9. 2004 s.w. rienstra & a. hirschberg-acoustics

8/13/2019 9. 2004 S.W. Rienstra & a. Hirschberg-Acoustics

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An Introduction to Acoustics

S.W. Rienstra & A. Hirschberg

Eindhoven University of Technology

11 July 2013

This is an extended and revised edition of IWDE 92-06.

Comments and corrections are gratefully accepted.

This le may be used and printed, but for personal or educational purposes only.

c S.W. Rienstra & A. Hirschberg 2004.


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Contents

Preface

1 Some uid dynamics 1

1.1 Conservation laws and constitutive equations . . . . . . . . . . . . . . . . . . . . . 1

1.2 Approximations and alternative forms of the conservation laws for ideal uids . . . . . 4

2 Wave equation, speed of sound, and acoustic energy 8

2.1 Order of magnitude estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Wave equation for a uniform stagnant uid and compactness . . . . . . . . . . . . . 11

2.2.1 Linearization and wave equation . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Simple solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.3 Bubbly liquid at low frequencies . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Inuence of temperature gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Inuence of mean ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Sources of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6.1 Inverse problem and uniqueness of sources . . . . . . . . . . . . . . . . . . . 19

2.6.2 Mass and momentum injection . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6.3 Lighthills analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6.4 Vortex sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Acoustic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7.2 Kirchhoffs equation for quiescent uids . . . . . . . . . . . . . . . . . . . . 26

2.7.3 Acoustic energy in a non-uniform ow . . . . . . . . . . . . . . . . . . . . . 29

2.7.4 Acoustic energy and vortex sound . . . . . . . . . . . . . . . . . . . . . . . . 30


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ii Contents

3 Greens functions, impedance, and evanescent waves 33

3.1 Greens functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.2 Remarks on nding Greens functions . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Acoustic impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Impedance and acoustic energy . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2 Impedance and reection coefcient . . . . . . . . . . . . . . . . . . . . . . 37

3.2.3 Impedance and causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.4 Impedance and surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.5 Acoustic boundary condition in the presence of mean ow . . . . . . . . . . . 41

3.2.6 Surface waves along an impedance wall with mean ow . . . . . . . . . . . . 433.2.7 Instability, ill-posedness, and a regularization . . . . . . . . . . . . . . . . . . 45

3.3 Evanescent waves and related behaviour . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 An important complex square root . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.2 The Walkman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.3 Ill-posed inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.4 Typical plate pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.5 Snells law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.6 Silent vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 One dimensional acoustics 53

4.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Basic equations and method of characteristics . . . . . . . . . . . . . . . . . . . . . 54

4.2.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.3 Linear behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.4 Non-linear simple waves and shock waves . . . . . . . . . . . . . . . . . . . 59

4.3 Source terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Reection at discontinuities and abrupt changes . . . . . . . . . . . . . . . . . . . . 65

4.4.1 Jump in characteristic impedance c . . . . . . . . . . . . . . . . . . . . . . 65

4.4.2 Smooth change in pipe cross section . . . . . . . . . . . . . . . . . . . . . . 67

4.4.3 Orice and high amplitude behaviour . . . . . . . . . . . . . . . . . . . . . . 68

4.4.4 Multiple junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.5 Reection at a small air bubble in a pipe . . . . . . . . . . . . . . . . . . . . 72

4.5 Attenuation of an acoustic wave by thermal and viscous dissipation . . . . . . . . . . 75

4.5.1 Reection of a plane wave at a rigid wall . . . . . . . . . . . . . . . . . . . . 75

4.5.2 Viscous laminar boundary layer . . . . . . . . . . . . . . . . . . . . . . . . 78

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Contents iii

4.5.3 Damping in ducts with isothermal walls. . . . . . . . . . . . . . . . . . . . . 79

4.6 One dimensional Greens function . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6.1 Innite uniform tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.6.2 Finite uniform tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.7 Aero-acoustical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.7.1 Sound produced by turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.7.2 An isolated bubble in a turbulent pipe ow . . . . . . . . . . . . . . . . . . . 84

4.7.3 Reection of a wave at a temperature inhomogeneity . . . . . . . . . . . . . . 86

5 Resonators and self-sustained oscillations 91

5.1 Self-sustained oscillations, shear layers and jets . . . . . . . . . . . . . . . . . . . . 91

5.2 Some resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.2 Resonance in duct segment . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.3 The Helmholtz resonator (quiescent uid) . . . . . . . . . . . . . . . . . . . 102

5.2.4 Non-linear losses in a Helmholtz resonator . . . . . . . . . . . . . . . . . . . 105

5.2.5 The Helmholtz resonator in the presence of a mean ow . . . . . . . . . . . . 105

5.3 Greens function of a nite duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4 Self-sustained oscillations of a clarinet . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.4.2 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4.3 Rayleighs Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4.4 Time domain simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.5 Some thermo-acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5.2 Modulated heat transfer by acoustic ow and Rijke tube . . . . . . . . . . . . 113

5.6 Flow induced oscillations of a Helmholtz resonator . . . . . . . . . . . . . . . . . . 117

6 Spherical waves 125

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2 Pulsating and translating sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3 Multipole expansion and far eld approximation . . . . . . . . . . . . . . . . . . . . 130

6.4 Method of images and inuence of walls on radiation . . . . . . . . . . . . . . . . . 134

6.5 Lighthills theory of jet noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.6 Sound radiation by compact bodies in free space . . . . . . . . . . . . . . . . . . . . 139

6.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.6.2 Tailored Greens function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.6.3 Curles method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.7 Sound radiation from an open pipe termination . . . . . . . . . . . . . . . . . . . . 144



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7 Duct acoustics 149

7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2 Cylindrical ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.3 Rectangular ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.4 Impedance wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.4.1 Behaviour of complex modes . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.4.2 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.5 Annular hard-walled duct modes in uniform mean ow . . . . . . . . . . . . . . . . . 159

7.6 Behaviour of soft-wall modes and mean ow . . . . . . . . . . . . . . . . . . . . . . 162

7.7 Source expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.7.1 Modal amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.7.2 Rotating fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.7.3 Tyler and Sofrin rule for rotor-stator interaction . . . . . . . . . . . . . . . . . 165

7.7.4 Point source in a lined ow duct . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.7.5 Point source in a duct wall . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.7.6 Vibrating duct wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.8 Reection and transmission at a discontinuity in diameter . . . . . . . . . . . . . . . 171

7.8.1 The iris problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.9 Reection at an unanged open end . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8 Approximation methods 178

8.1 Websters horn equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.2 Multiple scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.3 Helmholtz resonator with non-linear dissipation . . . . . . . . . . . . . . . . . . . . 185

8.4 Slowly varying ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.5 Reection at an isolated turning point . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.6 Ray acoustics in temperature gradient . . . . . . . . . . . . . . . . . . . . . . . . . 195

