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[资源] An_Introduction_to_Acoustics

Contents
Preface
1 Some fluid dynamics 3
1.1 Conservation laws and constitutive equations . . . . . . . . . . . 3
1.2 Approximations and alternative forms of the conservation laws for
ideal fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Wave equation, speed of sound, and acoustic energy 13
2.1 Order of magnitude estimates . . . . . . . . . . . . . . . . . . . 13
2.2 Wave equation for a uniform stagnant fluid and compactness . . . 18
2.2.1 Linearization and wave equation . . . . . . . . . . . . . . 18
2.2.2 Simple solutions . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 Bubbly liquid at low frequencies . . . . . . . . . . . . . . 24
2.4 Influence of temperature gradient . . . . . . . . . . . . . . . . . 26
2.5 Influence of mean flow . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Sources of sound . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6.1 Inverse problem and uniqueness of sources . . . . . . . . . 29
2.6.2 Mass and momentum injection . . . . . . . . . . . . . . . 29
2.6.3 Lighthill’s analogy . . . . . . . . . . . . . . . . . . . . . 31
2.6.4 Vortex sound . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7 Acoustic energy . . . . . . . . . . . . . . . . . . . . . . . . . . 36
ii Contents
2.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7.2 Kirchhoff’s equation for quiescent fluids . . . . . . . . . . 38
2.7.3 Acoustic energy in a non-uniform flow . . . . . . . . . . . 41
2.7.4 Acoustic energy and vortex sound . . . . . . . . . . . . . . 42
3 Green’s functions, impedance, and evanescent waves 46
3.1 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1 Integral representations . . . . . . . . . . . . . . . . . . . 46
3.1.2 Remarks on finding Green’s functions . . . . . . . . . . . . 49
3.2 Acoustic impedance . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Impedance and acoustic energy . . . . . . . . . . . . . . . 52
3.2.2 Impedance and reflection coefficient . . . . . . . . . . . . 52
3.2.3 Impedance and causality . . . . . . . . . . . . . . . . . . 53
3.2.4 Impedance and surface waves . . . . . . . . . . . . . . . . 56
3.2.5 Acoustic boundary condition in the presence of mean flow . 57
3.2.6 Surface waves along an impedance wall with mean flow . . 59
3.3 Evanescent waves and related behaviour . . . . . . . . . . . . . . 62
3.3.1 An important complex square root . . . . . . . . . . . . . 62
3.3.2 The Walkman . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.3 Ill-posed inverse problem . . . . . . . . . . . . . . . . . . 64
3.3.4 Typical plate pitch . . . . . . . . . . . . . . . . . . . . . . 65
3.3.5 Snell’s law . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.6 Silent vorticity . . . . . . . . . . . . . . . . . . . . . . . 66
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Contents iii
4 One dimensional acoustics 70
4.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Basic equations and method of characteristics . . . . . . . . . . . 72
4.2.1 The wave equation . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.3 Linear behaviour . . . . . . . . . . . . . . . . . . . . . . 74
4.2.4 Non-linear simple waves and shock waves . . . . . . . . . 79
4.3 Source terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Reflection at discontinuities . . . . . . . . . . . . . . . . . . . . 86
4.4.1 Jump in characteristic impedance c . . . . . . . . . . . . 86
4.4.2 Monotonic change in pipe cross section . . . . . . . . . . 88
4.4.3 Orifice and high amplitude behaviour . . . . . . . . . . . . 88
4.4.4 Multiple junction . . . . . . . . . . . . . . . . . . . . . . 93
4.4.5 Reflection at a small air bubble in a pipe . . . . . . . . . . 94
4.5 Attenuation of an acoustic wave by thermal and viscous dissipation 98
4.5.1 Reflection of a plane wave at a rigid wall . . . . . . . . . . 98
4.5.2 Viscous laminar boundary layer . . . . . . . . . . . . . . 102
4.5.3 Damping in ducts with isothermal walls. . . . . . . . . . . 103
4.6 One dimensional Green’s function . . . . . . . . . . . . . . . . . 106
4.6.1 Infinite uniform tube . . . . . . . . . . . . . . . . . . . . 106
4.6.2 Finite uniform tube . . . . . . . . . . . . . . . . . . . . . 107
4.7 Aero-acoustical applications . . . . . . . . . . . . . . . . . . . . 108
4.7.1 Sound produced by turbulence . . . . . . . . . . . . . . . 108
4.7.2 An isolated bubble in a turbulent pipe flow . . . . . . . . . 111
4.7.3 Reflection of a wave at a temperature inhomogeneity . . . . 113
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iv Contents
5 Resonators and self-sustained oscillations 120
5.1 Self-sustained oscillations, shear layers and jets . . . . . . . . . . 120
5.2 Some resonators . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2.2 Resonance in duct segment . . . . . . . . . . . . . . . . . 129
5.2.3 The Helmholtz resonator (quiescent fluid) . . . . . . . . . 136
5.2.4 Non-linear losses in a Helmholtz resonator . . . . . . . . . 138
5.2.5 The Helmholtz resonator in the presence of a mean flow . . 139
5.3 Green’s function of a finite duct . . . . . . . . . . . . . . . . . . 141
5.4 Self-sustained oscillations of a clarinet . . . . . . . . . . . . . . . 143
5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4.2 Linear stability analysis . . . . . . . . . . . . . . . . . . . 144
5.4.3 Rayleigh’s Criterion . . . . . . . . . . . . . . . . . . . . . 146
5.4.4 Time domain simulation . . . . . . . . . . . . . . . . . . 146
5.5 Some thermo-acoustics . . . . . . . . . . . . . . . . . . . . . . . 148
5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 148
5.5.2 Modulated heat transfer by acoustic flow and Rijke tube . . 150
5.6 Flow induced oscillations of a Helmholtz resonator . . . . . . . . 156
