideration. These tokens can be simply rays, cones, or beams that carry a certain amount of sound energy. Upon propagation within the medium, and through reflections and scatterings, this energy content is spectrally filtered according to the intervening absorption mechanisms of sound energy. The propagated sound energy is then collected at a receiver, whereby one obtains energy responses of the room under study.
In addition to the referred ray-tracing approaches, some theories considering energy balance relationships have been derived with application to specific cases of room acoustics, where a simplified and idealised description of the underlying physical phenomena is portrayed. An example of such a theoretical approach is given by considering a linear inhomogeneous integral equation for the intervening energy quantities. Such integral equations have been derived in numerous fields of physics, for example, in radiative heat transfer, illumination engineering, optics, and in acoustics. In the case of room acoustics, an important integral equation is the well-known Kuttruff’s Integral Equation (132), which governs the propagation of sound energy inside enclosures with walls that reflect sound in a purely diffuse manner, given by the Lambert’s cosine law. Since its formulation, Kuttruff’s Integral Equation has seen some minor additions, some of which by a different treatment of the effects of the sound sources [Carrol and Chien, 1977] and some, due to Kuttruff itself, regarding the more exact mathematical formulation in terms of integration kernels. Miles showed some general properties of Kuttruff’s Integral Equation [Miles, 1984], and an extension regarding the relaxation of the strong assumption of pure diffuse reflections has been suggested by Joyce [1978]. However, a complete general formalism incorporating all these “ideas” has not been given to date. In addition, to the author’s knowledge, no exact derivation of this equation exists, from the point of view of more fundamental physical and mathematical principles. This general derivation will be presented in this chapter.
112 In addition, some work on the acoustical radiosity method has been carried out, many times without the conscious knowledge that this method approximates the underlying integral equations. Even the extensively applied energy ray tracing method is just one method for solving the integral formalism, a fact that is many times overlooked by some authors.
Therefore, it seems that a great deal of confusion exists regarding energy-based methods in acoustics. As an example, it is noteworthy to highlight the many different definitions used for energy-based quantities. Some authors define the intervening energy quantity as an “irradiation strength” [Kuttruff, 1979], while others define it by considering phonon fluxes that carry a certain amount of energy [Joyce, 1978]. Others use intensity as the basic energy quantity, but this intensity is not the same as the usual definition given for waves in acoustics [Miles, 1984]. Still, others use simply the term radiance, as directly imported from the field of radiative heat engineering and illumination engineering, to define the basic energy quantity [Cianfrini et al., 1998]. Finally, Le Bot and Bocquillet [2000] use as the basic quantity power per volume.
In this chapter, a new model for describing the propagation of sound energy inside arbitrary enclosures is presented. This model considers that sound energy is transported by a large ensemble of sound particles, which are emitted from sound sources and which propagate through the enclosure, being subject to different absorption and scattering mechanisms, until they are detected at some receiver. Therefore, one takes the point of view of some authors [Joyce, 1975] that energy propagation is associated with the motion of sound particles. It is thus natural to adopt some well- established concepts from the field of Transport Theory as the basis for the elaboration of this particle-based model. It must be underlined that this approach parallels some treatments that have been conducted in other fields, such as Radiative Transfer and Neutron Transport, [Davison, 1957; Barnett, 2000] and Illumination Engineering [Arvo and Kirk, 1990]
First, a rigorous treatment of the concept of sound particles is given, where one resorts to classical notions of phase space and to the mathematical framework of measure theory in order to develop a theoretical framework that will then be used throughout this chapter. The theoretical framework will serve to rigorously define the necessary energy-based quantities used for describing sound energy propagation inside enclosures. Secondly, a general transport equation for particles is derived, which is then adapted to the case of sound particles. This results in the most general boundary integral equation for the sound energy inside enclosures, which generalises conceptually Kuttruff’s
113 Integral Equation and its extensions. Thirdly, new theoretical tools for the study of sound energy fields in rooms are introduced. The focus is on a new way to express the governing equations in terms of linear operators with very convenient properties, which will be mathematically detailed. Finally, some approximate solution methods that can be used to solve general integral equations, highlighting application benefits and problems will also be presented.
5.2 Sound Particles and Sound Energy In this section, a number of fundamental quantities for describing the propagation of acoustical sound energy inside enclosures are defined. Instead of adopting standard vague definitions, such as sound ray, wave front ray, etc, the important concepts are deduced starting from “highly expected” behaviours of particle-based phenomena that are introduced as a kind of axioms.
The underlying concept throughout this chapter consists on that sound energy may be adequately modelled as a flow of a large number of non-interacting, neutral particles, called sound particles. The old concept of a particle-based description for sound energy, which was also adopted more recently by some authors as for example Joyce [1975] is here reconsidered. From elementary principles that operate at the macroscopic scale, one then constructs functions that correspond to usual energy quantities, which will be essential building blocks in what follows.
5.2.1 Measure Spaces, Measures and Domains In this section, some important mathematical concepts arising in the fields of Set Theory and Functional Analysis that will be used in the next sections are described.
A measure space is a triple (,Γ P ,)ρ , where Γ is the underlying set of the measure space, P is a collection of subsets of Γ , called the measurable sets, and ρ :0,P [ ∞] is a non-negative, countable additive set function, called the measure on the set [Boccara, 1990]. The countably additive property means that
⎛⎞∞ ∞ ρρ⎜⎟∪ DDii= ∑ () (133) ⎝⎠i=1 i=1
whenever the sets Di are mutually disjoint measurable sets. The symbol ∪ stands for the union of sets.
114 The measurable sets form a σ-algebra, meaning that P contains the set Γ , and is closed under the operations of complementation and countable unions [Boccara, 1990]. For technical reasons, P is generally a proper subset of 2Γ (the set of all subsets of Γ ), that is, some sets are not measurable. However, for the measure spaces one is interested in, which will be those constructed as the product of Lebesgue measures, the unmeasurable sets represent pathological situations that can be ignored in practice.
Sometimes, the measures to be considered will not be finite, meaning that ρ()Γ=∞. However, they will always have the weaker property of being σ-finite, meaning that there is an infinite sequence DD12, ... of measurable sets such that
∞ ∪ Di = Γ (134) i=1
and ρ()Di is finite for all i. That is, a σ-finite measure space is one that can be decomposed into countable many regions, each with a finite measure.
The enclosure’s geometry will be described in terms of a finite set of surfaces in R3 , whose union is denoted by M . Formally, each surface is a piecewise differentiable two-dimensional manifold in R3 . In addition, it’s required that each manifold be orientable and a closed set, meaning that each manifold must include its boundary ∂M [Spivak, 1965]. This prevents the existence of gaps between abutting surfaces. M itself is not necessarily a manifold.
Let one define an area measure σ on M in the following way: given that M is the union of manifolds MM1 ,..., M , one defines σ()D as the sum of the areas σii()D ∩M , where σi is the usual
Lebesgue area measure on the manifold Mi . The measurable sets D ⊂ M are defined by the requirement that all D ∩Mi are measurable. One also requires that the intersection between any pair of surfaces Mi and M j is a set of measure zero. In practice, this means that when the intersection between two surfaces has non-zero area, then one must discard the common surface area from one of the intersecting surfaces. This ensures that almost every point of M (up to a set
115 of area measure zero) has a unique set of surface properties. σ()D then denotes the Lebesgue area of a region D ⊂ M . The notation
∫ fd()s σ ()s (135) M denotes the Lebesgue integral of the function f :M with respect to surface area. The Lebesgue integral is based on the concept of measure, where dσ()s signifies both the measure σ and the dummy variable s . The Lebesgue integral is an appropriate abstraction for functional analysis, where the emphasis is on integrals as transformations rather than the numerical aspects of integration. While the simpler concept of the Riemann integral is equivalent when it exists, the Lebesgue integral is more robust with respect to limiting operations, which makes it a valuable tool for defining abstract spaces such as Banach spaces [Kolmogorov and Fomin, 1999].
The surfaces divide the Euclidean space R3 into a multiply connected region V , which is filled with some propagation medium. The same concepts about measure spaces apply to regions in R3 . In this case, the adopted measure is the Lebesgue volume measure V . Therefore, one can write:
fd()rVr () (136) ∫ V which denotes the Lebesgue integral of the function f :V with respect to volume content.
Directions will be represented as unit-length vectors Ω∈ R3 . The set of all directions is denoted by S2 , the unit sphere in R3 . However, usually the set of all directions is written in local angular parameterisation, given in spherical coordinates ΩΩ= (,θ ϕππ );∈×[ 0,] [ 0,2]. Let σ be the usual surface area measure on S2 . Then, given a set of directions χ ⊂ S2 , the solid angle occupied by χ is simply σ()χ . Similarly, the solid angle subtended by a surface P from a point rR∈ 3 is determined by projecting P onto the unit sphere centred at r , and computing the measure of the resulting set of directions.
116 Another useful concept is the projected solid angle, which arises in determining power fluxes received by some surface. Given a point s ∈M , let ns() be the surface normal at s (the convention that the normal to the surface will be the inward-pointing normal is adopted, meaning that the normal vector points into the considered volume V , bounded by M ). Given a set of
2 ⊥ directions χ ⊂ S , the “projected solid angle” measure σs ()χ is defined by:
⊥ σΩs ()χ = ins ()dσΩ () (137) ∫ χ
The factor Ωins()is often written as cosθ , where θ is the polar angle of Ω , i.e. the angle between Ω and the surface normal.
The name “projected solid angle” arises from the following geometric interpretation. Let T ()s be M the tangent space at the point s , i.e. the space of vectors in R3 that are perpendicular to the surface normal:
3 TM ()srRrns= { ∈= :i () 0} (138)
Unlike the more familiar tangent plane, the tangent space passes through the origin, thus being a linear space rather than an affine one. This tangent space divides the unit sphere S2 into two hemispheres, namely the positive hemisphere
2 2 H + ()s = {Ω∈>S :Ωins () 0} (139) and the negative hemisphere
2 2 H − ()s = {Ω∈
Now, given a set of directions ϖ contained by just one hemisphere, the projected solid angle can be obtained by simply projecting ϖ orthogonally onto the tangent space, and then finding the area of the resulting planar region. For example, one assumes that ϖ is the entire positive hemisphere
2 ⊥ 2 H + , then the corresponding projected region is a unit disk, so we have σs ()H + = π .
117 5.2.2 Sound Particles and the Phase Space Most of the large-scale behaviour of particles can be translated in terms of abstract particles with minimal semantics. This typifies the point of view taken in Transport Theory, which is the study of abstract particles and their interactions with matter. Transport Theory applies classical notions of physics at the level of discrete particles to predict large-scale statistical behaviour. The resulting distributions are expressed purely in terms of geometrical and physical properties of the medium within which the particles propagate. Essential to the theory are several fundamental simplifying assumptions about the particles. For example, it is usually assumed that:
1. The particles are so small and numerous that their statistical distribution can be treated as a continuum.
2. At any time instant t , a particle is completely described by its position, velocity, and internal states, such as polarization, frequency, charge, or spin.
These assumptions lead naturally to the adoption of the classical concept of phase space [Reif, 1965; Goldstein 1980]. Phase space is a mathematical abstraction used in representing the spatial and dynamical distribution and internal states of a collection of particles, called an ensemble. Frequently, the term means a 6N-dimensional Euclidean space with each point encoding the position and velocity of N distinct particles. A single point in phase space completely specifies the dynamic configuration of the ensemble of N particles. The time evolution of the particles defines a space-curve.