8.7 Refraction in shear ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

8.8 Matched asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

8.9 Duct junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8.10 Co-rotating line-vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209



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9 Effects of ow and motion 213

9.1 Uniform mean ow, plane waves and edge diffraction . . . . . . . . . . . . . . . . . 213

9.1.1 Lorentz or Prandtl-Glauert transformation . . . . . . . . . . . . . . . . . . . 213

9.1.2 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

9.1.3 Half-plane diffraction problem . . . . . . . . . . . . . . . . . . . . . . . . . 214

9.2 Moving point source and Doppler shift . . . . . . . . . . . . . . . . . . . . . . . . . 216

9.3 Rotating monopole and dipole with moving observer . . . . . . . . . . . . . . . . . 218

9.4 Ffowcs Williams & Hawkings equation for moving bodies . . . . . . . . . . . . . . . 220

Appendix 224

A Integral laws and related results 224

A.1 Reynolds transport theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

A.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

A.3 Normal vectors of level surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

A.4 Vector identities and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

B Order of magnitudes: O and o. 227

C Fourier transforms and generalized functions 228

C.1 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

C.1.1 Causality condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231C.1.2 Phase and group velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

C.2 Generalized functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

C.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

C.2.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

C.2.3 The delta function and other examples . . . . . . . . . . . . . . . . . . . . . 236

C.2.4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

C.2.5 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

C.2.6 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

C.2.7 Higher dimensions and Greens functions . . . . . . . . . . . . . . . . . . . . 239

C.2.8 Surface distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

C.3 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

C.3.1 The Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

D Bessel functions 246

E Free eld Greens functions 254



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F Summary of equations for uid motion 255

F.1 Conservation laws and constitutive equations . . . . . . . . . . . . . . . . . . . . . . 255

F.2 Acoustic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

F.2.1 Inviscid and isentropic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

F.2.2 Perturbations of a mean ow . . . . . . . . . . . . . . . . . . . . . . . . . . 258

F.2.3 Myers Energy Corollary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

F.2.4 Zero mean ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

F.2.5 Time harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

F.2.6 Irrotational isentropic ow . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

F.2.7 Uniform mean ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

F.2.8 Parallel mean ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

G Answers to exercises. 263

Bibliography 274

Index 285

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Preface

Acoustics was originally the study of small pressure waves in air which can be detected by the humanear: sound . The scope of acoustics has been extended to higher and lower frequencies: ultrasound andinfrasound. Structural vibrations are now often included in acoustics. Also the perception of soundis an area of acoustical research. In our present introduction we will limit ourselves to the originaldenition and to the propagation in uids like air and water. In such a case acoustics is a part of uid dynamics .

A major problem of uid dynamics is that the equations of motion are non-linear. This implies that anexact general solution of these equations is not available. Acoustics is a rst order approximation inwhich non-linear effects are neglected. In classical acoustics the generation of sound is considered tobe a boundary condition problem. The sound generated by a loudspeaker or any unsteady movementof a solid boundary are examples of the sound generation mechanism in classical acoustics. In thepresent course we will also include some aero-acoustic processes of sound generation: heat transferand turbulence. Turbulence is a chaotic motion dominated by non-linear convective forces. An ac-curate deterministic description of turbulent ows is not available. The key of the famous Lighthilltheory of sound generation by turbulence is the use of an integral equation which is much more suit-able to introducing approximations than a differential equation. We therefore discuss in some detailthe use of Greens functions to derive integral equations.

Next to Lighthills approach which leads to order of magnitude estimate of sound production bycomplex ows we also describe briey the theory of vortex sound which can be used when a simpledeterministic description is available for a ow at low Mach numbers (for velocities small comparedto the speed of sound).

In contrast to most textbooks we have put more emphasis on duct acoustics, both in relation to itsgeneration by pipe ows, and with respect to more advanced theory on modal expansions and approx-imation methods. This is particular choice is motivated by industrial applications like aircraft enginesand gas transport systems.

This course is inspired by the book of Dowling and Ffowcs Williams: Sound and Sources of Sound[52]. We also used the lecture notes of the course on aero- and hydroacoustics given by Crighton,

Dowling, Ffowcs Williams, Heckl and Leppington [42].Among the literature on acoustics the book of Pierce [175] is an excellent introduction available for alow price from the Acoustical Society of America.

In the preparation of the lecture notes we consulted various books which cover different aspects of theproblem [14, 16, 18, 37, 48, 70, 87, 93, 99, 113, 122, 145, 160, 168, 171, 217, 230].


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2 1 Some uid dynamics

The uid stress tensor is related to the pressure p and the viscous stress tensor = ( i j ) by therelationship:

P = p I , or Pi j = p i j i j (1.4)where I = (i j ) is the unit tensor, and i j the Kronecker 4 delta. In most of the applications whichwe consider in the sequel, we can neglect the viscous stresses. When this is not the case one usuallyassumes a relationship between and the deformation rate of the uid element, expressed in the rate-of-strain tensor v +( v)T. It should be noted that a characteristic of a uid is that it opposes a rateof deformation, rather than the deformation itself, as in the case of a solid. When this relation is linearthe uid is described as Newtonian and the resulting momentum conservation equation is referred toas the Navier-Stokes equation. Even with such a drastic simplication, for compressible uids as weconsider in acoustics, the equations are quite complicated. A considerable simplication is obtainedwhen we assume Stokes hypothesis, that the uid is in local thermodynamic equilibrium, so that the

pressure p and the thermodynamic pressure are equivalent. In such a case we have:

= ( v +( v)T) 23 ( v) I , or i j = v i x j +

v j xi

23

vk xk

i j (1.5)

where is the dynamic viscosity. Equation (1.5) is what we call a constitutive equation. The viscosity is determined experimentally and depends in general on the temperature T and the pressure p.At high frequencies the assumption of thermodynamic equilibrium may partially fail resulting in adissipation related to volume changes v which is described with a volume viscosity parameter notsimply related to [240, 175]. These effects are also signicant in the propagation of sound in dustygases or in air over large distances [230].

In general ( m

= 0) the energy conservation law is given by ([14, 168, 230]):

t

e + 12 v2 + v(e + 12 v2) = q ( pv ) + ( v) + f v (1.6)or

t

e + 12 v2 + xi

v i (e + 12 v2) = qi xi

xi

( pvi ) + xi

( i j v j ) + f i viwhere v = |v|, e is the internal energy per unit of mass 5 and q is the heat ux due to heat conduction.A commonly used linear constitutive equation for q is Fouriers law:

q = K T , (1.7)

where K is the heat conductivity which depends on the pressure p and temperature T . Using thefundamental law of thermodynamics for a reversible process:

T ds = de + p d( 1) (1.8)and the equation for mechanical energy, obtained by taking the inner product of the momentum con-servation law (equation 1.2) with v, we obtain the equation for the entropy 6

T s t +v s = q + : v, or T

s t +vi

s xi =

qi xi + i j

v j xi

(1.9)

4 i j = 1 if i = j, i j = 0 if i = j.5We call this the specic internal energy , and simply the energy when there is no ambiguity.6 : v = ( v) v( ) since is symmetric. Note the convention ( v) i j =

xi v j .



