An Introduction to Acoustics S.W. Rienstra & A. Hirschberg Eindhoven University of Technology 2 Oct 2021 This is an extended and revised edition of IWDE 92-06. Comments and corrections are gratefully accepted. This file may be used and printed, but for personal or educational purposes only. © S.W. Rienstra & A. Hirschberg 2004. Contents Page Preface 1 Some fluid dynamics 1 1.1 Conservation laws and constitutive equations ..................... 1 1.2 Approximations and alternative forms of the conservation laws for ideal fluids ..... 4 2 Wave equation, speed of sound, and acoustic energy 9 2.1 Order of magnitude estimates ............................. 9 2.2 Wave equation for a uniform stagnant fluid and compactness ............. 13 2.2.1 Linearization and wave equation ........................ 13 2.2.2 Simple solutions ................................. 14 2.2.3 Compactness .................................. 16 2.3 Speed of sound ..................................... 17 2.3.1 Ideal gas ..................................... 17 2.3.2 Water ...................................... 19 2.3.3 Bubbly liquid at low frequencies ........................ 19 2.4 Influence of temperature gradient ........................... 20 2.5 Influence of mean flow ................................. 22 2.6 Sources of sound .................................... 22 2.6.1 Inverse problem and uniqueness of sources ................... 22 2.6.2 Mass and momentum injection ......................... 23 2.6.3 Lighthill’s analogy ............................... 24 2.6.4 Vortex sound .................................. 27 2.7 Acoustic energy .................................... 29 2.7.1 Introduction ................................... 29 2.7.2 Kirchhoff’s equation for quiescent fluids .................... 30 2.7.3 Acoustic energy in a non-uniform flow ..................... 33 2.7.4 Acoustic energy and vortex sound ........................ 35 ii Contents 3 Green’s functions, impedance, and evanescent waves 38 3.1 Green’s functions .................................... 38 3.1.1 Integral representations ............................. 38 3.1.2 Remarks on finding Green’s functions ...................... 41 3.2 Acoustic impedance .................................. 41 3.2.1 Impedance and acoustic energy ......................... 43 3.2.2 Impedance and reflection coefficient ...................... 43 3.2.3 Impedance and causality ............................ 44 3.2.4 Impedance and surface waves .......................... 47 3.2.5 Acoustic boundary condition in the presence of mean flow ........... 48 3.2.6 Surface waves along an impedance wall with mean flow ............ 50 3.2.7 Instability, ill-posedness, and a regularization .................. 52 3.3 Evanescent waves and related behaviour ........................ 54 3.3.1 An important complex square root ....................... 54 3.3.2 The Walkman .................................. 55 3.3.3 Ill-posed inverse problem ............................ 56 3.3.4 Typical plate pitch ................................ 56 3.3.5 Snell’s law ................................... 56 3.3.6 Silent vorticity ................................. 59 4 One dimensional acoustics 62 4.1 Plane waves ...................................... 62 4.2 Basic equations and method of characteristics ..................... 64 4.2.1 The wave equation ............................... 64 4.2.2 Characteristics .................................. 65 4.2.3 Linear behaviour ................................ 66 4.2.4 Non-linear simple waves and shock waves ................... 70 4.3 Source terms ...................................... 72 4.4 Reflection at discontinuities and abrupt changes .................... 76 4.4.1 Jump in characteristic impedance ρc ...................... 76 4.4.2 Smooth change in pipe cross section ...................... 79 4.4.3 Orifice and high amplitude behaviour ...................... 79 4.4.4 Multiple junction ................................ 83 4.4.5 Reflection at a small air bubble in a pipe .................... 84 4.5 Attenuation of an acoustic wave by thermal and viscous dissipation .......... 88 4.5.1 Reflection of a plane wave at a rigid wall .................... 88 4.5.2 Viscous laminar boundary layer ........................ 92 Contents iii 4.5.3 Damping in ducts with isothermal walls. .................... 93 4.6 One dimensional Green’s function ........................... 95 4.6.1 Infinite uniform tube .............................. 95 4.6.2 Finite uniform tube ............................... 96 4.7 Aero-acoustical applications .............................. 97 4.7.1 Sound produced by turbulence ......................... 97 4.7.2 An isolated bubble in a turbulent pipe flow ................... 100 4.7.3 Reflection of a wave at a temperature inhomogeneity .............. 