For a large class of physical problems, a phase space of far fewer dimensions suffices. This has conceptually and computationally advantages. For example, when the particles can not influence one another, their aggregate behaviour may be determined by characterising the behaviour of a single particle. Each phase space dimension then represents one possible degree of freedom of a particle, with the space as a whole representing all possible particle states.
By assuming that the particles do not interact with each other, one adopts a picture where all interference effects are ignored, an assumption inherent to geometrical acoustics on which this sound particle model rests. It shall further be assumed that a particle-matter interaction, or scattering, consists on collisions that are either perfectly elastic or perfectly inelastic. That is, either a
118 particle retains its original internal state after scattering, such as energy or wavelength, or it is completely absorbed. Therefore, one assumes that an original sound field is composed of a very large number of sound particles that propagate through an enclosure. These sound particles can be inelastically absorbed as they propagate through an absorbing medium such as air. They can also be scattered through reflections at the enclosure’s boundary, whereby they are either elastically reflected or completely absorbed at the boundary. Thus, the original large number of sound particles tends to be reduced with time, in the case when no sources of sound particles are present, as in the case of reverberant decay. This can be viewed as a correspondence with the annihilation method adopted by some ray-tracing techniques [Vorländer, 1989].
The only internal state that will be considered here is the sound frequency, an attribute carried by each particle and that remains invariant in time, since only non-interacting particles are considered. In an ensemble of sound particles, one then can find numerous particles with the same internal state, corresponding to a certain sound frequency. Therefore, within the ensemble one can define families, whose members are all the particles that have the same invariant internal state. Each family will therefore possess an intrinsic physical behaviour differentiating it from the whole ensemble.
Since a particle suffers either perfectly elastic scattering or is completely absorbed, its energy is constant throughout its lifespan. By the classical kinetic energy relation
1 E = mc2 ;[] J (141) 2 whereby one neglects any external forces acting on the particle with mass m , this means that the velocity of each particle is also constant, and equal to the sound speed c . Therefore, each particle is characterised only by its position in space, by its travelling direction, and by its internal state attribute of sound frequency. This internal state attribute can be made function of the particle’s energy by the Einstein relation E = hf , where hJs= 6.626×⋅ 10−34 is Planck’s constant. This is however not necessary for the sound particle model that is adopted here.
In Transport Theory, this assumption is known as the one-speed model, since speed is proportional to energy for most particles, and energy is assumed to remain constant despite multiple collisions. Under these assumptions, each particle has only six degrees of freedom: three for position, two for
119 direction (in terms of spherical coordinates) and one for the internal state. The corresponding 6- dimensional phase space is then:
Γ ≡××RS32 (142) where R3 is Euclidean 3-space, S2 is the unit sphere in R3 , and is the set of real numbers (corresponding to the range of allowable frequencies).
In order to quantify ensembles of particles and define various transformations, it is necessary to endow the space with a set-theoretic measure. One introduces therefore the phase space measure, consisting of the triple (,Γ P ,ρΓ ), where ρΓ is a non-negative set function, or measure, defined on the elements of a set P , which is a σ-algebra of subsets of the space Γ , as previously defined in section 5.2.1.
3 The phase space measure (,Γ P ,ρΓ ) is then built up from the natural Lebesgue measures on R ,
S2 , and , which correspond to the standard notions of volume, surface area, and length, and we denote these measures by V , σ , and l respectively. The positive set measure ρΓ is defined as the product measure V ××σ l .
5.2.3 Particle Measures and the Phase Space Density To characterise exactly a configuration of particles one must specify all degrees of freedom of each particle at an instant in time. This corresponds to discrete points in phase space. However, since it was considered that a very large number of particles exist, the complete accounting for discrete particles can be translated into a continuum of particles.
Phase space measure applies only to the space on which the particles are defined and not to the distributions themselves. To quantify the distributions and relate them to phase space measure several additional properties are required to apply to an ensemble of particles.
Let one begin by assuming the existence of a field of non-interacting, neutral, one-speed particles in space. Every region of space then has associated definite particle content, a number indicating how many particles exist within that region. This number may be further partitioned according to the
120 directions in which the particles move and according to the internal states that they possess. By associating values to subsets of space, direction and internal states, one defines a non-negative, real- valued set function over phase space, which will be denoted by particle measure N . This particle measure is defined over the same σ-algebra, as is the phase space measure. In addition, the following two physically plausible properties for N are required:
P1:ρΓ (AA )< ∞⇒N ( ) <∞ (143) P2:ρΓ (AA )= 0⇒=N ( ) 0
These properties establish the only connection between the particle measure and the phase space through which the particles propagate. Specifically, property P1 states that every finite region of phase space has finite particle content, and property P2 states that a region of phase space with zero volume can not contain a meaningful number of particles. These properties influence the nature of energy-based calculations since any meaningful transfer of energy entails both spatial and directional integration.
With this interpretation of the particle measure N , property P2 states that the particle measure is absolutely continuous with respect to the phase space measure ρΓ [Kolmogorov and Fomin, 1999]. Absolute continuity defines an order relation on the set of measures over the same σ-algebra, which is usually denoted by “ ”. Thus, the notation N ρΓ simply means that N()A = 0 whenever ρΓ ()A = 0. Therefore, the three-tuple (,Γ P ,N )will be called the particle measure space, with the associated particle measure N .
In order to characterise the distribution of particles in a continuous phase space, one can also use the well-known concept of phase space density n , defined over the space, and that emerges automatically from the concept of particle measure through the so-called Radon-Nikodym theorem [Kolmogorov and Fomin, 1999]. This theorem states that if (,,B B m )is a measure space over the set B , and if µ is a σ-finite, non-negative measure defined on B , such that µ m , then there exists a non-negative m -integrable function f over B , such that
121 µm()Afd= ∫ ()γγ () (144) A for every measurable set A∈B . In addition, this function f , called the Radon-Nikodym derivative of µ with respect to m , is unique to within a set of m -measure zero. The Radon-Nikodym theorem constitutes an obvious generalization of the Lebesgue’s theorem, which states that an absolutely continuous function is the integral of its derivative.
Therefore, this theorem can be applied to the particle measure and to the phase space measure. The following statement can be obtained:
Theorem 1: (Existence of Phase Space Density). For every configuration of non-interacting, one-speed
+ particles in phase space Γ with measure ρΓ , there exits a ρΓ -measurable function n : Γ , which is unique to within a set of ρΓ -measure zero, satisfying the Lebesgue integral
N()And= ∫ ()γ ρ ()γ (145) A
where N()A denotes the particle content of the measurable subset A∈P .
The function n , whose existence is guaranteed by the above theorem, is called the phase space density, and in order to emphasize the similarity with usual differentiation, the Radon-Nikodym derivative is denoted as
dN −−31 − 1 n=× ;[.º n of particles m sr Hz ] (146) dρΓ
Note that the dimension of the phase space density is just equal to number of particles times inverse phase space volume.
This phase space density is therefore a real-valued function defined over phase space, n : Γ + , such that
ndddl(,r Ω ,ω )Vr ()σΩ ( ) (ω ) (147)
122 yields the number of particles in an elementary volume dVr()about the point rR∈ 3 with directions within an elementary solid angle dσΩ() about Ω∈S2 having values of the internal state (sound frequency attribute) in the range dl ()ω . Therefore,
Nn= ∫∫∫ (,r Ω ,ω ) dddlVr ()σΩ ( ) (ω ) (148) ∆ χ V gives the total number of particles inside some region AV= ××∆χ of phase space. Similarly, the expression
Nndd()ωω= ∫∫ (,,)r Ω Vr ()σΩ () (149) χ V equals the total number of particles inside the region AV= × χ , possessing an internal attribute ω .
The abstract nature of phase space density makes it universal, underlying virtually all particle transport problems from gas dynamics to neutron diffusion to sound particles propagation.
5.2.4 The Trajectory Space and Particle Events If the phase space positions of all particles are graphed over time, then a set of one-dimensional space-curves defined in the trajectory space can be obtained. This trajectory space is defined by:
Ψ ≡×Γ (150) where the set of real numbers represents time. One endows the trajectory space with a set- theoretic measure, consisting of the triple (,,Ψ G ρΨ ), where ρΨ is a non-negative measure, defined on the elements of a set G , which is a σ-algebra of subsets of the space Ψ . Analogously to what was defined for the phase space, one also assumes the existence of a particle measure N associated to the measure space (,,)Ψ GN , that gives the particle content existing in some region of the trajectory space Ψ .
A particle event is defined to be a single point in trajectory space Ψ . Some events have natural definitions. For example, each emission, absorption, or scattering event corresponds to a single
123 point along a particle trajectory. In fact, each scattering event corresponds to two points along the particle trajectory, since the parameter Ω has different values before and after the collision, which corresponds to a discontinuity in the trajectory. Therefore, one must distinguish between in- scattering and out-scattering events, according to whether one defines the particle state before or after the collision. Other events can be defined artificially, usually by specifying a surface in Ψ that intersects the particle trajectories at a set of points. For example, one could define the events to be the particle states at a particular time t0 . This corresponds to intersecting the trajectories with the
3 plane tt= 0 in the trajectory space Ψ . Similarly, given an arbitrary plane P in R , one could define a particle event to be a crossing of P , corresponding to an intersection with the surface ××P S2 × in trajectory space.
Once the particle events have been defined, one is left with a set of points in trajectory space. These points may be distributed throughout the whole space Ψ , or they may lie on some lower- dimensional manifold, as for example in the case the particle events were defined as an intersection with a surface. In order to define energy quantities, one then has to quantify the distribution of these events with respect to a suitable geometric measure.
The consideration of particle events leads to a definition of the particle event space. This is the subset IP ⊂Ψof the trajectory space containing all possible locations of the particle events that one wishes to count. Thus, IP depends on the definition of a particle event. By defining particle events in different ways, different energy-based quantities will be obtained. For example, consider the case of particle emission by some source of particles, which is located in a volume V of R3 . In this case, the particle event space is equal to (IP= ×V ××S2 ) ⊂Ψ. In the case that particle events are defined as crossings of a hypothetical surface P in R3 , then the particle event space would be ()IP= ××P S2 × ⊂Ψ. In this example, IP is a 6-dimensional manifold within the 7- dimensional trajectory space Ψ .
In order to account for the time evolution of the distributions of the particles in space, consider time must be considered. Analogous to the phase space density introduced in last section, one can
124 guarantee the existence of a trajectory space density through the Radon-Nikodym theorem. More specifically, one can enunciate the following
Theorem 2: (Existence of Trajectory Space Density). For every configuration of non-interacting, one-
+ speed particles in trajectory space Ψ with measure ρΨ , there exits a ρΨ -measurable function ν : Γ , which is unique to within a set of ρΨ -measure zero, satisfying the Lebesgue integral
()Ad= ν ()γγρ () N ∫ Ψ (151) A
where N ()A denotes the particle content of the measurable subset A∈G .
The trajectory space density is then defined as the Radon-Nikodym derivative of the particle content measure N with respect to the trajectory space measure ρΨ
dN ν =× ;[.ºn of particles s−13 m−− sr 1 Hz − 1 ] (152) dρΨ
This trajectory space density is therefore a real-valued function defined over phase space and time, meaning ν :()Ψ= ×Γ + , such that
ν (,r Ω ,ωω ,)tdVr () dσΩ ( ) dl ( ) (153) is the number of particles in an elementary volume dVr()about the point rR∈ 3 moving in a direction within an elementary solid angle dσΩ() about Ω∈S2 having values of the internal state (sound frequency attribute) in the range dl ()ω , at time t . Therefore,
N ()ttdddl= ∫∫∫ν (,r Ω ,ωω ,)Vr ( )σΩ ( ) ( ) (154) ∆ χ V gives the total number of particles inside some volume AV= ××∆χ , at time t . Similarly, the expression
125 N (,)ωνωttdd= ∫∫ (,,,)r Ω Vr ()σΩ () (155) χ V equals the total number of particles inside the volume AV= × χ , possessing an internal attribute ω , at time t .