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1.1 Conservation laws and constitutive equations 3

where s is the specic entropy or entropy per unit of mass. When heat conduction q and viscousdissipation : v may be neglected, the ow is isentropic 7 . This means that the entropy s of a uidparticle remains constant:

s t +v s = 0 . (1.10)

Except for regions near walls this approximation will appear to be quite reasonable for most of theapplications considered. If initially the entropy is equal to a constant value s0 throughout the uid, itretains this value, and we have simply a ow of uniform and constant entropy s = s0 . Note that someauthors dene this type of ow isentropic.

Equations (1.11.10) still contain more unknowns than equations. As closure condition we introducean additional constitutive equation, for example e = e( , s) , which implies with equation (1.8):

p

= 2

e

s

(1.11a)

T =es

(1.11b)

In many cases we will specify an equation of state p = p( , s) rather than e = e( , s) . In differentialform this becomes:

d p = c2d + ps

ds (1.12)

where

c2 = p s

(1.13)

is the square of the isentropic speed of sound c. While equation (1.13) is a denition of the thermody-namic variable c( , s) , we will see that c indeed is a measure for the speed of sound. When the sameequation of state c( , s) is valid for the entire ow we say that the uid is homogeneous . When thedensity depends only on the pressure we call the uid barotropic . When the uid is homogeneous andthe entropy uniform (d s = 0) we call the ow homentropic .In the following chapters we will use the heat capacity at constant volume C V which is dened for areversible process by

C V =e T V

. (1.14)

For an ideal gas the energy e is a function of the temperature only

e(T ) = T

0C V d T . (1.15)

For an ideal gas with constant heat capacities we will often use the simplied relation

e = C V T . (1.16)We call this a perfect gas . Expressions for the pressure p and the speed of sound c will be given insection 2.3. A justication for some of the simplications introduced will be given in chapter 2 wherewe will consider the order of magnitude of various effects and derive the wave equation. Before goingfurther we consider some useful approximations and some different notations for the basic equationsgiven above.

7When heat transfer is negligible, the ow is adiabatic . It is isentropic when it is adiabatic AN D reversible.



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1.2 Approximations and alternative forms of the conservation laws forideal uids

Using the denition of convective (or total) derivative 8 D/ Dt :

DDt =

t +v (1.17)

we can write the mass conservation law (1.1) in the absence of a source (m = 0) in the form:1

DDt = v (1.18)

which clearly shows that the divergence of the velocity v is a measure for the relative changein density of a uid particle. Indeed, the divergence corresponds to the dilatation rate 9 of the uidparticle which vanishes when the density is constant. Hence, if we can neglect density changes, themass conservation law reduces to:

v = 0 . (1.19)This is the continuity equation for incompressible uids. The mass conservation law (1.18) simplyexpresses the fact that a uid particle has a constant mass.

We can write the momentum conservation law for a frictionless uid ( negligible) as:

DvDt = p + f . (1.20)

This is Eulers equation, which corresponds to the second law of Newton (force = mass accelera-tion) applied to a specic uid element with a constant mass. The mass remains constant because weconsider a specic material element. In the absence of friction there are no tangential stresses actingon the surface of the uid particle. The motion is induced by the normal stresses (pressure force) pand the bulk forces f . The corresponding energy equation for a gas is

DsDt = 0 (1.10)

which states that the entropy of a particle remains constant. This is a consequence of the fact that heatconduction is negligible in a frictionless gas ow. The heat and momentum transfer are governed bythe same processes of molecular collisions. The equation of state commonly used in an isentropic ow

isD pDt = c

2 DDt

(1.21)

where c = c(, s ), a function of and s, is measured or derived theoretically. Note that in thisequation

c2 = p s

(1.13)

8The total derivative D f / Dt of a function f = f ( xi , t ) and velocity eld vi denotes just the ordinary time derivatived f / dt of f ( xi (t ), t ) for a path xi = xi (t ) dened by

. xi = vi , i.e. moving with a particle along xi = xi ( t ) .9

Dilatation rate = rate of relative volume change.



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1.2 Approximations and alternative forms of the conservation laws for ideal uids 5

is not necessarily a constant.

Under reasonably general conditions [144, p.53] the velocity v, like any vector eld, can be split into

an irrotational part and a solenoidal part:

v = +, = 0 , or vi = xi + i j k

k x j

, j x j = 0 , (1.22)

where is a scalar velocity potential, = ( i ) a vectorial velocity potential or vector stream func-tion, and i jk the permutation symbol 10 . A ow described by the scalar potential only ( v = ) iscalled a potential ow. This is an important concept because the acoustic aspects of the ow are linkedto . This is seen from the fact that ( ) = 0 so that the compressibility of the ow is describedby the scalar potential . We have from (1.18):

1

D

Dt = 2. (1.23)

From this it is obvious that the ow related to the acoustic eld is an irrotational ow. A usefuldenition of the acoustic eld is therefore: the unsteady component of the irrotational ow eld .The vector stream function describes the vorticity = v in the ow, because = 0. Hencewe have 11 :

= ( ) = 2 . (1.24)It can be shown that the vorticity corresponds to twice the angular velocity of a uid particle.When = ( p) is a function of p only, like in a homentropic ow (uniform constant entropy d s = 0),and in the absence of tangential forces due to the viscosity (

= 0), we can eliminate the pressure and

density from Eulers equation by taking the curl of this equation 12 , to obtain

t +v = v v +( f /). (1.25a)

If we apply the mass conservation equation (1.1) we get

t +v

= v

m +

f

. (1.25b)

We see that vorticity of the particle is changed either by stretching 13 , by a mass source in the presenceof vorticity, or by a non-conservative external force eld [229, 110]. In a two-dimensional incom-pressible ow (

v

= 0), with velocity v

= (v

x, v

y, 0) , the vorticity

= (0, 0,

z) is not affected

by stretching because there is no ow component in the direction of . Apart from the source terms

m/ and ( f /) , the momentum conservation law reduces to a purely kinematic law. Hencewe can say that (and ) is linked to the kinematic aspects of the ow.

