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An_Introduction_to_Acoustics
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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 RienstraHirschberg 11th March 2010 12:00 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 RienstraHirschberg 11th March 2010 12:00 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 RienstraHirschberg 11th March 2010 12:00 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 RienstraHirschberg 11th March 2010 12:00 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 RienstraHirschberg 11th March 2010 12:00 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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