101 5 Resonators and self-sustained oscillations 107 5.1 Self-sustained oscillations, shear layers and jets .................... 107 5.2 Some resonators .................................... 113 5.2.1 Introduction .................................. 113 5.2.2 Resonance in duct segment ........................... 114 5.2.3 The Helmholtz resonator (quiescent fluid) ................... 121 5.2.4 Non-linear losses in a Helmholtz resonator ................... 123 5.2.5 The Helmholtz resonator in the presence of a mean flow ............ 124 5.3 Green’s function of a finite duct ............................ 125 5.4 Self-sustained oscillations of a clarinet ......................... 128 5.4.1 Introduction .................................. 128 5.4.2 Linear stability analysis ............................. 128 5.4.3 Rayleigh’s Criterion ............................... 130 5.4.4 Time domain simulation ............................ 130 5.5 Some thermo-acoustics ................................. 132 5.5.1 Introduction .................................. 132 5.5.2 Modulated heat transfer by acoustic flow and Rijke tube ............ 134 5.6 Flow induced oscillations of a Helmholtz resonator .................. 138 6 Spherical waves 147 6.1 Introduction ...................................... 147 6.2 Pulsating and translating sphere ............................ 147 6.3 Multipole expansion and far field approximation .................... 153 6.4 Method of images and influence of walls on radiation ................. 158 6.5 Lighthill’s theory of jet noise .............................. 161 6.6 Sound radiation by compact bodies in free space .................... 164 6.6.1 Introduction .................................. 164 6.6.2 Tailored Green’s function ............................ 165 6.6.3 Curle’s method ................................. 166 6.7 Sound radiation from an open pipe termination .................... 169 iv Contents 7 Duct acoustics 176 7.1 General formulation .................................. 176 7.2 Cylindrical ducts .................................... 178 7.3 Rectangular ducts .................................... 182 7.4 Impedance wall .................................... 183 7.4.1 Behaviour of complex modes .......................... 183 7.4.2 Attenuation ................................... 186 7.5 Annular hard-walled duct modes in uniform mean flow ................. 188 7.6 Behaviour of soft-wall modes and mean flow ...................... 192 7.7 Source expansion .................................... 194 7.7.1 Modal amplitudes ................................ 194 7.7.2 Rotating fan ................................... 194 7.7.3 Tyler and Sofrin rule for rotor-stator interaction ................. 195 7.7.4 Point source in a lined flow duct ......................... 197 7.7.5 Point source in a duct wall ........................... 200 7.7.6 Vibrating duct wall ............................... 202 7.8 Reflection and transmission .............................. 203 7.8.1 A discontinuity in diameter ........................... 203 7.8.2 The iris problem ................................. 207 7.8.3 The edge condition ............................... 209 7.9 Reflection at an unflanged open end .......................... 211 8 Approximation methods 216 8.1 Webster’s horn equation ................................ 217 8.2 Multiple scales ..................................... 220 8.3 Helmholtz resonator with non-linear dissipation .................... 224 8.4 Slowly varying ducts .................................. 229 8.5 Reflection at an isolated turning point ......................... 232 8.6 Ray acoustics in temperature gradient ......................... 236 8.7 Refraction in shear flow ................................ 241 8.8 Matched asymptotic expansions ............................ 242 8.9 Duct junction ...................................... 249 8.10 Co-rotating line-vortices ................................ 254 Contents v 9 Effects of flow and motion 259 9.1 Uniform mean flow, plane waves and edge diffraction ................. 259 9.1.1 Lorentz or Prandtl-Glauert transformation ................... 259 9.1.2 Plane waves ................................... 260 9.1.3 Half-plane diffraction problem ......................... 261 9.2 Moving point source and Doppler shift ......................... 263 9.3 Rotating monopole and dipole with moving observer ................. 265 9.4 Ffowcs Williams & Hawkings equation for moving bodies ............... 268 Appendix 273 A Integral laws and related results 273 A.1 Reynolds’ transport theorem .............................. 273 A.2 Conservation laws ................................... 273 A.3 Normal vectors of level surfaces ............................ 275 A.4 Vector identities and theorems