10i j k =

+1 if i j k = 123 , 231 , or 312 ,1 if i j k = 321 , 132 , or 213 ,

0 if any two indices are alike

Note that vw = ( i j k v j wk ) .11 For any vector eld A: ( A) = ( A) 2 A.12(v v) = v v +v , ( 1 p) = 2 ( p) = 1 ( p)( p p) = 0 .13 The stretching of an incompressible particle of uid implies by conservation of angular momentum an increase of

rotation, because the particles lateral dimension is reduced. In a viscous ow tangential forces due to the viscous stress dochange the uid particle angular momentum, because they exert a torque on the uid particle.



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Using the denition of the specic enthalpy i :

i

= e

+ p

(1.26)

and the fundamental law of thermodynamics (1.8) we nd for a homentropic ow (homogeneous uidwith d s = 0):

di = d p

. (1.27)

Hence we can write Eulers equation (1.20) as:

DvDt = i +

1

f . (1.28)

We dene the total specic enthalpy B (Bernoulli constant) of the ow by:

B =i + 12 v

2. (1.29)

The total enthalpy B corresponds to the enthalpy which is reached in a hypothetical fully reversibleprocess when the uid particle is decelerated down to a zero velocity (reservoir state). Using the vectoridentity 14 :

(v )v = 12 v2 +v (1.30)we can write Eulers equation (1.20) in Croccos form:

v t = B v +

1

f (1.31)

which will be used when we consider the sound production by vorticity. The acceleration v cor-responds to the acceleration of Coriolis experienced by an observer moving with the particle which isrotating at an angular velocity of = 12 .When the ow is irrotational in the absence of external force ( f = 0) , with v = and hence = = 0, we can rewrite (1.28) into:

t + B = 0 ,

which may be integrated to Bernoullis equation:

t + B = g ( t ), (1.32a)or

t + 1

2v2

+ d p

= g ( t ) (1.32b)

where g( t ) is a function determined by boundary conditions. As only the gradient of is important(v = ) we can, without loss of generality, absorb g( t ) into and use g(t ) = 0. In acoustics theBernoulli equation will appear to be very useful. We will see in section 2.7 that for a homentropicow we can write the energy conservation law (1.10) in the form:

t

( B p) + ( v B) = f v , (1.33a)or t

( e + 12 v2) + ( v B) = f v . (1.33b)14

[(v )v]i = j v j

x j vi



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1.2 Approximations and alternative forms of the conservation laws for ideal uids 7

Exercises

a) Derive Eulers equation (1.20) from the conservation laws (1.1) and (1.2).

b) Derive the entropy conservation law (1.10) from the energy conservation law (1.6) and the second lawof thermodynamics (1.8).

c) Derive Bernoullis equation (1.32b) from Croccos equation (1.31).

d) Is the trace 13 Pii of the stress tensor P i j always equal to the thermodynamic pressure p = ( e/ 1 )s ?e) Consider, as a model for a water pistol, a piston pushing with a constant acceleration a water from a tube

1 with surface area A1 and length 1 through a tube 2 of surface A2 and length 2 . Calculate the forcenecessary to move the piston if the water compressibility can be neglected and the water forms a free

jet at the exit of tube 2. Neglect the non-uniformity of the ow in the transition region between the twotubes. What is the ratio of the pressure drop over the two tubes at t = 0?



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2 Wave equation, speed of sound, and acoustic energy

2.1 Order of magnitude estimates

Starting from the conservation laws and the constitutive equations given in section 1.2 we will obtainafter linearization a wave equation in the next section. This implies that we can justify the approx-imation introduced in section 1.2, (homentropic ow), and that we can show that in general, soundis a small perturbation of a steady state, so that second order effects can be neglected. We there-

fore consider here some order of magnitude estimates of the various phenomena involved in soundpropagation.

We have dened sound as a pressure perturbation p which propagates as a wave and which is de-tectable by the human ear. We limit ourselves to air and water. In dry air at 20 C the speed of soundc is 344 m / s, while in water a typical value of 1500 m / s is found. In section 2.3 we will discuss thedependence of the speed of sound on various parameters (such as temperature, etc. ). For harmonicpressure uctuations, the typical range of frequency of the human ear is:

20 Hz f 20 kHz . (2.1)

The maximum sensitivity of the ear is around 3 kHz, (which corresponds to a policemans whistle!).Sound involves a large range of power levels:

when whispering we produce about 10 10 Watts, when shouting we produce about 10 5 Watts, a jet airplane at take off produces about 10 5 Watts.

In view of this large range of power levels and because our ear has roughly a logarithmic sensitivitywe commonly use the decibel scale to measure sound levels. The Sound Power Level (PWL) is givenin decibel (dB) by:

PWL = 10 log 10 (Power / 1012 W). (2.2)The Sound Pressure Level (SPL) is given by:

SPL = 20 log 10 ( prms / pref ) (2.3)where prms is the root mean square of the acoustic pressure uctuations p , and where pref = 2 105 Pain air and pref = 10 6 Pa in other media. The sound intensity I is dened as the energy ux (powerper surface area) corresponding to sound propagation. The Intensity Level (IL) is given by:

IL = 10 log 10 ( I / 1012 W / m2). (2.4)The reference pressure level in air pref = 2105 Pa corresponds to the threshold of hearing at 1 kHz fora typical human ear. The reference intensity level I ref = 1012 W / m2 is related to this pref = 2 105 Pain air by the relationship valid for progressive plane waves:

I = p2

rms / 0c0 (2.5)


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2.1 Order of magnitude estimates 9

where 0c0 = 4 102 kg / m2s for air under atmospheric conditions. Equation (2.5) will be derived later.The threshold of pain 1 (140 dB) corresponds in air to pressure uctuations of prms = 200 Pa. Thecorresponding relative density uctuations / 0 are given at atmospheric pressure p0 = 10

5Pa by:

/ 0 = p / p0 103 (2.6)where = C P / C V is the ratio of specic heats at constant pressure and volume respectively. Ingeneral, by dening the speed of sound following equation 1.13, the relative density uctuations aregiven by:

0 =

1 0c20

p = 1 0

p s

p . (2.7)

The factor 1 / 0c20 is the adiabatic bulk compressibility modulus of the medium. Since for water 0 =10

3kg / m

3and c0 = 1 .5 10

3m / s we see that 0c

20 2 .2 10

9Pa, so that a compression wave of

10 bar corresponds to relative density uctuations of order 10 3 in water. Linear theory will thereforeapply to such compression waves. When large expansion waves are created in water the pressure candecrease below the saturation pressure of the liquid and cavitation bubbles may appear, which resultsin strongly non-linear behaviour. On the other hand, however, since the formation of bubbles in purewater is a slow process, strong expansion waves (negative pressures of the order of 10 3 bar!) can besustained in water before cavitation appears.