5.2.5 The Trajectory Space Flux It is often more convenient to characterise the density of particles by their rate of flow across a real or imaginary unit surface, thereby introducing the concept of trajectory space flux ψ . In terms of the usual definitions of infinitesimals and limit arguments, the trajectory space flux ψ is defined as
ψ (,r Ω ,ωνω ,)t≡× c (,r Ω , ,) t [.º n of particles s−−12 m sr − 1 Hz − 1 ] (156) where c is the speed of sound. In terms of this trajectory space flux,
ψ (,,,)()()()()r Ωσωωtdltdr dσΩ dl (157) gives the number of particles with attribute in the range dl ()ω , with directions in infinitesimal solid angle dσΩ(), inside the volume dl() t dσ (r ) c located at point r , with the infinitesimal surface dPσ()orientated along the direction Ω .
The trajectory space flux can also be derived formally as a measure-theoretic quantity. For that sake, let one consider, as already introduced in the last section, that particle events are defined as crossings of a hypothetical surface P in R3 . In this case, the particle event space would be ()IP= ××P S2 × ⊂Ψ. The geometric space measure is defined by:
ρσσ= ll××⊥ × (158)
where σ⊥ corresponds to the projected solid angle measure (which is a function of the point on the surface P ) as introduced in section 5.2.1. One can then define the Radon-Nikodym derivative of the particle measure N with respect to this geometric measure ρ , on the considered particle event space, as:
126 ddNN −12−− 1 − 1 ψ == [.ºn of particles × s m sr Hz ] (159) dρ dldσσ d⊥ dl
Trajectory space flux thus gives the number of particles crossing a real or imaginary area, per unit area, per unit time, per unit-projected solid angle and per unit of internal state attributes (in this case, frequency).
The concept of trajectory space flux will be the fundamental quantity of interest, since it can be used to express virtually any quantity relating to the macroscopic distribution of particles.
One must distinguish between incident and exitant trajectory space flux functions, according to the interpretation of the Ω variable. An incident function ψ (,r Ω ,ω ,)t measures the flux arriving at i point r from the direction Ω , while an exitant function ψ (,r Ω ,ω ,)t measures the flux departing o from r in the direction Ω . In free space, these two functions are simply related by:
ψ (,r Ω ,ωψ ,)tt= (,r −Ω , ω ,) (160) io
However, at real surfaces the distinction is more fundamental. ψ i and ψ o measure different sets of sound particle events, corresponding to the particle states just before their arrival at the surface, or just after their departure respectively. The relation between ψ i and ψ o can be rather complex, depending on the scattering properties of the surface.
Recall that each sound particle traces out a one-dimensional curve in the trajectory space, namely the graph of the function (,r Ω ,ω )()t over all values of t . To quantify the flux, one must define kkk the particle event space to be the surface (IP= ×M ××S2 ) ⊂Ψin trajectory space. The traced one-dimensional curve is not continuous at the surface IP , since scattered sound particles instantaneously change their direction, in the case that they are not absorbed at the boundary M .
ψ i and ψ o quantify events that are limit points of trajectories on opposite sides of the surface IP . Each event (,t r ,Ω ,ω )measured by ψ is the limit of a trajectory defined for tt< , while an kk k k i k event measured by ψ o is the limit of a trajectory defined for tt> k , with tk being the instant when
127 the sound particle k hits the boundary M . This gives a simple and precise way to differentiate between incident and exitant trajectory space flux.
5.3 Transport Equations In this section, a transport equation for neutral, non-interacting particles is presented. The derived equation translates the evolution of an ensemble of abstract particles through the description of the balance between the gains and losses in the number of the existing particles. By adopting the required semantics for the description of sound energy transport inside an enclosure, this transport equation will then describe also the evolution of an ensemble of sound particles.
The goal of the present work is to derive an equation for the trajectory space flux ψ based on particle conservation, thereby requiring the characterisation of the phenomena, which are responsible for the perturbation of their motion through the medium. In Transport Theory, all particle behaviours fall into one of three categories: emission, streaming and collisions.
Emission refers to any process capable of injecting new particles into the system under consideration. In the case of sound particles, emission is accomplished by the sound sources that introduce new particles into the medium. Once emitted, particles travel unaffected until they suffer an instantaneous collision. For neutral, non-interacting particles, like in the case of sound particles, the path of propagation is straight and the behaviour of following such a path is called streaming.
Collisions are derived by interactions of the particles with matter, such as the propagation medium, namely air, or the materials composing the boundaries of the enclosure. They can be subdivided into absorption and scattering. Absorption removes particles from the system by capturing them and by converting their kinetic energy into some other form of energy, for example heat. Scattering is a process that instantaneously changes the direction of travel of the particles. In the case of the one-speed assumption, all scattering is assumed perfectly elastic. In this model, all interactions are considered to take place instantaneously, meaning that the time particles spend interacting with the medium or with the boundaries is much smaller than the mean typical times involved in the propagation processes.
128 5.3.1 A Transport Equation for Particles To derive a transport equation for the particles, let one start by observing the particles within some fixed small region in trajectory space at some instant t0 . This corresponds to a particle event space
3 given by IP=×Γ {t0 } . One thus defines the region as the product V × χ ×∆, where V ⊂ R is the spatial extent of the considered volume, χ ⊂ S2 is an arbitrary solid angle, and ∆⊂ is the range of considered internal states. This product is a set of all pairs (,r Ω ,ω )such that r ∈V , Ω∈ χ , and ω ∈∆ at the instant t0 . In order to derive a balance equation one has to determine how the number of particles inside this region of trajectory space changes with time. At any time t , the number of particles possessing an internal state attribute ω inside the region is obtained by integrating the trajectory space density over the 6-dimensional volume as given by (155).
If the time-rate-of-change of the quantity N (,)ω t is considered, then it is clear that a balance of particles is established by considering the gains and the losses of particles inside the volume per unit time. More specifically, the following must be true:
∂N (,)ω t ⎡⎤⎡⎤⎡⎤change change change =+⎢⎥⎢⎥⎢⎥ + (161) ∂t ⎣⎦⎣⎦⎣⎦due to emission due to streaming due to collisions
The first term on the right accounts for the change per unit time in the number of particles inside the volume due to the creation of new particles. One assumes that the only process capable of injecting new particles into the system is generated by sources of particles. The nature of these sources does not need to be specified in detail. For the moment, one just adopts the concept of a trajectory space source function q , which is a real-valued function defined on the trajectory space Ψ . This function gives the number of particles injected per unit time, per unit volume, per unit solid angle, and per unit internal state attribute. Its units are therefore [.ºn of particles× s−−13 m sr − 1 Hz − 1 ]. The magnitude of the emission of particle can thus be written in terms of this trajectory space source function, yielding the emission function
Et(,)ωω≡ ∫∫ q (,,,)r Ω tddVr ()σΩ () (162) χ V
129 The trajectory space source function injects particles into regions of trajectory space, the injected particles having an intrinsic internal state of ω . The regions in which the particles are injected can be either confined regions or vast extended regions, depending on the specific characteristics of the source.
The changes due to streaming are the ones that refer to particles with directions in χ that either escape from or enter into the volume V because of their straight path motion. More precisely, the change due to streaming is the net flow of particles with directions in χ that pass through the confining surface ∂V of volume V . By subdividing the confining surface into elementary areas, each possessing a surface normal directed inwards to the volume, then the changes due to streaming can be calculated by integrating the normal component of the flux due to particles in χ over the entire surface ∂V , yielding the streaming function
St(ωψω ,)≡ ∫∫ (,s ΩΩ , ,) tins () dσ ()s dσΩ ( ) (163) χ ∂V where s refers to a point located on the confining surface ∂V of the volume V and ns()is the inward-pointing normal at s . In general, some of the elementary surfaces will have positive flows and some negative, so (163) accounts for particles flowing into and out of the volume V .
The last term on the right of expression (161) translates the changes due to collisions of the particles during their motion. This term can be split into changes due to absorption of particles and into changes due to elastic scattering of particles that redirect their motion. In the case of non- interacting particles, the collision rate is independent of the trajectory space flux ψ since they can only interact with the propagation medium. The probability of a particle being absorbed will be assumed proportional to the distance of travel through the medium. In fact, this assumption derives from the consideration of a mean collision time, or relaxation time. It can be easily shown that the probability that a particle, after surviving without collisions for a time t , suffers a collision in the time interval between t and tdt+ is given by [Reif, 1965]:
t 1 − Ptdt() = eτ dt (164) τ
130 where τ is the mean time between collisions.
The constant of proportionality between the probability of being absorbed and the distance of
-1 travel will be denoted as the particle absorption coefficient σ A , with units of [m ]. This time independent coefficient can vary with the position in phase space for an anisotropic medium and can be a function of the internal state of the particles. However, the propagation medium will be considered as isotropic, whereby the particle absorption coefficient is only a function of the position of the particle in physical space. The absorption coefficient is a scalar quantity with values between zero and one. The changes due to particle absorption can thus be written as:
Ct(,)ωσωψω≡ (,)(,,,)rrΩ tddVr ()σΩ () AA∫∫ (165) χ V
As with absorption, one assumes that the probability of an elastic collision occurring is proportional to the distance the particle travels. Upon collision, however, the new direction of travel of the particle must also be accounted for. Because only elastic collisions are considered, no other changes to the particle need be considered. The process of scattering through elastic collisions can be characterised by introducing the volume scattering function κ , defined in R3 ×−[1,1] × such that:
κω(,r ΩΩi '' , )d σΩ ( ) (166) gives the probability that a particle possessing internal state ω at r , moving in the direction Ω , will de deflected into the solid angle dσΩ()' around the new direction Ω' , per unit distance of travel. The units of the volume scattering function are therefore [m-1sr-1]. The isotropy of the medium is incorporated in the dot product of expression (166), whereby the volume scattering function depends only on the relative orientation of the direction vectors.
The changes due to scattering from elastic collisions are best calculated by considering the two types of scattering that can either make particles to leave or to enter the volume V × χ . These two types are called particle out-scattering and in-scattering. In out-scattering, a particle remains within
131 the volume V while its direction is instantaneously changed to something outside of χ because of an elastic collision. In order to account for this out-scattering let one define
Ct(,)ωκωψω≡ (,r ΩΩi '' ,)(,,,)r ΩσΩ tddd ( )Vr ()σΩ () (167) OUT ∫∫∫ χ V S2 which gives the total number of scattering collisions suffered by the particles in V × χ at time t . Analogously one defines a quantity for in-scattering as:
Ct(,)ωκωψω≡ (,r ΩΩ'''i ,)(,r ΩσΩ ,,) tddd ( )Vr ()σΩ () (168) IN ∫∫∫ χ V S2 which is equal to the number of particles at time t in V , with directions of travel in χ after elastic scattering. The net change due to elastic scattering will then be given by the difference between the in-scattering quantity and the out-scattering quantity.