6 Spherical waves 166
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.2 Pulsating and translating sphere . . . . . . . . . . . . . . . . . . 166
6.3 Multipole expansion and far field approximation . . . . . . . . . . 173
6.4 Method of images and influence of walls on radiation . . . . . . . 178
6.5 Lighthill’s theory of jet noise . . . . . . . . . . . . . . . . . . . . 181
6.6 Sound radiation by compact bodies in free space . . . . . . . . . . 185
6.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 185
6.6.2 Tailored Green’s function . . . . . . . . . . . . . . . . . . 186
6.6.3 Curle’s method . . . . . . . . . . . . . . . . . . . . . . . 187
6.7 Sound radiation from an open pipe termination . . . . . . . . . . 191
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Contents v
7 Duct acoustics 199
7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . 199
7.2 Cylindrical ducts . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.3 Rectangular ducts . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.4 Impedance wall . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.4.1 Behaviour of complex modes . . . . . . . . . . . . . . . . 207
7.4.2 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.5 Annular hard-walled duct modes in uniform mean flow . . . . . . . 212
7.6 Behaviour of soft-wall modes and mean flow . . . . . . . . . . . . 216
7.7 Source expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.7.1 Modal amplitudes . . . . . . . . . . . . . . . . . . . . . . 217
7.7.2 Rotating fan . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.7.3 Tyler and Sofrin rule for rotor-stator interaction . . . . . . . 220
7.7.4 Point source in a lined flow duct . . . . . . . . . . . . . . . 222
7.7.5 Point source in a duct wall . . . . . . . . . . . . . . . . . 225
7.7.6 Vibrating duct wall . . . . . . . . . . . . . . . . . . . . . 227
7.8 Reflection and transmission at a discontinuity in diameter . . . . . 228
7.8.1 The iris problem . . . . . . . . . . . . . . . . . . . . . . . 231
7.9 Reflection at an unflanged open end . . . . . . . . . . . . . . . . 231
8 Approximation methods 236
8.1 Regular Perturbations . . . . . . . . . . . . . . . . . . . . . . . 237
8.1.1 Webster’s horn equation . . . . . . . . . . . . . . . . . . 237
8.2 Multiple scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8.3 Helmholtz resonator with non-linear dissipation . . . . . . . . . . 244
8.4 Slowly varying ducts . . . . . . . . . . . . . . . . . . . . . . . . 247
8.5 Reflection at an isolated turning point . . . . . . . . . . . . . . . 250
8.6 Ray acoustics in temperature gradient . . . . . . . . . . . . . . . 254
8.7 Refraction in shear flow . . . . . . . . . . . . . . . . . . . . . . 258
8.8 Matched asymptotic expansions . . . . . . . . . . . . . . . . . . 260
8.9 Duct junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
8.10 Co-rotating line-vortices . . . . . . . . . . . . . . . . . . . . . . 272
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vi Contents
9 Effects of flow and motion 278
9.1 Uniform mean flow, plane waves and edge diffraction . . . . . . . 278
9.1.1 Lorentz or Prandtl-Glauert transformation . . . . . . . . . 278
9.1.2 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.1.3 Half-plane diffraction problem . . . . . . . . . . . . . . . 280
9.2 Moving point source and Doppler shift . . . . . . . . . . . . . . . 281
9.3 Rotating monopole and dipole with moving observer . . . . . . . 284
9.4 Ffowcs Williams & Hawkings equation for moving bodies . . . . . 288
Appendix 293
A Integral laws and related results 293
A.1 Reynolds’ transport theorem . . . . . . . . . . . . . . . . . . . . 293
A.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . 293
A.3 Normal vectors of level surfaces . . . . . . . . . . . . . . . . . . 295
B Order of magnitudes: O and o. 296
C Fourier transforms and generalized functions 297
C.1 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . 297
C.1.1 Causality condition . . . . . . . . . . . . . . . . . . . . . 301
C.1.2 Phase and group velocity . . . . . . . . . . . . . . . . . . 305
C.2 Generalized functions . . . . . . . . . . . . . . . . . . . . . . . 306
C.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 306
C.2.2 Formal definition . . . . . . . . . . . . . . . . . . . . . . 306
C.2.3 The delta function and other examples . . . . . . . . . . . 307
C.2.4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 309
C.2.5 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . 310
C.2.6 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
C.2.7 Higher dimensions and Green’s functions . . . . . . . . . . 311
C.2.8 Surface distributions . . . . . . . . . . . . . . . . . . . . . 312
C.3 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
C.3.1 The Fast Fourier Transform . . . . . . . . . . . . . . . . . 318
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Contents vii
D Bessel functions 320
E Free field Green’s functions 328
F Summary of equations for fluid motion 329
F.1 Conservation laws and constitutive equations . . . . . . . . . . . . 329
F.2 Acoustic approximation . . . . . . . . . . . . . . . . . . . . . . . 331
F.2.1 Inviscid and isentropic . . . . . . . . . . . . . . . . . . . 331
F.2.2 Perturbations of a mean flow . . . . . . . . . . . . . . . . 332
F.2.3 Myers’ Energy Corollary . . . . . . . . . . . . . . . . . . 333
F.2.4 Zero mean flow . . . . . . . . . . . . . . . . . . . . . . . 334
F.2.5 Time harmonic . . . . . . . . . . . . . . . . . . . . . . . 334
F.2.6 Irrotational isentropic flow . . . . . . . . . . . . . . . . . 334
F.2.7 Uniform mean flow . . . . . . . . . . . . . . . . . . . . . 335
G Answers to exercises. 337
Bibliography 349
Index 361
RienstraHirschberg

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