For acoustic waves in a stagnant medium, a progressive plane wave involves displacement of uidparticles with a velocity u which is given by (as we will see in equations 2.20a, 2.20b):

u

= p / 0c0 . (2.8)

The factor 0c0 is called the characteristic impedance of the uid. By dividing (2.8) by c0 we see byusing (1.13) in the form p = c20 that the acoustic Mach number u / c0 is a measure for the relativedensity variation / 0. In the absence of mean ow (u0 = 0) this implies that a convective term suchas ( v )v in the momentum conservation (1.20) is of second order and can be neglected in a linearapproximation.The amplitude of the uid particle displacement corresponding to harmonic wave propagation at acircular frequency = 2 f is given by:

= |u |/. (2.9)Hence, for f

=1 kHz we have in air:

SPL = 140 dB, prms = 2 102 Pa, u = 5 101 m/s, = 8 105 m,SPL = 0 dB, prms = 2 105 Pa, u = 5 108 m/s, = 1 1011 m.

In order to justify a linearization of the equations of motion, the acoustic displacement should besmall compared to the characteristic length scale L in the geometry considered. In other words, theacoustical Strouhal number Sr a = L/ should be large. In particular, if is larger than the radius of curvature R of the wall at edges the ow will separate from the wall resulting into vortex shedding.So a small acoustical Strouhal number R/ implies that non-linear effects due to vortex shedding areimportant. This is a strongly non-linear effect which becomes important with decreasing frequency,because increases when decreases.

1The SPL which we can only endure for a very short period of time without the risk of permanent ear damage.



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10 2 Wave equation, speed of sound, and acoustic energy

We see from the data given above that the particle displacement can be signicantly smaller thanthe molecular mean free path which in air at atmospheric pressure is about 5 108 m. It shouldbe noted that a continuum hypothesis as assumed in chapter 1 does apply to sound even at such lowamplitudes because is not the relevant length scale. The continuum hypothesis is valid if we candene an air particle which is small compared to the dimensions of our measuring device (eardrum,diameter D = 5 mm) or to the wave length , but large compared to the mean free path = 5 108 m.It is obvious that we can satisfy this condition since for f = 20 kHz the wave length:

= c0/ f (2.10)is still large ( 1 .7 cm ) compared to . In terms of our ear drum we can say that although adisplacement of = 10 11 m of an individual molecule cannot be measured, the same displacementaveraged over a large amount of molecules at the ear drum can be heard as sound.

It appears that for harmonic signals of frequency f =

1 kHz the threshold of hearing pref =

2

10

5 Pa

corresponds to the thermal uctuations pth of the atmospheric pressure p0 detected by our ear. Thisresult is obtained by calculating the number of molecules N colliding within half an oscillation periodwith our eardrum 2 : N n D 2c0 / 2 f , where n is the air molecular number density 3 . As N 10 20 and pth p0/ N we nd that pth 105 Pa.In gases the continuum hypothesis is directly coupled to the assumption that the wave is isentropicand frictionless. Both the kinematic viscosity = / and the heat diffusivity a = K / C P of a gasare typically of the order of c , the product of sound speed c and mean free path . This is relatedto the fact that c is in a gas a measure for the random (thermal) molecular velocities that we knowmacroscopically as heat and momentum diffusion. Therefore, in gases the absence of friction goestogether with isentropy. Note that this is not the case in uids. Here, isothermal rather than isentropic

wave propagation is common for normal frequencies.As a result from this relation c , the ratio between the acoustic wave length and the mean freepath , which is an acoustic Knudsen number, can also be interpreted as an acoustic Fourier number:

= c

= 2 f

. (2.11)

This relates the diffusion length (/ f )1/ 2 for viscous effects to the acoustic wave length . Moreover,this ratio can also be considered as an unsteady Reynolds number Re t :

Re t =

u t

2u x2

2 f , (2.12)

which is for a plane acoustic wave just the ratio between inertial and viscous forces in the momentumconservation law. For air = 1 .5105 m2/ s so that for f = 1kHz we have Re t = 4 107 . We thereforeexpect viscosity to play a signicant rle only if the sound propagates over distances of 10 7 wavelengths or more (3 103 km for f = 1 kHz). In practice the kinematic viscosity appears to be a ratherunimportant effect in the attenuation of waves in free space. The main dissipation mechanism is the

2The thermal velocity of molecules may be estimated to be equal to c0 .3n is calculated for an ideal gas with molar mass M from: n = N A / M = N A p/ M RT = p/ R T (see section 2.3)

where N A is the Avogadro number



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In a quiescent uid the equations of motion given in chapter 1 simplify to:

t + 0 v = 0 (2.14a) 0

v t + p = 0 (2.14b)

s t = 0 (2.14c)

where second order terms in the perturbations have been neglected. The constitutive equation (1.13)becomes:

p = c20 . (2.15)By subtracting the time derivative of the mass conservation law (2.14a) from the divergence of themomentum conservation law (2.14b) we eliminate v to obtain:

2 t 2

2 p = 0 . (2.16)Using the constitutive equation p = c20 (2.15) to eliminate either or p yields the wave equations:

2 p t 2 c

20 2 p = 0 (2.17a)

or

2

t 2 c2

0

2

= 0 . (2.17b)

Using the linearized Bernoulli equation:

t +

p 0 = 0 (2.18)

which should be valid because the acoustic eld is irrotational 4 , we can derive from (2.17a) a waveequation for / t . We nd therefore that satises the same wave equation as the pressure and thedensity:

2

t 2 c20

2

= 0 . (2.19)

Taking the gradient of (2.19) we obtain a wave equation for the velocity v = . Although a ratherabstract quantity, the potential is convenient for many calculations in acoustics. The linearizedBernoulli equation (2.18) is used to translate the results obtained for into less abstract quantitiessuch as the pressure uctuations p .

4In the case considered this property follows from the fact that ( 0 t v + p) = 0 t (v ) = 0. In general thisproperty is imposed by the denition of the acoustic eld.



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2.2 Wave equation for a uniform stagnant uid and compactness 13

2.2.2 Simple solutions

Two of the most simple and therefore most important solutions to the wave equation are dAlembertssolution in one and three dimensions. In 1-D we have the general solution

p = f ( x c0 t ) +g( x +c0 t ), (2.20a)v =

1 0c0

f ( x c0t ) g ( x +c0 t ) , (2.20b)where f and g are determined by boundary and initial conditions, but otherwise they are arbitrary.The velocity v is obtained from the pressure p by using the linearized momentum equation (2.14b).As is seen from the respective arguments x c0t , the f -part corresponds to a right-running wave(in positive x-direction) and the g-part to a left-running wave. This solution is especially useful todescribe low frequency sound waves in hard-walled ducts, and free eld plane waves. To allow for a

general orientation of the coordinate system, a free eld plane wave is in general written as

p = f ( n x c0 t ), v = n 0c0

f ( n x c0 t ), (2.21)where the direction of propagation is given by the unit vector n. Rather than only left- and right-running waves as in the 1-D case, in free eld any sum (or integral) over directions n may be taken.A time harmonic plane wave of frequency is usually written in complex form 5 as

p = Ae it i k x , v = k 0 A e it i k x , c20| k|2 = 2 , (2.22)where the wave-number vector, or wave vector, k

= n k

= n c

0

, indicates the direction of propagationof the wave (at least, in the present uniform and stagnant medium).