With these definitions, the balance equation for the particles, namely expression (161), can then be written as:
∂N (,)ω t =−−E(,)ω tS (,)ωω tC (,) tC + (,) ω tC − (,) ω t (169) ∂t A IN OUT
By replacing the explicit expressions for each of the terms yields:
∂ ∫∫νω(,r Ω , ,)tdVr () dσΩ ( )= ∂t χ V ∫∫qtdd(,r Ω ,ωψω ,)Vr ()σΩ ( )− ∫∫ (,s ΩΩ , ,) tddins ()σ ()s σΩ ( )− χχVV ∂ σωψω(,rr ) (,Ω , ,)tdVr () dσΩ ( )+ ∫∫ A (170) χ V κωψω(,r ΩΩ'''i , ) (,r ΩσΩ , ,)td ( ) dVr () dσΩ ( ) − ∫∫∫ χ V S2 κωψω(,r ΩΩi '' ,)(,,,)()r ΩσΩtd dVr ()() dσΩ ∫∫∫ χ V S2
132 The second term on the right of the equal sign in this expression can be rewritten in terms of a volume integral using Gauss’s Theorem:
∫∫ψω(,s ΩΩ , ,)tddins ()σ ()s σΩ ( )= χ ∂V (171) ∫∫∇=∇ii()ψω(,r ΩΩ , ,)tdVr () dσΩ ( ) ∫∫ Ω ψω (,r Ω , ,) tddVr ()σΩ ( ) χχVV and where Ω is fixed with respect to the divergence operator. Therefore, all the terms in equation (170) incorporate an integration over the small volume of phase space V × χ , which is completely arbitrary. Therefore, it follows that equality must hold for all the integrands, yielding for the balance of particles:
1(,,,)∂ψωr Ω t +∇Ωi ψω(,,,)r Ω tq = (,,,)r Ω ωσωψω t − (,)(,,,)rrΩ t − ct∂ A (172) κωψω(,r ΩΩii''''' , ) (,r ΩσΩ , ,)td ( )+ κωψω (,r ΩΩ , ) (,r ΩσΩ , ,) td ( ) ∫∫ SS2 2 where we used relation (156) in order to eliminate the trajectory space density.
This equation determines the evolution in time of the distribution of the particles, taking into account their motion (via the advection term Ωi∇ψ ) and their interaction with the medium (via the scattering terms on the right hand side). The first integral on the left hand side of equation (172) can be simplified since the trajectory space flux ψ does not depend on the variables of integration. Therefore, one can write
σω(,rr )= κ (,ΩΩi '' , ω )d σΩ ( ) (173) S ∫ S2 and denote this quantity as the particle out-scattering coefficient. This term gives the probability that a particle will suffer an out-scattering collision per unit distance travelled. Its units are thus equal to the units of the particle absorption coefficient, namely [m-1]. Adopting this new definition, the balance equation can be finally stated as:
133 1(,,,)∂ψωr Ω t +∇Ωi ψωσωψω(,,,)r Ω tt + (,)(,,,)rrΩ = ct∂ (174) qt(,r Ω ,ωκ ,)+ (,r ΩΩ'''i , ωψω ) (,r ΩσΩ , ,) td ( ) ∫ S2 where one introduces the particle extinction coefficient, with units of []m−1 :
σ (,rrrωσ )= (, ωσ )+ (, ω ) (175) AS
Equation (174) is the derived transport equation for neutral, non-interacting particles propagating within an isotropic medium under the one-speed assumption, where all the particle-matter interactions are either perfectly elastic or perfectly inelastic. This integro-differential equation can be viewed as a special instance of the well-known general Boltzmann Equation [Barnett, 2000; Bellomo and Pulvirenti, 2000]. It establishes the time evolution of an ensemble of abstract particles in terms of losses and gains, where these are attributable to sources and to collisions. Losses and gains are expressed in terms of three independent quantities: σ AS,σ and q . These quantities are of macroscopic nature, depending on the physics of the interactions and on the properties of the sources of particles. They can be obtained by first considering the laws that govern the scattering, the absorption, and the emission of a single particle, and then by solving the statistical problem of determining the result of a large number of such interactions governed by these laws. Here, however, one will not bother about these details. One just adopts the position that all three independent quantities can be obtained by some way or another, or that they are just predetermined. In this case, the transport equation is a linear equation on the trajectory space flux. Despite being linear, this equation is rather complex since it poses a mixed problem, in the sense that the unknown time-dependent trajectory space flux, defined in a 6-dimensional space, is differentiated with respect to the spatial variables and integrated with respect to the direction variables. In the next sections, some simplifications will thus be assumed to obtain more tractable and computationally friendly expressions.
This transport equation serves as the basis for describing the evolution of sound particles and for determining the evolution of sound energy inside enclosures. To this end, one needs to consider the effect of the boundaries. This will be dealt with in the next section.
134 5.3.2 Boundary Conditions The transport equation derived in last section is a linear first-order differential equation in both time and space variables. As such, it requires both initial values and boundary conditions to eliminate the arbitrary constants of integration [Sokolnikoff and Redheffer, 1958]. The transport equation is only valid away from surfaces, which constitute the boundaries of the system. In the present case, these boundaries, which are collectively denoted by M , will be identified with the walls of the specific room under consideration.
The boundary M introduces a partitioning of S2 into two hemispheres at each point s ∈M , as already mentioned in section 5.2.1. Each elementary surface area of the boundary located at s is supposed to have a normal vector ns(), orientated inwards to the volume under study. In addition, a set of all corresponding inwards-orientated rays impinging on the boundary will be defined by:
2 2 ϒ≡IN {(,s Ω ) ∈×MHS :Ω ∈− ()s } (176)
In a similar way, the set of all outwards-pointing rays originating at the boundary are defined by:
2 2 ϒ≡OUT {(,s Ω ) ∈×MHS :Ω ∈+ ()s } (177)
These definitions satisfy the following trivial equalities:
HH22()ssS∪= () 2 +− (178) 2 ϒ+ϒ=×OUT IN M S
2 2 where the positive and the negative hemispheres H + and H − respectively were previously defined in section 5.2.1.
There are different ways of specifying boundary conditions, and different physical situations call for alternative formulations. The formally simpler are the so-called explicit boundary conditions, also known as the prescribed boundary conditions, where the boundary flux is completely predefined:
ψ (,s Ω ,ωψ ,)tt= (,s Ω , ω ,) (179)
135 for all (,s Ω )∈ϒ . Explicit boundary conditions are independent of the trajectory space flux OUT itself, accounting for particles that are generated by independent processes and subsequently enter the system through ∂M . On the contrary, the so-called implicit boundary conditions are a function of the trajectory space flux:
ψ (,s Ω ,ωϕψω ,)t = ( ,,s Ω , ) (180) for all s ∈M , and where they are assumed to be time-independent. The most general form for implicit boundary conditions expresses the exitant flux ψ o in terms of the incident flux ψ i . The physical interpretation is that of boundary scattering, whereby particles originating at the boundary result directly of those impinging on it. This general form can be written as an integral transformation such as:
()ψω(,s Ω , ,)|ttdsss∈=M ρωκωψω (, ) (,ΩΩ''' , , ) (,s ΩσΩ , ,) ( ) (181) oBBi ∫ S2 where κω(,s ΩΩ' , , ) denotes the surface scattering function, giving the probability that an B incident particle at s from the elementary solid angle dσΩ()' around the incident direction Ω' is scattered into an exitant direction Ω . The units of the surface scattering function are [sr-1]. The function ρ (,s ω ) is called the surface transparency coefficient, giving the probability that an B incident particle from any incident direction is not absorbed at the boundary. The integral transformation in equation (181) portrays the density of particles scattered in any direction as a weighted sum of the incoming densities, and the weighting can be strongly dependent on both the incoming and outgoing directions. If the surface scattering function includes a Dirac delta term such as κωδ(,s ΩΩ'' , , )= ( Ω− Ω ), then the integral transformation is simplified into: B
()ψ (,s Ω ,ω ,)|ttsss=−ρ (,ω )ψ (,Ω ,ω ,) (182) oBi translating the specular reflection behaviour of the boundary. In this case, the particle is reflected into a direction whose angle with the surface’s normal is equal to the incident angle. The other extreme is denoted by the ideal diffusely reflecting boundary, where the probability that the particle is scattered into any direction is uniform. In this case, the surface scattering function is written as:
136 κωκω(,,,)s ΩΩ' == (,,)s Ω const . (183) BB
Note, however, that the surface scattering function accounts not only for reflection at the boundary but also for transmission across the boundary.
5.3.3 The Integral Version of the Transport Equation for Particles In this section, an integral version of the integro-differential transport equation (174) is derived. The derivation takes into consideration the boundary conditions as defined in the previous section. The resulting pure integral equation is equivalent to equation (174) together with a set of boundary conditions. However, this integral version of the transport equation yields a much friendlier form to handle both conceptually and computationally. The basic idea consists on integrating equation (174) along a parametric straight-line segment until a boundary point is reached [Williams, 1971]. This integration will eliminate the gradient operator and incorporate the boundary value at the point of intersection with this parametric line. In order to eliminate the time derivative a Laplace Transform is used.
Let one write the integro-differential transport equation (174) as:
1(,,,)∂ψωr Ω t +∇Ωi ψωσωψω(,,,)r Ω ttQt + (,)(,,,)rrΩ = (,,,);r Ω ω ct∂ (184) Qtqt(,r Ω ,ωωκωψω ,)=+ (,r Ω , ,) (,r ΩΩ''i , ) (,r ΩσΩ , ,) td (' ) ∫ S2 where both gain terms were incorporated into a single quantity Qt(,r Ω ,ω ,). One starts by applying to this equation a one-sided Laplace Transform, defined as [Bracewell, 1986]:
∞ L[]()f pftedt≡ ∫ ()− pt (185) 0 and taking into account the fundamental property of the Laplace Transform that
⎡⎤df LL()p = pfp () (186) ⎣⎦⎢⎥dt
137 the transport equation can be rewritten in terms of the Laplace Transforms of ψ and Q as:
1 pLr[](,,,)ψωΩΩpp+∇i Lr [](,,,) ψωσωψωΩ + (,)[](,,,)rL rΩ pQp =Lr [](,,,)Ω ω (187) c
The second term on the left can be reformulated since the dot product of the direction vector with the gradient operator is equal to the directional derivative in the direction Ω . One introduces a parametric straight-line segment defined by r + γ Ω , where γ ∈ . This yields:
⎛⎞⎛⎞∂∂LL[]ψψ [] Ωi∇=+=−−Lr[](,,,)ψΩ ωpp⎜⎟⎜⎟ (r γωΩΩ ,,,) (r γωΩΩ ,,,) p (188) ⎝⎠⎝⎠∂∂γγ γ =00 γ =
Through the introduction of the three single-variable functions
ψˆ ()γ ≡−Lr [ψγ ](ΩΩ , , ω ,p ); p QLrˆ ()γγω≡− [Qp ](ΩΩ , , , ); (189) p σγˆ()≡− σ (r γΩ , ω ) where all other quantities are for now considered as fixed, together with the equality (188), one can cast equation (187) into a first order linear equation for ψˆ p ()γ :
dψˆ ()γ p ⎛⎞p ˆ −+⎜⎟σˆ()γγγψˆ pp () =−Q () (190) dcγ ⎝⎠
ˆ It should be noted that although ψˆ p ()γ is included in the gain term Q p ()γ , its contribution is restricted to a set of measure zero. Therefore, the equation is still a linear one.
