Propagation of sound in porous media : modelling sound absorbing materials

Propagation of Sound in Porous Media......Page 3
Contents......Page 7
Preface to the second edition......Page 15
1.2 Notation – vector operators......Page 17
1.3 Strain in a deformable medium......Page 18
1.4 Stress in a deformable medium......Page 20
1.5 Stress–strain relations for an isotropic elastic medium......Page 21
1.6 Equations of motion......Page 24
1.7 Wave equation in a fluid......Page 26
1.8 Wave equations in an elastic solid......Page 27
References......Page 29
2.2.1 Travelling waves......Page 31
2.2.3 Attenuation......Page 32
2.3.1 Impedance variation along a direction of propagation......Page 33
2.3.2 Impedance at normal incidence of a layer of fluid backed by an impervious rigid wall......Page 34
2.4.1 Reflection coefficient......Page 35
2.5.1 Porosity and flow resistivity in porous materials......Page 36
2.5.3 The Laws of Delany and Bazley and flow resistivity......Page 38
2.6 Examples......Page 39
References......Page 42
3.2 Inhomogeneous plane waves in isotropic fluids......Page 45
3.3 Reflection and refraction at oblique incidence......Page 47
3.4.1 Impedance variation along a direction perpendicular to an impedance plane......Page 49
3.4.2 Impedance at oblique incidence for a layer of finite thickness backed by an impervious rigid wall......Page 50
3.4.3 Impedance at oblique incidence in a multilayered fluid......Page 51
3.5 Reflection coefficient and absorption coefficient at oblique incidence......Page 52
3.6 Examples......Page 53
3.7 Plane waves in fluids equivalent to transversely isotropic porous media......Page 55
3.8 Impedance at oblique incidence at the surface of a fluid equivalent to an anisotropic porous material......Page 57
References......Page 59
4.2 Viscosity effects......Page 61
4.3 Thermal effects......Page 66
4.4 Effective density and bulk modulus for cylindrical tubes having triangular, rectangular and hexagonal cross-sections......Page 70
4.5 High- and low-frequency approximation......Page 71
4.6.1 Effective density and bulk modulus in cylindrical pores having a circular cross-section......Page 73
4.6.2 Effective density and bulk modulus in slits......Page 75
4.6.3 High- and low-frequency limits of the effective density and the bulk modulus for pores of arbitrary cross-sectional shape......Page 76
4.7.1 Similarity between Gc and Gs......Page 77
4.7.2 Bulk modulus of the air in slits......Page 78
4.7.3 Effective density and bulk modulus of air in cylindrical pores of arbitrary cross-sectional shape......Page 80
4.8.1 Normal incidence......Page 81
4.9 Tortuosity and flow resistivity in a simple anisotropic material......Page 83
4.10.1 Effective density......Page 85
Effective density and bulk modulus......Page 87
References......Page 88
5.1 Introduction......Page 89
5.2.1 Definitions......Page 90
5.2.2 Direct measurement of the static permeabilities......Page 92
5.3.1 Classical tortuosity......Page 94
5.3.2 Viscous characteristic length......Page 95
5.3.4 Characteristic lengths for fibrous materials......Page 96
5.3.5 Direct measurement of the high-frequency parameters, classical tortuosity and characteristic lengths......Page 97
5.3.6 Static tortuosity......Page 98
5.4.2 Simplified Lafarge model for the bulk modulus......Page 99
5.5.2 The Champoux–Allard model......Page 100
5.5.5 Prediction of the bulk modulus with the simplified Lafarge model and the Champoux-Allard model......Page 101
5.5.6 Prediction of the surface impedance......Page 103
5.6.2 Practical considerations......Page 104
5.7 Fluid layer equivalent to a porous layer......Page 105
5.8 Summary of the semi-phenomenological models......Page 106
5.9 Homogenization......Page 107
5.10.1 Definitions......Page 111
5.10.2 Orders of magnitude for realistic double porosity media......Page 112
5.10.3 Asymptotic development method for double porosity media......Page 113
5.10.4 Low permeability contrast......Page 114
5.10.5 High permeability contrast......Page 115
5.10.6 Practical considerations......Page 118
Appendix 5.A: Simplified calculation of the tortuosity for a porous material having pores made up of an alternating sequence of cylinders......Page 119
Appendix 5.B: Calculation of the characteristic length......Page 120
Appendix 5.C: Calculation of the characteristic length   for a cylinder perpendicular to the direction of propagation......Page 122
References......Page 123
6.2.1 Stress......Page 127
6.2.2 Stress–strain relations in the Biot theory: The potential coupling term......Page 128
6.2.3 A simple example......Page 131
6.2.4 Determination of P, Q and R......Page 132
6.3 Inertial forces in the Biot theory......Page 133
6.4 Wave equations......Page 135
6.5.1 The two compressional waves......Page 136
6.5.2 The shear wave......Page 138
6.5.4 Example......Page 139
6.6.2 Prediction of the surface impedance at normal incidence......Page 142
6.6.3 Example: Fibrous material......Page 145
Appendix 6.A: Other representations of the Biot theory......Page 147
References......Page 150
7.2 Sommerfeld representation of the monopole field over a plane reflecting surface......Page 153
7.3 The complex sinθ plane......Page 155
7.4 The method of steepest descent (passage path method)......Page 156
7.5.1 Definitions......Page 161
7.5.2 Planes waves associated with the poles......Page 162
7.5.3 Contribution of a pole to the reflected monopole pressure field......Page 166
7.6 The pole subtraction method......Page 167
7.7.1 Localization from the r dependence of the reflected field......Page 169