In 3-D we have a general solution for spherically symmetric waves ( i.e. depending only on radialdistance r ). They are rather similar to the 1-D solution, because the combination r p (r , t ) happens tosatisfy the 1-D wave equation (see section 6.2). Since the outward radiated wave energy spreads outover the surface of a sphere, the inherent 1 / r -decay is necessary from energy conservation arguments.

It should be noted, however, that unlike in the 1-D case, the corresponding radial velocity vr is rathermore complicated. The velocity should be determined from the pressure by time-integration of themomentum equation (2.14b), written in radial coordinates.

We have for pressure and radial velocity

p = 1r f (r c0 t ) + 1r g(r +c0 t ), (2.23a)vr =

1 0c0

1r

f (r c0 t ) 1r 2

F (r c0 t ) 1 0c0

1r

g (r +c0 t ) 1r 2

G (r +c0t ) , (2.23b)

where F ( z) = f ( z)d z and G ( z) = g( z)d z. Usually we have only outgoing waves, which meansfor any physical solution that the eld vanishes before some time t 0 (causality). Hence, f ( z) = 0 for z =r c0 t r c0 t 0 c0 t 0 because r 0, and g( z) = 0 for any z =r +c0t r +c0 t 0 . Since r is not restricted from above, this implies that

g( z) 0 for all z.5The physical quantity considered is described by the real part.



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is the specic-heat ratio. Comparison of (2.33) with the denition of the speed of sound c2 = ( p/) syields:

c = ( p/) 1/ 2 or c = ( RT )1/ 2 . (2.35)We see from this equation that the speed of sound of an ideal gas of given chemical compositiondepends only on the temperature. For a mixture of ideal gases with mole fraction X i of component ithe molar mass M is given by:

M =i

M i X i (2.36)

where M i is the molar mass of component i . The specic-heat ratio of the mixture can be calculatedby:

= X i i /( i 1) X i /( i 1)

(2.37)

because i /( i 1) = M i C p, i / R and i = C p, i / C V , i . For air = 1 .402, whilst the speed of soundat T = 273 .15 K is c = 331 .45 m / s. Moisture in air will only slightly affect the speed of sound butwill drastically affect the damping, due to departure from thermodynamic equilibrium [230].

The temperature dependence of the speed of sound is responsible for spectacular differences in soundpropagation in the atmosphere. For example, the vertical temperature stratication of the atmosphere(from colder near the ground to warmer at higher levels) that occurs on a winter day with fresh fallensnow refracts the sound back to the ground level, in a way that we hear trafc over much largerdistances than on a hot summer afternoon. These refraction effects will be discussed in section 8.6.

2.3.2 Water

For pure water, the speed of sound in the temperature range 273 K to 293 K and in the pressure range105 to 10 7 Pa can be calculated from the empirical formula [175]:

c = c0 +a (T T 0) +bp (2.38)where c0 = 1447 m / s, a = 4 .0 m / sK, T 0 = 283 .16 K and b = 1 .6 106 m / sPa. The presence of saltin sea water does signicantly affect the speed of sound.

2.3.3 Bubbly liquid at low frequencies

Also the presence of air bubbles in water can have a dramatic effect on the speed of sound ([114, 42]).The speed of sound is by denition determined by the mass density and the isentropic bulk modulus:

K s = p s

(2.39)

which is a measure for the stiffness of the uid. The speed of sound c, given by:

c = (K s /)12

(2.40)



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2.3 Speed of sound 17

increases with increasing stiffness, and decreases with increasing inertia (density ). In a one-dimensional model consisting of a discrete mass M connected by a spring of constant K , we can

understand this behaviour intuitively. This mass-spring model was used by Newton to derive equation(2.40), except for the fact that he used the isothermal bulk modulus K T rather than K s . This resultedin an error of 1/ 2 in the predicted speed of sound in air which was corrected by Laplace [230].

A small fraction of air bubbles present in water considerably reduces the bulk modulus K s , while at thesame time the density is not strongly affected. As the K s of the mixture can approach that for pureair, one observes in such mixtures velocities of sound much lower than in air (or water). The behaviourof air bubbles at high frequencies involves a possible resonance which we will discuss in chapter 4and chapter 6. We now assume that the bubbles are in mechanical equilibrium with the water, whichallows a low frequency approximation. Combining this assumption with (2.40), following Crighton[42], we derive an expression for the soundspeed c of the mixture as a function of the volume fraction of gas in the water. The density of the mixture is given by:

= (1 ) + g , (2.41)where and g are the liquid and gas densities. If we consider a small change in pressure d p weobtain:

dd p = (1 )

dd p +

d gd p +( g )

dd p

(2.42)

where we assume both the gas and the liquid to compress isothermally [42]. If no gas dissolves in theliquid, so that the mass fraction ( g /) of gas remains constant, we have:

gdd p +

d gd p

g

dd p = 0 . (2.43)

Using the notation c2 = d p/ d , c2g = d p/ d g and c2 = d p/ d , we nd by elimination of d / d pfrom (2.42) and (2.43):

1 c2 =

1 c2 +

g c2g

. (2.44)

It is interesting to see that for small values of the speed of sound c drops drastically from c at = 0towards a value lower than cg . The minimum speed of sound occurs at = 0 .5, and at 1 bar we ndfor example in a water/air mixture c 24 m / s! In the case of not being close to zero or unity, wecan use the fact that g c

2g c

2

and g , to approximate (2.44) by: c2

g c2g

, or c2 g c2g

( 1 ). (2.45)

The gas fraction determines the bulk modulus g c2g / of the mixture, while the water determines thedensity (1 ) . Hence, we see that the presence of bubbles around a ship may dramatically affectthe sound propagation near the surface. Air bubbles are also introduced in sea water near the surfaceby surface waves. The dynamics of bubbles involving oscillations (see chapter 4 and chapter 6) appearto induce spectacular dispersion effects [42], which we have ignored here.



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2.4 Inuence of temperature gradient

In section 2.2 we derived a wave equation (2.17a) for an homogeneous stagnant medium. We haveseen in section 2.3 that the speed of sound in the atmosphere is expected to vary considerably as aresult of temperature gradients. In many cases, when the acoustic wave length is small compared tothe temperature gradient length (distance over which a signicant temperature variation occurs) wecan still use the wave equation (2.17a). It is however interesting to derive a wave equation in the moregeneral case: for a stagnant ideal gas with an arbitrary temperature distribution.