The solution of a first order linear equation as
dy +=M ()xy Nx () (191) dx is readily obtained by introducing a so-called integrating factor, yielding [Sokolnikoff and Redheffer, 1958]
138 − Mxdx() Mxdx () yx()=+ e∫∫⎡ Nxe () dx C⎤ (192) ⎣⎢∫ 1 ⎦⎥
ˆ Therefore, identifying Mx() −+()σˆ ()γγ pc ; Nx () −Q p ()and defining the integrating factor
γ p −−γ σηˆ ()d η c ∫ µγ()= e 0 (193) the solution of equation (190) is given by:
1 ⎡ γ ⎤ ψˆ ()γηµηη=−Qˆ ()()d + C (194) pp⎢ ∫ 1 ⎥ µγ()⎣ 0 ⎦
The integration constant is readily determined by calculating ψˆ p (0) :
1 ψˆ (0)= C= C (195) p 11µ(0) yielding for the solution (194):
1 ⎡ γ ⎤ ψˆˆ()γηµηη=−Qˆ ()()d +ψ (0) ppp⎢ ∫ ⎥ µγ()⎣ 0 ⎦ (196) γ ⇔=ψψˆˆ(0)µ (γγ ) ( ) +Qˆ ( ηµηη ) ( )d ppp∫ 0
The correspondence ψˆ (0)= Lr [ψ ]( ,Ω ,ω ,p ) allows the calculation of the value of p Lr[](,ψω' Ω ,,)p at any arbitrary point rr'*=−γ Ω along the parametric straight-line by using equation (196) and ψˆ ()γψ*'= Lr [](,,,)Ω ωp . One imposes the arbitrary point r' to lie on the p boundary. The boundary conditions can then be incorporated into the solution (196). That is, the line parameter is set equal to γ * =−d (,r Ω ) where the boundary distance function M
139 d (,r Ω )≡>+∈ inf{γγ 0|r Ω M} (197) M gives the distance to the nearest intersection point on the boundary along the parametric straight- line. The intersection point will then be defined as
sr(,Ω )= rr−−d (,ΩΩ ) (198) M and by its definition s ∈M necessarily. In order to apply equation (196) to any arbitrary straight-line segment, the integrating factor can be written in vector notation as:
rr'− p −−−rr' σ (,) r +ηωη u d c ∫ µγ() µ (,rr' , ω )= e 0 ; (199) rr' − u = rr' − and the integral in equation (196) can also be written as a function of vectors r and r' :
rr' − γ ι()γηµηηι==−−QrrLrˆ ()()dQpd (,' ) [ ]( γωµγωγΩΩ , , , )(r Ω ,,r ) ∫∫p (200) 00
With this new notation, the integral form for the transport equation, given by (196), can be written as:
sr− Lr[](,,,)ψ Ω ωµωψωp =+−− (,,)[](,,,)sr L sΩ pQpd∫ µγω (r Ω ,,)[](rL r γωγΩΩ , ,,) (201) 0 where µ(,sr ,ω )is given by (199) and the boundary point is obtained from sr(,Ω )=−rrd (, −ΩΩ ) together with definition (197). M
The last step in the transformation consists in incorporating the implicit general boundary conditions into equation (201), thereby using relation (181), which yields:
140 Lr[](,,,)ψ Ω ω p = µωρωκ(,sr , ) (, s ) (, sΩΩ''' , , ωψ )Ls [ ](,ΩσΩ , ω ,pd ) ( )+ (202) BB∫ S2 sr− ∫ µγ(,,)[](,,,)r −−Ω rL ωQpd r γΩΩ ω γ 0
This is the complete integral form of the transport equation for the Laplace Transform of the trajectory space flux incorporating the implicit general boundary conditions. This equation holds for the evolution of an ensemble of neutral, non-interacting abstract particles, and according to the assumptions of the sound particle model, it will be valid for describing also the sound energy propagation inside an arbitrary enclosure. The translation into energy quantities, as describing energy sound fields will be presented in section 5.3.5.
In order to retrieve the temporal behaviour of equation (202), one must apply to it an inverse Laplace Transform defined by [Bracewell, 1986]:
1 i∞+δ L−1[]()f tfpedp≡ ∫ ()pt (203) 2πi −∞+i δ where δ is a suitably chosen positive constant (normally δ → 0 ), yielding:
i∞ ⎡⎤ ψ (,,,)r Ω ωµωρωκωψωtpdedp=+ (,,)sr (,) s (, sΩΩ''' ,,)[](,LsΩσΩ ,,)( ) pt ∫∫⎢⎥BB 2 −∞i ⎣⎦⎢⎥S (204) i∞ sr− ⎡⎤ ⎢⎥µγ(,,)[](,,,)r −−Ω rL ωQpdedp r γΩΩ ω γpt ∫∫ −∞i ⎣⎦⎢⎥0 and by replacing the integration factor µ and combining the exponentials one obtains the equivalent expression:
141 ψ (,r Ω ,ω ,)t = rs− ⎡⎤⎛⎞rs− i∞ pt⎜⎟− −+∫ σηωη(,)su d ⎢⎥⎜⎟c ''' ee⎝⎠0 ρωκ(,ss ) (,ΩΩ , , ωψ )Ls [ ](,ΩσΩ , ω , pddp ) ( ) + (205) ∫∫⎢⎥BB −∞i 2 ⎣⎦⎢⎥S
γ i∞ ⎡⎤sr− ⎛⎞γ −−+σγηωη(,)r Ω u' d pt⎜⎟− ∫ ⎢⎥ee⎝⎠c 0 Lr[]( Q−γωγΩΩ , ,,) pddp ∫ ⎢⎥∫ −∞i ⎣⎦⎢⎥0
The integrations in p can be performed immediately, whereby one retrieves the original non- transformed quantities, but with a time shift corresponding to the phase factor of the exponential. Therefore, the time-dependent transport equation for neutral, non-interacting particles is obtained:
rs− −+σηωη(,)su d ∫ ⎛−⎞rs ψ (,r Ω ,ωρωκωψω ,)te=−0 (,ss ) (,ΩΩ'' , , )s ,ΩΩ , , t d ' BB∫ ⎜⎟ 2 S ⎝⎠c (206) γ sr− −+−σηγωη((r ),)Ω d ∫ ⎛⎞γ +−−∫ eQ0 ⎜⎟r γωΩΩ,,, td γ 0 ⎝⎠ c together with the definitions (175), (173), (181) and (184).
Finally, introducing the total extinction function defined by the relation:
rr− ' ⎛⎞rr− ' ξ (,rr'' ,ωσηωη )=+⎜⎟ r , d (207) ∫ ⎜⎟ rr− ' 0 ⎝⎠ equation (206) can be written in a more compact form as:
−ξω(,,rs ) '⎛−⎞ 'rs ' ψ (,r Ω ,ωρωκωψω ,)te=− (,ss ) (,ΩΩ , , )s ,ΩσΩ , , t d ( ) BB∫ ⎜⎟ 2 S ⎝⎠c (208) sr− −−ξγω(,rr Ω , ) ⎛⎞γ +−−∫ eQ ⎜⎟r γωΩΩ,,, td γ 0 ⎝⎠ c
The interpretation of this equation is straightforward. The equation states that the trajectory space flux at some given point at some time t is equal to two terms. The first term arises from the
142 trajectory space flux that was reflected by the boundary at some intersection point at an earlier instant corresponding to the propagation delay between the boundary and the observation point. This boundary flux is multiplied by an exponential factor, whose exponent is equal to minus the total extinction, meaning the total losses that occurred during the propagation from the boundary until the observation point. The second term arises from the sources of particles, whereby all particles injected by the sources (be it real sources of particles or the in-scattered particles) are integrated along the path between the boundary point and the observation point. Note that the integration takes also in consideration the finite propagation delays existing between the moment of particle injection and the moment of observation. Finally, an exponential factor accounting for the total extinction is also included in this second term.
5.3.4 Definition of Energy Quantities for describing Sound Fields It must be stated, as already pointed out in the beginning of this chapter, that there are no standard definitions and nomenclature for acoustical energy-based quantities. Therefore, in this section several rigorous definitions for acoustical energy-based quantities are presented, which are formally derived from a set-theoretic perspective. The resulting definitions parallel some definitions that are usually used for radiometric quantities in the fields of radiative heat transfer and illumination engineering.
One starts by considering again the trajectory space and the particle event space, as introduced in section 5.2.4, in which the existence of definite particle content measures was assumed. These measures are set functions describing some characteristics of regions of the trajectory space. In the sound particle model presented herein, each sound particle was assumed to propagate with constant sound speed through a medium. Associated to the propagation there is a certain defined kinetic energy. Therefore, one assumes that the propagating sound energy is due to the kinetic energies of the sound particles of an ensemble, each having a definite energy. Therefore, one can introduce another non-negative, real-valued set function over trajectory space that can be denoted by energy measure E . This energy measure is defined over the same σ-algebra, as is the trajectory space measure, and gives the energy content existing inside some region of trajectory space. The three-tuple (,,)Ψ GE is called the energy measure space, with the associated energy measure E . In addition, one can enunciate the following:
143 Theorem 3: (Existence of Energy Density). For every configuration of non-interacting, one-speed particles
+ in trajectory space Ψ with measure ρΨ , there is a ρΨ -measurable function ε : Γ , which is unique to within a set of ρΨ -measure zero, satisfying the Lebesgue integral
()Ad= ε ()γγρ () E ∫ Ψ (209) A
where E()A denotes the energy content of the measurable subset A∈G .
The units of the energy content measure are simply [J ] , whereby the energy density is given by
dE 13 1 1 3 1 1 ε == [][]JsmsrHz− −− − WmsrHz −− − (210) dρΨ
The existence of an energy content measure will allow the derivation of several important acoustic energy-based quantities that will be used later.
• Acoustic Power
To consider the quantity of acoustic power, the events resulting from the intersection of the trajectory space with a surface can be defined by
{()r,,,Ω ω t ∈Ψ :rr =00 ;ΩΩ = } (211)
In this case, the particle event space is two-dimensional and equal to (IP= ×⊂Ψ) , with a geometrical measure that accounts for the presence of the surface:
ρIP = ll××δδr Ω × (212) 00 where
⎧⎧1:r00∈∈DD 1:Ω δδr ()DD==⎨⎨ ;Ω () ; (213) 000:r ∉∉DD 0:Ω ⎩⎩ 00
144 Then, the Radon-Nikodym derivative of the energy content measure with respect to this geometric measure is the quantity
ddEE −−11 − 1 Π= = ; ⎣⎡Js Hz⎦⎣⎤⎡ = WHz ⎦⎤ (214) dρIP dldl which measures the energy content with respect to time and frequency at the trajectory space point ()r ,,,Ω ω t . Therefore, the measured quantity is just spectral power. 00
• Acoustic Angular Power Flux (or Acoustic Radiance)
This quantity is derived by considering a particle event space (IP= ××P S2 ×) ⊂Ψ corresponding to particles crossing an arbitrary plane P in R3 . The geometric measure is defined ⊥ as ρσσIP =××ll ×, where the Radon-Nikodym derivative of the energy content measure with respect to this geometric measure is
dE β ≡= ; ⎡Js−−12 m sr − 1 Hz − 1⎤⎡ Wm − 2 sr − 1 Hz − 1 ⎤ ⊥ ⎣ ⎦⎣ ⎦ (215) dldσσ dr dl
The angular acoustic power flux depends on the location of the arbitrary plane and on the direction vector of the projected solid angle measure. Therefore, this dependence can be written more explicitly as:
β ==ββθϕ(,s Ω ) (,s , ) (216)
The angular acoustic power flux is the amount of acoustic sound energy at some point s into a specified direction given by Ω = (,θ ϕ )(related to a local spherical coordinate system located at s ), per unit time, per unit area perpendicular to the direction of motion, per unit solid angle, and per unit frequency. Notice that although this definition is valid only on the plane P , the choice of P was arbitrary. Therefore, one can use equation (215) to define β anywhere in the trajectory space Ψ . The chosen plane P can thus represent a small planar region of some imaginary surface or it can also represent a small planar region of a real surface, meaning the boundary of the enclosure.