7.7.2 Localization from the vertical dependence of the total pressure......Page 171
7.8 The modified version of the Chien and Soroka model......Page 172
Appendix 7.A Evaluation of N......Page 176
Appendix 7.B Evaluation of pr by the pole subtraction method......Page 177
Appendix 7.C From the pole subtraction to the passage path: locally reacting surface......Page 180
References......Page 181
8.1 Introduction......Page 183
8.2.1 Excitation with a given wave number component parallel to the faces......Page 184
8.2.2 Circular and line sources......Page 188
8.3 Semi-infinite layer – Rayleigh wave......Page 189
8.4 Layer of finite thickness – modified Rayleigh wave......Page 192
8.5.1 Modes and resonances for an elastic solid layer and a poroelastic layer......Page 193
8.5.2 Excitation of the resonances by a point source in air......Page 195
Appendix 8.A Coefficients rij and Mi,j......Page 198
Appendix 8.B Double Fourier transform and Hankel transform......Page 199
References......Page 201
9.2.1 Inertial effect......Page 203
9.2.2 Calculation of the added mass and the added length......Page 204
9.2.3 Flow resistance......Page 207
9.2.4 Apertures having a square cross-section......Page 208
9.3.1 Evaluation of the impedance for the case of circular holes......Page 210
9.3.2 Evaluation at normal incidence of the impedance for the case of square holes......Page 214
9.3.3 Examples......Page 215
9.3.4 Design of stratified porous materials covered by perforated facings......Page 218
9.3.5 Helmholtz resonators......Page 219
9.4.1 Evaluation of the impedance in a hole at the boundary surface between the facing and the material......Page 221
9.4.2 Evaluation of the external added length at oblique incidence......Page 224
9.4.3 Evaluation of the impedance of a faced porous layer at oblique incidence......Page 225
9.4.4 Evaluation of the surface impedance at oblique incidence for the case of square perforations......Page 226
References......Page 227
10.1 Introduction......Page 229
10.2 Frame in vacuum......Page 230
10.3.1 Stress–strain equations......Page 231
10.3.2 Wave equations......Page 232
10.4.1 General equations......Page 233
10.4.4 Nature of the different waves......Page 235
10.4.5 Illustration......Page 236
10.5.1 Description of the problems......Page 238
10.5.2 Plane field in air......Page 239
10.5.3 Decoupling of the air wave......Page 242
10.6 Mechanical excitation at the surface of the porous layer......Page 243
10.7.1 Prediction of the slowness vector components of the different waves......Page 244
10.7.3 Description of the different waves......Page 246
10.8 Rayleigh poles and Rayleigh waves......Page 248
10.8.1 Example......Page 250
10.9 Transfer matrix representation of transversally isotropic poroelastic media......Page 252
Appendix 10.A: Coefficients Ti in Equation (10.46)......Page 254
Appendix 10.B: Coefficients Ai in Equation (10.97)......Page 255
References......Page 256
11.1 Introduction......Page 259
11.3.1 Fluid layer......Page 260
11.3.2 Solid layer......Page 261
11.3.3 Poroelastic layer......Page 263
11.3.4 Rigid and limp frame limits......Page 267
11.3.5 Thin elastic plate......Page 270
11.3.6 Impervious screens......Page 271
11.3.8 Other media......Page 272
11.4.1 Two layers of the same nature......Page 273
11.4.2 Interface between layers of different nature......Page 274
11.5 Assembling the global transfer matrix......Page 276
11.5.2 Semi-infinite fluid termination condition......Page 277
11.6.2 Transmission coefficient and transmission loss......Page 279
11.6.3 Piston excitation......Page 281
11.7.1 Materials with porous screens......Page 282
11.7.2 Materials with impervious screens......Page 287
11.7.3 Normal incidence sound transmission through a plate–porous system......Page 290
11.7.4 Diffuse field transmission of a plate–foam system......Page 291
Appendix 11.A The elements Tij of the Transfer Matrix [T]......Page 293
References......Page 296
12.2.1 Transmitted power......Page 299
12.2.2 Transmission coefficient......Page 303
12.3.1 Surface pressure......Page 304
12.3.2 Absorption coefficient......Page 305
12.3.3 Examples......Page 307
12.4.1 Formulation......Page 311
12.4.2 The TMM, SEA and modal methods......Page 313
12.4.3 Examples......Page 314
12.5 Point source excitation......Page 319
12.6 Other applications......Page 320
Appendix 12.A: An algorithm to evaluate the geometrical radiation impedance......Page 321
References......Page 322
13.1 Introduction......Page 325
13.2 Displacement based formulations......Page 326
13.3 The mixed displacement–pressure formulation......Page 327
13.4.1 Poroelastic–elastic coupling condition......Page 329
13.4.2 Poroelastic–acoustic coupling condition......Page 330
13.4.4 Poroelastic–impervious screen coupling condition......Page 331
13.4.5 Case of an imposed pressure field......Page 332
13.4.7 Coupling with a semi-infinite waveguide......Page 333
13.6 Numerical implementation......Page 336
13.7 Dissipated power within a porous medium......Page 339
13.8 Radiation conditions......Page 340
13.9.1 Normal incidence absorption and transmission loss of a foam: finite size effects......Page 343
13.9.2 Radiation effects of a plate–foam system......Page 345
13.9.3 Damping effects of a plate–foam system......Page 347
13.9.4 Diffuse transmission loss of a plate–foam system......Page 349
13.9.5 Application to the modelling of double porosity materials......Page 351
13.9.6 Modelling of smart foams......Page 355
13.9.7 An industrial application......Page 359
References......Page 363
Index......Page 367