We start from the linearized equations for the conservation of mass, momentum and energy for astagnant gas:

t + ( 0v ) = 0 (2.46a)

0 v t + p = 0 (2.46b)s t +v s0 = 0 , (2.46c)

where 0 and s0 vary in space. The constitutive equation for isentropic ow (Ds / Dt = 0) :D pDt = c

2 DDt

can be written as 7 :

p

t +v

p0

= c20

t +v

0 . (2.47)

Combining (2.47) with the continuity equation (2.46a) we nd:

p t +v p0 + 0c20 v = 0 . (2.48)

If we consider temperature gradients over a small height (in a horizontal tube for example) so that thevariation in p0 can be neglected ( p0 / p0 T 0/ T 0) , we can approximate (2.48) by:

v = 1 0c20

p t

.

Taking the divergence of the momentum conservation law (2.46b) yields:

t

( v ) + 1 0 p = 0 .

By elimination of v we obtain:2 p t 2 c

20 0

1 0 p = 0 . (2.49)

For an ideal gas c20 = p0 / 0 , and since we assumed p0 to be uniform, we have that 0c20 , given by: 0c20 = p0

7Why do we not use (2.15)?



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2.5 Inuence of mean ow 19

is a constant so that equation (2.49) can be written in the form:

2 p t 2 (c

20 p ) = 0 . (2.50)

This is a rather complex wave equation, since c0 is non-uniform. We will in section 8.6 considerapproximate solutions for this equation in the case ( c0 /) 1 and for large propagation distances.This approximation is called geometrical or ray acoustics.

It is interesting to note that, unlike in quiescent ( i.e. uniform and stagnant) uids, the wave equation(2.50) for the pressure uctuation p in a stagnant non-uniform ideal gas is not valid for the densityuctuations. This is because here the density uctuations not only relate to pressure uctuations butalso to convective effects (2.47). Which acoustic variable is selected to work with is only indifferentin a quiescent uid. This will be elaborated further in the discussion on the sources of sound in section2.6.

2.5 Inuence of mean ow

See also Appendix F. In the presence of a mean ow that satises

0v0 = 0 , 0v0 v0 = p0 , v0 s0 = 0 , v0 p0 = c20 v0 0 ,the linearized conservation laws, and constitutive equation for isentropic ow, become (withoutsources):

t +v0

+v

0

+ 0

v

+

v0

= 0 (2.51a)

0v t +v0 v +v v0 + v0 v0 = p (2.51b)s t +v0 s +v s0 = 0 . (2.51c) p t +v0 p +v p0 = c20

t +v0 +v 0 +c20 v0 0

p p0

0

(2.51d)

A wave equation can only be obtained from these equations if simplifying assumptions are introduced.For a uniform medium with uniform ow velocity v0 = 0 we obtain

t +v0

2 p c20 2 p = 0 (2.52)

where t +v0 denotes a time derivative moving with the mean ow.2.6 Sources of sound

2.6.1 Inverse problem and uniqueness of sources

Until now we have focused our attention on the propagation of sound. As starting point for the deriva-

tion of wave equations we have used the linearized equations of motion and we have assumed that the



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mass source term m and the external force density f in (1.1) and (1.2) were absent. Without these re-strictions we still can (under specic conditions) derive a wave equation. The wave equation will now

be non-homogeneous, i.e. it will contain a source term q. For example, we may nd in the absence of mean ow:

2 p t 2 c

20 2 p = q . (2.53)

Often we will consider situations where the source q is concentrated in a limited region of spaceembedded in a stagnant uniform uid. As we will see later the acoustic eld p can formally bedetermined for a given source distribution q by means of a Greens function. This solution p is unique.It should be noted that the so-called inverse problem of determining q from the measurement of poutside the source region does not have a unique solution without at least some additional informationon the structure of the source. This statement is easily veried by the construction of another soundeld, for example [64]: p

+ F , for any smooth function F that vanishes outside the source region

(i.e. F = 0 wherever q = 0), for example F q itself! This eld is outside the source region exactlyequal to the original eld p . On the other hand, it is not the solution of equation (2.53), because itsatises a wave equation with another source:

2

t 2 c20 2 ( p + F ) = q +

2

t 2 c20 2 F . (2.54)

In general this source is not equal to q . This proves that the measurement of the acoustic eld outsidethe source region is not sufcient to determine the source uniquely [52].

2.6.2 Mass and momentum injection

As a rst example of a non-homogeneous wave equation we consider the effect of the mass sourceterm m on a uniform stagnant uid. We further assume that a linear approximation is valid. Considerthe inhomogeneous equation of mass conservation

t

+ ( v) = m (2.55)and a linearized form of the equation of momentum conservation

t

( v) + p = f . (2.56)The source m consists of mass of density m of volume fraction

= ( x , t ) injected at a rate

m = t

( m ). (2.57)

The source region is where = 0. Since the injected mass displaces the original mass f by the same(but negative) amount of volume, the total uid density is

= m +(1 ) f (2.58)where the injected matter does not mix with the original uid. Substitute (2.58) in (2.55) and eliminate m

t f + ( v) =

t ( f ). (2.59)



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2.6 Sources of sound 21

Eliminate v from (2.56) and (2.59)

2

t 2 f 2 p =

2

t 2 ( f ) f . (2.60)If we assume, for simplicity, that p = c20 f everywhere, where f is the uctuating part of f whichcorresponds to the sound eld outside the source region, then

1c20

2

t 2 p 2 p =

2

t 2( f ) f (2.61)

which shows that mass injection is a source of sound, primarily because of the displacement of a vol-ume fraction of the original uid f . Hence injecting mass with a large density m is not necessarilyan effective source of sound.

We see from (2.61) that a continuous injection of mass of constant density does not produce sound,because 2 f / t 2 vanishes. In addition, it can be shown in an analogous way that in linear approxi-mation the presence of a uniform force eld (a uniform gravitational eld, for example) does not affectthe sound eld in a uniform stagnant uid.

2.6.3 Lighthills analogy

We now indicate how a wave equation with aerodynamic source terms can be derived. The mostfamous wave equation of this type is the equation of Lighthill.

The notion of analogy refers here to the idea of representing a complex uid mechanical process

that acts as an acoustic source by an acoustically equivalent source term. For example, one may modela clarinet as an idealized resonator formed by a closed pipe, with the effect of the ow through themouth piece represented by a mass source at one end. In that particular case we express by this analogythe fact that the internal acoustic eld of the clarinet is dominated by a standing wave correspondingto a resonance of the (ideal) resonator.