145 Therefore, the acoustic angular power flux is defined over all the boundary of the enclosure as well as over its volume. The dependence of the acoustic angular power flux on frequency will be implicitly assumed in the expression β (,s Ω ) in order to simplify the notation.
Under the one-speed assumption, every sound particle has the same energy. By denoting this energy by EP , it is obvious that the energy content measure and the particle content measure over trajectory space are simply related by:
E=EP N (217) where one has that
ddEN β ≡= ⊥⊥EEPP =ψ (218) dldσσ drr dl dld σσ d dl
Therefore, the angular acoustic power flux is just equal to the product of the individual sound particle energy and the trajectory space flux, which in turn equals the sound speed times the trajectory space density:
β = EEcPPψν= (219)
• Incident and Exitant Acoustic Angular Power Flux (or Incident and Exitant Acoustic Radiance)
The acoustic angular power flux is then directly proportional to the trajectory space flux. This flux ψ , defined in section 5.2.5, can be viewed as a function ψ :(Ψ= ×RS32 × × ) + .
32 + Therefore, the quantity β = EPψ is also a function β :(Ψ= ×RS × × ) . Most often, however, the domain of β will be restricted to ×M ××S2 , where M denotes the set of the surfaces that constitute the enclosure under consideration. In addition, one can expand the image range of β in order to allow also negative values, which have no physical meaning, but that will ensure that the set of all angular power flux functions forms a linear space. Therefore, functions β :(Ψ= ×RS32 × × ) will be allowed.
146 One will distinguish between incident and exitant angular power flux functions, according to the interpretation of the Ω variable, as already done in relation to the incident and exitant trajectory space fluxes in section 5.1.5. An incident function β (,r Ω )measures the flux arriving at point r i from the direction Ω , whereas an exitant function β (,r Ω )measures the flux departing from r in o the direction Ω . In free space, these two functions are simply related by:
β (,r Ω )= β (,r −Ω ) (220) io
However, at real surfaces the distinction is more fundamental: βi and βo measure different sets of sound particle events, corresponding to the particle states just before their arrival at the surface, or just after their departure, respectively. The relation between βi and βo can be rather complex, depending on the scattering properties of the surface.
The difference between incident and exitant angular power fluxes can be more precisely understood in terms of the trajectory space ψ . Recall that each sound particle traces out a one- dimensional curve in this space, namely the graph of the function (,r Ω ,ω )()t over all values of kkk t . To quantify the angular power flux, one can define the particle event space to be the surface ()IP= ×××M S2 ⊂Ψin trajectory space. The traced one-dimensional curve is not continuous at the surface IP , since scattered sound particles instantaneously change their direction, in the case that they are not absorbed at the boundary M . Note that βi and βo quantify events that are limit points of trajectories on opposite sides of the surface IP . Each event (,t r ,Ω ,ω )measured by kk k k
βi is the limit of a trajectory defined for tt< k , while an event measured by βo is the limit of a trajectory defined for tt> k , with tk being the instant when the sound particle k hits the boundary M . This gives a simple and precise way to differentiate between incident and exitant acoustic angular power flux.
• Acoustic Power Flux into direction Ω
dE ⊥−−−−− ⎡ 12 1⎤⎡ 2 1 ⎤ BΩ ()rr≡=β (,ΩσΩ ) dr ( ) ⎣ JsmHz⎦⎣ = WmHz ⎦ (221) dldσ dl
147 ⊥ gives the amount of acoustic sound energy in the small solid angle dσΩr (), per unit time, per unit area perpendicular to the direction of motion, and per unit frequency. The physical quantity defined by the acoustic power flux is analogous to the usual definition of sound intensity, namely power per unit area. However, sound intensity, as given by (21), is defined through wave-based quantities, whereas (221) is defined just on the basis of energy and geometrical quantities. This is the reason why both terms are differentiated and why the term sound intensity is not used in this chapter.
The explicit dependence of B ()r on frequency will be omitted for the sake of simplicity in Ω notation.
• Acoustic Radiosity
Acoustic radiosity is defined as the total acoustic power leaving from a surface point s (imaginary surface or real surface), per unit projected area, and results from integrating the exitant acoustic angular power flux over the unit sphere S2 surrounding point s :
2ππ ⊥ −−21 B()ss==ββθϕθθθϕoo (,ΩσΩ )dddWmHzs ( ) (,s , )sincos ; ⎡⎤ (222) ∫∫∫ 00 ⎣⎦ S2
• Scattering of Sound Energy by Walls
A passive wall, which scatters acoustic energy, such that when hit by an incident angular acoustic power flux, β (,s Ω )= βθϕ (,s , )from direction Ω , returns a scattered angular intensity, iiiii i β (,s Ω ) in direction Ω , can be characterized by a Wall Scattering Function oo o ϒ=ϒ(,s ΩΩ , ,ω ) (,s θϕθϕω , , , , )determined by: Soi Sooii
⊥ βωβoo(,s Ω )=ϒ Soiii (,s ΩΩ , , ) (,s ΩσΩ )d s ( i ) = ∫ S2 2ππ (223) ϒ (,ssθ ,ϕθϕωβ , , , ) (, θϕ , )sin θ cos θθϕdd ∫∫ Sooiiiiii iii 00
This expression allows the definition for the wall scattering function (in short, the WSF), as the ratio between the acoustic angular power flux in the outgoing direction (subscript o) and the
148 acoustic power flux in the incident direction (subscript i). It is therefore a function of both incoming and outgoing directions, and the preceding definition gives the following equation for the WSF:
βoo(,s Ω )β oo (,s Ω ) −1 ϒ=Soi(,s ΩΩ , ,ω ) ⊥ = ; [sr ] (224) ββθii(,s ΩσΩ )dds ( i ) ii (,s ΩΩ )cos ii
The function ϒS is the most general expression for the scattering characteristics of acoustic energy by a passive wall. The WSF is not necessarily a number between zero and one. By its definition, it can have any value between zero and infinity.
The scattered sound energy can be subdivided into two components. The first one corresponds to the energy that is reflected by the wall while the second component corresponds to the energy that is transmitted across the wall. This leads to the definition of the Wall Reflection Function (WRF) and the Wall Transmittance Function (WTF), denoted by ϒ (,s ΩΩ , ,ω ) and ϒ (,s ΩΩ , ,ω ) Roi Toi respectively. The WRF is obtained by simply restricting the WSF to a smaller domain:
22 ϒ×Ri: HH r (225)
2 2 where H i and H r are called the incident and reflected hemispheres respectively. In fact, both
22 symbols refer to the same set of directions, meaning that HHir= , which can be either the
2 2 positive hemisphere H + , or its complement H − .
The WTF is defined similarly to the WRF, by restricting the WSF to a domain of the form:
22 ϒ×Ti: HH t (226)
22 where the transmitted hemisphere HHti=− equals the complement of the incident hemisphere.
2 As before H i can represent either the positive hemisphere, or its complement.
The WSF is the union of two WRF’s, one for each side of the surface, and two WTF’s, one for sound energy transmitted in each direction. Therefore, purely reflective or purely transmissive walls
149 are simply special cases of the general formulation with the WSF. In addition, the WSF is actually easier to define, since one does not need to specify the hemispherical domains needed by the WRF and the WTF.
In the most usual cases with physical meaning, the WRF obeys the “principle of detailed balance” with ϒ=ϒ(,,,)s ΩΩω (,,,)s ΩΩω , meaning that the WRF is symmetric in relation to Roi Rio the incident and exitant directions
• Directional-spherical Surface Scattering
β (,s ΩΩ )dσ⊥ ( ) ∫ oos o 2 B()s Ss() ⊥ RdΩ (,s ΩiSoio ,ωω )==i ⊥ =ϒΩΩΩ (,s , , )σs ( ) (227) ∫ BdΩ ()ssβii (,ΩΩ )σs ( i ) Ss2 () is a dimensionless quantity obeying 0(,,)1≤ R s Ω ω < , and expressing the ratio between the Ω i acoustic radiosity and the incoming acoustic power flux ats from direction Ω . i
• Directional-hemispherical Surface Reflectivity
⊥ βoo(,s ΩΩ )dσs ( o ) ∫ B ()s H 2 ()s ⊥ H RdH (,s ΩiRoio ,ωω )==i ⊥ =ϒΩΩΩ (,s , , )σs ( ) (228) Bd()ssβ (,ΩΩ )σ ( ) ∫ Ω iis i H 2 ()s is a dimensionless quantity obeying 0(,,)1≤ R s Ω ω < , and expressing the ratio between the H i acoustic radiosity over the hemi-sphere H and the incoming acoustic power flux at s from direction Ωi .
5.3.5 A General Transport Equation for Sound Energy The general transport equation derived in the previous sections, either in integro-differential or pure integral form, is not connected with a particular physical situation. In this section, the general transport equation will be used to describe the transport of sound energy inside enclosures. For that sake, the bridge between the previously introduced particle-based quantities and the physical quantities that were defined for measuring sound energy will be made.
150 One can recast the integro-differential transport equation (174) in order to describe the propagation of sound energy inside enclosures. In the previous section, one showed that the acoustic angular power flux β is linearly related to the trajectory space flux ψ by equation (218). Therefore, by multiplying equation (174) by EP , yields:
1 ∂ ()Etψω(,r Ω , ,) P +∇Ωi ()EtEtψωσωψω(,r Ω , ,) + (,rr ) (,Ω , ,) = ct∂ PP (229) Eq(,r Ω ,ωκ ,) t+ (,r ΩΩ'''i , ωψω ) E (,r ΩσΩ , ,) td ( ) PP ∫ S2
If the quantities σ and κ are reconsidered in the light of energy transport and if the spectral angular power density function Q (,r Ω ,ω ,)tEq= (,r Ω ,ω ,) tis introduced, with units of P []Wm−−31 sr Hz − 1, then equation (229) is recast into:
1(,,,)∂βωr Ω t +∇Ωi βωσωβω(,r Ω , ,)tt + (,rr ) (,Ω , ,) = ct∂ β (230) Q (,r Ω ,ωκ ,)ttd+ (,r ΩΩ'''i , ωβω ) (,r ΩσΩ , ,) ( ) ∫ β S2 where σ (,r ω )means the distance attenuation factor that gives the energy attenuation at r , per β unit length of distance of propagation within the medium, for an acoustic angular power flux of frequency ω . The quantity κω(,r ΩΩ' i , ) means the energy volume scattering factor, which gives β the amount of energy flux travelling in direction Ω' that is redirected into the direction Ω , per unit length of propagation distance and per unit solid angle.
In order to complete the description of energy propagation inside an enclosure one must consider the relevant boundary conditions for the acoustic angular power flux. This is achieved by multiplying equation (181) by EP and making the correspondence
ρωκ(,ss ) (,ΩΩ , , ω )=ϒ (,s ΩΩ , , ω ) Ωins () (231) BBioSoii thus obtaining:
151 βωβ(,s Ω )=ϒ (,s ΩΩ , , ) Ωins () (, sΩσΩ )d ( ) oo∫ Soii ii i 2 S (232) ⊥ =ϒSoiii(,s ΩΩ , ,ωβ ) (,s ΩσΩ )d s ( i ) ∫ S2
where ϒS is just the wall scattering function as defined in equation (224).
Therefore, the general transport equation (230) together with the boundary conditions (232) describe completely the propagation and scattering of acoustic sound energy inside an arbitrary enclosure. If the four independent quantities are specified, namely σ BB,,κ Q and ϒS , then the set of equations can be solved for, yielding the solution of the time-dependent acoustic angular power flux inside the enclosure. Note that all intervening quantities are frequency dependent, where the solution has spectral character. The functions β and Q play the same roles as their abstract counterparts ψ and q , with the difference that they are now linked to physical units meaningful to the transfer of sound energy.
Instead of using the integro-differential transport equation (174), one can use as the starting point the integral form (208) to derive an equivalent equation for the transport of acoustic sound energy inside enclosures. As shown below, this alternative derivation leads to a more compact formulation for the evolution of the acoustic angular power flux. By multiplying equation (208) by EP , yields:
−ξω(,,rs ) '⎛−⎞ 'rs ' Eteψω(,,,)r Ω =− ρωκ (,)ss (,,,)ΩΩ ωψω Es ,ΩσΩ ,, td ( ) PBBP∫ ⎜⎟ 2 S ⎝⎠c (233) sr− −−ξγω(,rr Ω , ) ⎛⎞γ +−−eEQtd r γωΩΩ,,, γ ∫ P ⎜⎟ 0 ⎝⎠ c
By using the fact that β = EPψ , and by substituting Q by expression (184) and using (231) one obtains:
152 −ξω(,,rs ) '⎛−⎞ 'rs ⊥ ' βω(,r Ω , ,)te=ϒ (,s ΩΩ , , ωβω )s ,ΩσΩ , , t − d ( ) ∫ Si⎜⎟ s 2 S ⎝⎠c sr− −−ξγω(,rr Ω , ) ⎛⎞γ +−−∫ etd Q ⎜⎟r γωΩΩ,,, γ (234) 0 ⎝⎠ c sr− −−ξγω(,rr Ω , )⎡ '''⎛⎞γ ⎤ +−−−etd κγ(,,),,,()r ΩΩi Ω ωβγr ΩΩ ω σΩ dγ ∫ ⎢⎥∫ β ⎜⎟ 2 c 0 ⎣⎦⎢⎥S ⎝⎠ where all points s ∈M are located over the boundary of the enclosure, i.e. they are surface points. This is the general integral form of the transport and scattering equation for the acoustic angular power flux β inside an enclosure. Note that the general point r can also be located over the boundary M , meaning that equation (234) can be used to calculate the flux β on a specific surface point as a function of the existing flux function β over the entire enclosure’s boundary and of the existing sources inside the enclosure.
Two assumptions are now put forward regarding the properties of the propagation medium. The first assumption consists on adopting a homogeneous medium filled with air. Under this assumption one can adopt the air attenuation coefficient mT(,ω Celsius ,) has defined in (64), which is independent of the position inside the enclosure, and write
σ (,r ωω )= mT ( , , hm )= ( ω ) (235) Celsius
where in m()ω the dependence on the humidity h and the temperature TCelsius is implicit. Then, the total extinction function is simply given by
rr− ' ξω(,rr'' , )==−∫ md ( ωηω ) m ( ) r r (236) 0
The second assumption considers that the homogeneous propagation medium consisting of air exhibits very low in-scattering characteristics meaning that κ β ≈ 0.
Under these two assumptions, the transport equation in integral form for the acoustic angular power flux in air is given by
153 −−m()ω rs ''⎛−⎞rs ⊥ ' βω(,r Ω , ,)te=ϒ (,s ΩΩ , , ωβω )s ,ΩσΩ , , t − d ( ) ∫ Si⎜⎟ s 2 S ⎝⎠c (237) sr− −m()ωγ ⎛⎞γ +−−∫ etdQ ⎜⎟r γωΩΩ,,, γ 0 ⎝⎠ c
As mentioned earlier, one can choose the point r to lie on the boundary, whereby the transport equation can be written entirely in terms of the incident acoustic angular power flux over the boundary together with the sources inside the enclosure. It should be noted that this general formalism allows also the presence of surface sources. Therefore, one can write
' ' ⎛⎞ −−m()ω ss ss− βω(,s''''Ω , ,)te=ϒ− (,s ΩΩ , , ωβω )⎜⎟s ,ΩσΩ , , t − d⊥ ( ) iSi∫ s 2 ⎜⎟c S ⎝⎠ (238) ss− ' −m()ωγ ⎛⎞' γ ++−−∫ etdQ ⎜⎟s γωΩΩ,,, γ 0 ⎝⎠ c for the incident flux, and where ss=+''d (, sΩΩ ) is the boundary intersection point along the M ray (,s' Ω ), and where the boundary distance function d was previously defined in (197). M Explicitly, equation (238) can be written as
' −md()ω (,)s' Ω βω(,s Ω , ,)te=×M i ' '' '⎛⎞ '' 'dM (,s Ω ) ⊥ ' ϒ+Si((,),,,)(,),,,ssddtdMMΩΩ − ΩΩ ωβ⎜⎟ss +ΩΩΩ ω − σs () Ω (239) ∫ S2 ⎝⎠ c d (,)s' Ω M −m()ωγ ⎛⎞' γ ++−−∫ etdQ ⎜⎟s γωΩΩ,,, γ 0 ⎝⎠ c
A similar equation can also be written for the exitant flux βo over the boundary, although the resulting expression is more complex than equation (239).
154 5.3.6 Steady-state Regime Under steady-state conditions, the explicit time dependence no longer appears in the derived equations. In the case of propagation within an enclosure filled with air, the integral form of the transport equation (239) is therefore converted into
' −md()ω (,)s' Ω βω(,s Ω , )=×e M i '' ' '' ' ⊥ ' ϒ+Si((,),,,)(,),,()ssdddMMΩΩ − ΩΩ ωβ()ss + ΩΩΩ ω σs Ω (240) ∫ S2 d (,)s' Ω M ++−∫ ed−m()ωγQ ()s' γωγΩΩ,, 0
5.3.7 Time-dependent Regime The time-dependent regime is governed by equation (239). This equation can however be transformed by using a Laplace Transform, as defined in equation (185), yielding
∞ ' − ptˆ −md()ω (,)s' Ω βω(,s Ω , ,)te=× e M ∫ i 0 ∞ ' '' '⎛⎞ '' 'dM (,s Ω ) −⊥ptˆ ' ϒ+Si((,),,,)ssddtedtdMMΩΩ − ΩΩ ωβ⎜⎟ss + (,),,,ΩΩΩ ω − σs () Ω (241) ∫∫ S2 0 ⎝⎠ c ' d (,)s Ω ∞ M −−mpt()ωγ ⎛⎞' γ ++−−∫∫etedtdQ ⎜⎟s γωΩΩ,,, γ 0 0 ⎝⎠ c
By replacing
d (,s' Ω ) γ tt'''=−M ; tt =− (242) cc respectively in the first and in the second time integrals on the right side of (241), and collecting and rearranging terms, one can write the transformed equation
155 ⎡⎤pˆ ' −+⎢⎥md()ω M (,)s Ω Ls[]βω(,' Ω , ,)peˆ =×⎣⎦c i '' ' '' ' ⊥ ' ϒ+Si((,),,,)ssddpdMMΩΩ − ΩΩ ωβLs[]() + (,),,,() sΩΩΩ ωˆ σs Ω (243) ∫ S2 ' dM (,)s Ω ⎡⎤pˆ −+⎢⎥m()ωγ ++−∫ eQ⎣⎦c Ls[]()' γωγΩΩ,,, pdˆ 0 for the Laplace-transformed incident angular power flux. Equation (243) has the same functional structure as equation (240) that governs the steady-state regime for the acoustic angular power flux, the only difference residing in the modified exponential term.
5.4 Functional Analysis Formalism In this section, the governing equations for the acoustic angular power flux for the steady state and transient regimes are reformulated in terms of the mathematical language of linear operators. A similar analysis has been done for example in the field of light transport [Arvo, 1993]. The introduced linear operators are studied in detail in what their main properties are concerned, by using standard techniques of functional analysis. One begins by identifying appropriate function spaces for the acoustic angular power flux functions and then one can show that under these function spaces the operators exhibit determinate properties, which are of the most important nature to yield existing and valid solutions of the governing equations. For simplicity of notation, the explicit dependency of the quantities in the frequency parameter will be omitted throughout this section and in the following sub-sections.
Consider that the surfaces of the enclosure under study are adequately described mathematically by M , as already defined in section 5.2.1. The volume enclosed by M is denoted by V ⊂ R3 and the complete enclosure is therefore denoted by EMV= ∪ . One introduces the linear space F of real- valued acoustic angular power flux functions. These functions are defined on the set M ×S2 , meaning that they are defined over all surface points of the enclosure and over all the directions of the unit sphere S2 . A generic acoustic angular power flux function is denoted by β ∈ F .
Metric properties shall be imposed on F , thus making it a normed linear space, denoted by the ordered pair ()F, i , where i is a real-valued norm defined on the space F . In addition, one
156 imposes that the space F is complete, meaning that every Cauchy sequence in F is convergent. In this case, F is a Banach space [Kolmogorov and Fomin, 1999].
An infinite number of norms can be defined on Banach spaces, each possessing a different topology on the space as well as a different notion of size, distance, and convergence. Since the governing equations (240) and (243) require only that angular power flux functions (and its Laplace transformed functions) be integrable in order to be well defined, it is natural to consider the so- called Lp -norms and their corresponding function spaces [Boccara, 1990]. Therefore, normed linear spaces of integrable functions will be considered in the following. The appropriate definition
2 for the Lp -norm of a flux function β ∈ F defined on M ×S is given by
1 p ⎡⎤p ββ≡ s,()()ΩσΩσdd⊥ ' s (244) p ⎢⎥∫∫ () s 2 ⎣⎦⎢⎥M S where p ∈∞[1, ].
2 2 The function space Lp ()M ×S is defined as the set of measurable functions over M ×S with finite Lp -norms. Three of the Lp -norms are of particular interest. The L1 -norm and the L∞ -norm have immediate physical interpretations, while the L2 -norm has algebraic properties making it appropriate for introducing the structure of Hilbert Spaces, which are required to define projection- based methods [Kolmogorov and Fomin, 1999]. The limiting case of the L∞ -norm is more rigorously defined by [Boccara, 1999]
ββ≡ esssup ess sups ,Ω (245) ∞ 2 ( ) s∈M Ω∈S where “ esssup ” is the essential supremum, meaning the least upper bound attainable by ignoring a subset of the domain with measure zero. Thus, the L∞ -norm ignores isolated maximal points, for example. As the maximum acoustic angular power flux attained (or approached) over all surface β points and in all directions, ∞ has the same dimensions of the power flux, meaning
157 ⎡⎤Wm−−21 sr . In contrast, β is the total power of the power flux function β , and consequently ⎣⎦ 1 has the dimensions of power [W ].
Usual transformations on functions are most obviously characterised by linear operators. If U and W are two normed linear spaces, a linear mapping A of U into W is called a linear operator AU: → W. In connection with the above mentioned function norms, one should recall the concept of operator norms. A linear operator is bounded if
AAxx≤∀: x∈D (246) A
where DUA ⊂ is the domain of A and where the positive real-valued constant A denotes the norm of the operator A , given by
Ax AA==supx sup (247) xx≤∈1 D x A
Operator norms obey the same relations as function norms with the supplementary property
AB≤ A B (248) which makes operator norms compatible with the multiplicative structure of operators. The set of bounded operators whose domain is the normed linear space F is also a Banach space, more precisely it is a Banach algebra [Boccara, 1999]. The identity operator I , whose norm is equal to 1, is the unit element of this algebra.
5.4.1 Linear Operator Formalism of the Governing Equations Integral equations such as equations (240) and (243) can be expressed more abstractly as operator equations. Operator equations tend to be more concise than their integral counterparts are, while capturing essential algebraic properties such as linearity and associativity of composition, as well as topological properties such as boundedness and compactness. The abstraction afforded by the linear operators’ formalism is appropriate when the emphasis is on integrals as transformations rather than on the numerical aspects of integration. Operator notation was first introduced in
158 energy-based room acoustics of lambertian enclosures by Carrol and Miles [1978], although only in a very brief manner.
Equations (240) and (243) are essentially Fredholm equations of the second kind [Kolmogorov and Fomin, 1999; Smirnov, 1964] although they do not precisely conform to the standard definition because of the implicit relationship ss=+''d (, sΩΩ ) , which depends on the argument s' and the M dummy variable Ω . This difference is responsible for the important distinguishing feature of energy transport problems inside enclosures, namely the non-localness of the intervening interactions.
In this section, we recast the governing equations (240) and (243) in the language of linear operators, thus obtaining the basic operator equations intervening in the propagation of acoustic energy inside enclosures. Let one define a set of operators, which allow the rewriting of the governing equations.
• The Scattering Operator S
The local scattering operator, denoted by S , is introduced as an integral operator whose kernel accounts for the scattering of incident acoustic angular power flux at real walls. This operator is easily defined by specifying the action on an arbitrary incident flux function βi :
''' ⊥ ()(,)SdββiSis Ω ≡ϒ (,,)(,)()s ΩΩ s ΩσΩs (249) ∫ S2
When this scattering operator is applied to an incident flux function it returns the exitant flux
βoi= Sβ , that results from a single scattering operation.
• The Propagation Operator P
To define the propagation operator, the ray-casting function ss(,Ω ) is introduced. This is a M function with domain M ×S2 and range M that gives the first point of M that is visible from s in the direction Ω . This function is readily determined if one uses the already defined boundary
159 distance function d (,s Ω )[see section 5.3.3], which gives the distance from s to the nearest M intersection point on the boundary along the direction Ω . Therefore, the ray-casting function is defined by:
ss(,Ω )≡ ss+ d (,ΩΩ ) (250) MM
The propagation of acoustic sound energy inside the enclosure is then represented by the propagation operator P , defined by
−−md()ωω (,)s Ω md () (,)s Ω (Pedeββ )(,s Ω )≡− (ss (,ΩΩ ), )M =+− β (ss (,ΩΩ ) , Ω ) M (251) oo MM o
Therefore, the propagation operator P expresses the incident angular power flux βio= Pβ at each point of M in terms of the exitant power fluxes of surrounding visible surfaces.
• The Transport Operator T
The composition of the scattering operator S with the propagation operator P yields the transport operator,
TPS= (252)
This operator maps an incident angular power flux function βi into the incident function Tβi that results after a single scattering and propagation step.
• The Laplace-conjugated Scattering Operator SL
An analogous operator can be defined for the surface scattering of the Laplace-transformed angular power flux L[βi ] through the relation
''' ⊥ (SpLLs[ββiSi] )(,Ω ,ˆˆ )≡ϒ (,s ΩΩ , )Ls[ ] (,ΩσΩ , pd )s ( ) (253) ∫ S2
160 Operator SL applies a single scattering operation at the walls of the enclosure to the Laplace- transformed L[βi ] thus returning the Laplace-transformed exitant power flux LL[βoi] = SL [β ] .
• The Laplace-conjugated Propagation Operator PL
This operator is defined in a similar manner as the propagation operator P through
⎡⎤pˆ −+⎢⎥md()ω M (,)s Ω ()(,,)((,),,)PpLs[]ββΩ ˆˆ≡−Lss [] ΩΩ pe⎣⎦c (254) L oo M
Therefore, the Laplace-conjugated propagation operator PL expresses the incident Laplace- transformed angular power flux LL[βio] = PL [β ] at each point of M in terms of the exitant Laplace-transformed power fluxes of surrounding visible surfaces.
• The Laplace-conjugated Transport Operator TL
The composition of the Laplace-conjugated scattering operator SL with the Laplace-conjugated propagation operator PL yields the definition of the Laplace-conjugated transport operator
TPSLLL= (255)
This operator maps an incident Laplace-transformed angular power flux function L[βi ] into the incident function TLL[βi ] that results after a single scattering and propagation step in the Laplace domain.
5.4.1.1 Steady-state Regime With the above-introduced linear operators, the governing equation for the steady-state propagation of sound energy inside enclosures can be written in a compact form as
βββββ(,s Ω )=+=+ (PS )(,s Ω )Source (,s Ω ) ( T )(,s Ω ) Source (,s Ω ) (256) iiiii where
161 d (,s Ω ) M β Source(,s Ω )≡+−ed− m()ωγ s γωγΩΩ , , i ∫ Q () (257) 0 equals the direct component of the incident flux on s from direction Ω due to the sound sources inside the enclosure. Equation (256) can be finally written as an operator equation through
Source βii=+ββT i (258) which has the structure of a Fredholm equation of the second kind [Smirnov, 1964].
5.4.1.2 Time-dependent Regime The time-dependent regime is governed by equation (243). This equation can be rewritten with the aid of the Laplace-conjugated operators above as
ˆˆˆ⎡⎤Source Ls[βββiii](,Ω ,p )=+ (PSLLLs[ ] )(,Ω , p )Ls⎣⎦ (,Ω , p ) (259) =+(TpLs[]ββ )(,,)Ω ˆˆLs⎡⎤Source (,,)Ω p L ii ⎣⎦ or in a more compact form as an operator equation
⎡⎤Source LL[βii] =+⎣⎦ββTL L[ i] (260) which again has the structure of a Fredholm equation of the second kind.
5.4.2 Properties of the Operators Let one start by establishing the property that the directional-spherical surface scattering function R (,s Ω )is a dimensionless quantity less than one, as already mentioned in section 5.3.4. This Ω i property is determined from the principle of energy conservation.
Theorem 4: If ϒ (,s ΩΩ , )is the Wall Scattering Function (WSF) for a physically valid passive surface, Soi which is part of the boundary M of an enclosure, then
ϒ≡<∀∀(,s ΩΩ , )dR σΩ⊥ ( ) (,s Ω ) 1 Soiss oΩ i Ω ∈S2 ∈M (261) ∫ i S2
162 Proof: The relationships
⊥ Ed()ss≡ βii (,ΩσΩ )s ( i ); ∫ S2 ⊥ ββoo(,s Ω )≡ϒ Soiii (,s ΩΩ , ) (,s ΩσΩ )d s ( i ); (262) ∫ S2 ⊥ Md()ss≡ βoo (,ΩσΩ )s ( o ); ∫ S2 are introduced, where E and M denote respectively the incident power and the scattered power per unit area at surface point s . By conservation of energy, one requires that M < E for all possible incident angular flux functions βi , meaning that every physically valid surface always scatters less sound energy than it is incident on it. By fixing a particular direction Ω and i considering the incident flux β (,s Ω )= b ()s δ (ΩΩ− ), meaning that one lets the incident power ii be concentrated in a single direction Ω , then we have that i
⊥ Eb()ss=− ()δ (ΩΩii ) d σΩs ( ) = b ();s ∫ S2 ⊥ βδoo(,)s Ω =ϒ So (,,)()(s ΩΩbdbs ΩΩ − i ) σΩs ()()(,,); =ss ϒ SoiΩΩ (263) ∫ S2 ⊥ Mb()ss=ϒ ()Soi (, sΩΩ , ) d σΩs ( o ) ∫ S2 and from M < E the required result is readily obtained.
5.4.2.1 Norms of the Operators In this section, conditions on the norms of the previously defined operators are proved in order to ensure that the solutions of the steady state and transient problems of sound energy inside enclosures are well defined.
Theorem 5: The norm of the Scattering Operator S is less than one for every enclosure possessing physically valid passive walls.
Proof: By introducing the definition of the scattering operator (249) into the 1-norm expression, we have:
163 ''' ⊥⊥ Sdddββ=ϒS (,s ΩΩ , ) (,s ΩσΩ )ss ( ) σΩσ ( ) ()s ∫∫ ∫ M SS22 ''' ⊥⊥ ≤ϒS (,s ΩΩ , )β (,s ΩσΩσΩσ )dddss ( ) ( ) ()s (264) ∫∫∫ M SS22 ⎡⎤ =ϒ(,s ΩΩ ,''' )ddd σ⊥⊥ ( Ω )β (,s ΩσΩσ ) ( ) ()s ∫∫⎢⎥ ∫ S ss 22 M SS⎣⎦⎢⎥ where Fubini’s Theorem [Boccara, 1999] was used in order to change the order of integration in the last passage above. Define the constant
' ⊥ messess=ϒsup supS (s ,ΩΩ , ) d σs ( Ω ) (265) '2 ∫ s∈M Ω ∈S 2 S which bounds the effect of the bracketed expression in (264). According to Theorem 4, m <1 for every physically valid surface. Therefore, we have
'' ⊥ Smβ ≤=ββ(,s ΩσΩσ ) ds ( ) d ()s m (266) ∫∫ M S2 which shows that the norm of operator S is given by Sm= , and thus S <1.
Theorem 6: The ∞-norm of the Scattering Operator S is less than one for every enclosure possessing physically valid passive walls, with wall scattering functions (WSF) ϒS obeying the principle of detailed balance.
Proof: From the definition of the ∞-norm:
'' ⊥ ' Sessessββ=ϒsup supS (s ,ΩΩ , ) (s ,ΩσΩ ) ds ( ) ∞ 2 ∫ s∈M Ω∈S 2 S '' ⊥ ' ≤ϒesssup ess supS (,s ΩΩ , )β (,s ΩσΩ ) d s ( ) (267) 2 ∫ s∈M Ω∈S 2 S
⎡⎤'' ⊥ ≤ϒ⎢⎥esssup ess supS (s ,ΩΩ , ) d σs ( Ω ) β 2 ∫ ∞ s∈M Ω∈S 2 ⎣⎦⎢⎥ S
Sβ and thus the bracketed expression establishes the bound on ∞ . Therefore,
164 '' ⊥ Sessess=ϒsup supS (s ,ΩΩ , ) d σs ( Ω ) (268) ∞ 2 ∫ s∈M Ω∈S 2 S
If the WSF obeys the principle of detailed balance [Joyce, 1975], meaning that ϒ=ϒ(,s ΩΩ ,'' ) (,s ΩΩ , ), then SS
'' ⊥ Sessess=ϒsup supS (s ,ΩΩ , ) d σs ( Ω ) = m (269) ∞ 2 ∫ s∈M Ω∈S 2 S with m as already defined in Theorem 5. Thus m <1 for any physically valid passive surface and the theorem is proved.
Theorem 7: The p-norm of the Scattering Operator S is less than one for every enclosure possessing physically valid passive walls, with wall scattering functions(WSF) ϒS obeying the principle of detailed balance.
Proof: According to a well-established theorem stating that
fff≤ rp1− rp pr∞ (270) for 1<