While Lighthills equation is formally exact ( i.e. derived without approximation from the Navier-Stokes equations), it is only useful when we consider the case of a limited source region embedded ina uniform stagnant uid. At least we assume that the listener which detects the acoustic eld at a point x at time t is surrounded by a uniform stagnant uid characterized by a speed of sound c0 . Hence theacoustic eld at the listener should accurately be described by the wave equation:

2

t 2 c20 2 = 0 (2.17b)

where we have chosen as the acoustic variable as this will appear to be the most convenientchoice for problems like the prediction of sound produced by turbulence. The key idea of the so-called aero-acoustic analogy of Lighthill is that we now derive from the exact equations of motiona non-homogeneous wave equation with the propagation part as given by (2.17b). Hence the uniformstagnant uid with sound speed c0 , density 0 and pressure p0 at the listeners location is assumedto extend into the entire space, and any departure from the ideal acoustic behaviour predicted by(2.17b) is equivalent to a source of sound for the observer [118, 119, 178, 81].



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By taking the time derivative of the mass conservation law (1.1) and eliminating m / t as in (2.59)we nd:

2

t xi(v i ) =

m t

2 t 2 =

2 f t 2 +

2 f t 2

. (2.62)

By taking the divergence of the momentum conservation law (1.2) we nd:

2

t xi(v i ) =

2

xi x j( Pi j +v i v j ) +

f i xi

. (2.63)

Hence we nd from (2.62) and (2.63) the exact relation:

2 f t 2 =

2

xi x j( Pi j +v i v j ) +

2 f t 2

f i xi

. (2.64)

Because f = 0 + where only varies in time we can construct a wave equation for bysubtracting from both sides of (2.63) a term c20 (

2 / x2i ) where in order to be meaningful c0 is notthe local speed of sound but that at the listeners location .

In this way we have obtained the famous equation of Lighthill:

2 t 2 c

20

xi =

2 T i j xi x j +

2 f t 2

f i xi

(2.65)

where Lighthills stress tensor T i j is dened by:

T i j = Pi j +v i v j (c20 + p0 ) i j . (2.66)We used

c202

x2i = 2(c20 i j )

xi x j(2.67)

which is exact because c0 is a constant. Making use of denition (1.4) we can also write:

T i j = v i v j i j +( p c20 ) i j (2.68)

which is the usual form in the literature 8 . In equation (2.68) we distinguish three basic aero-acousticprocesses which result in sources of sound:

the non-linear convective forces described by the Reynolds stress tensor v i v j , the viscous forces i j , the deviation from a uniform sound velocity c0 or the deviation from an isentropic behaviour

( p c20 ) .8The perturbations are dened as the deviation from the uniform reference state ( 0 , p0 ): = 0 , and p = p p0 .



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2.6 Sources of sound 23

As no approximations have been made, equation (2.65) is exact and not easier to solve than the orig-inal equations of motion. In fact, we have used four equations: the mass conservation and the three

components of the momentum conservation to derive a single equation. We are therefore certainly notcloser to a solution unless we introduce some additional simplifying assumptions.

The usefulness of (2.65) is that we can introduce some crude simplications which yield an order of magnitude estimate for . Such estimation procedure is based on the physical interpretation of thesource term. However, a key step of Lighthills analysis is to delay this physical interpretation untilan integral equation formulation of (2.65) has been obtained. This is an efcient approach because anorder of magnitude estimate of 2T i j / xi x j involves the estimation of spatial derivatives which isvery difcult, while, as we will see, in an integral formulation we will need only an estimate for anaverage value of T i j in order to obtain some relevant information on the acoustic eld.

This crucial step was not recognized before the original papers of Lighthill [118, 119]. For a givenexperimental or numerical set of data on the ow eld in the source region, the integral formulationof Lighthills analogy often provides a maximum amount of information about the generated acousticeld.

Unlike in the propagation in a uniform uid the choice of the acoustic variable appeared already inthe presence of a temperature gradient (section 2.4) to affect the character of the wave equation. If wederive a wave equation for p instead of , the structure of the source terms will be different. In somecases it appears to be more convenient to use p instead of . This is the case when unsteady heatrelease occurs such as in combustion problems. Starting from equation (2.64) in the form:

2 p x2i =

2 t 2 +

2

xi x j( i j v i v j )

where we assumed that m = 0 and f = 0, we nd by subtraction of c2

0 ( 2 / t 2) p on both sides:

1c20

2 p t 2

2 p x2i =

2

xi x j(v i v j i j ) +

2 p0 x2i +

2

t 2 pc20 (2.69)

where the term 2 p0 / x2i vanishes because p0 is a constant.

Comparing (2.65) with (2.69) shows that the deviation from an isentropic behaviour leads to a sourceterm of the type ( 2/ x2i )( p c20 ) when we choose as the acoustic variable, while we nda term ( 2/ t 2)( p / c20 ) when we choose p as the acoustic variable. Hence is more appro-priate to describe the sound generation due to non-uniformity as for example the so-called acousticBremsstrahlung produced by the acceleration of a uid particle with an entropy different from the

main ow. The sound production by unsteady heat transfer or combustion is easier to describe in termsof p (Howe [81]).

We see that (/ t )( p / c20 ) acts as a mass source term m, which is intuitively more easily un-derstood (Crighton et al. [42]) when using the thermodynamic relation (1.12) applied to a movingparticle:

D pDt = c

2 DDt +

ps

DsDt

. (1.12)

We nd from (1.12) that:

D

Dt

p

c20

=c2

c20 1

D

Dt + 2

c20

T

s

Ds

Dt (2.70)



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where we made use of the thermodynamic relation:

ps =

2 T s (2.71)

derived from the fundamental law of thermodynamics (1.8) in the form:

de = T ds p d( 1). (1.8)As a nal result, using the mass conservation law, we nd

2 e t 2 =

t

c2

c20 1 + e

DDt +

2

c20

T s

DsDt + (v e) (2.72)

where the excess density e is dened as:

e = pc20

.

In a free jet the rst term in 2 e / t 2 vanishes for an ideal gas with constant heat capacity (becausec2/ c20 1 + e / = 0). We see that sound is produced both by spatial density variations (v e) andas a result of non-isentropic processes ( 2/ c20 )( T /) s (Ds / Dt ) , like combustion.2.6.4 Vortex sound

While Lighthills analogy is very convenient for obtaining order of magnitude estimates of the sound

produced by various processes, this formulation is not very convenient when one considers the soundproduction by a ow which is, on its turn, inuenced by the acoustic eld. In Lighthills procedure theow is assumed 9 to be known, with any feedback from the acoustic eld to the ow somehow alreadyincluded. When such a feedback is signicant, and in general for homentropic low Mach numberow, the aerodynamic formulation of Powell [178], Howe [81] and Doak [50] based on the conceptof vortex sound is most appropriate. This is due to the fact that the vorticity = v is a veryconvenient quantity to describe a low Mach number ow.

Considering a homentropic non-conductive frictionless uid, we start our derivation of a wave equa-tion from Eulers equation in Croccos form:

v

t + B

=

v (1.31)

where B =i + 12 v2 , and the continuity equation:1

DDt = v . (1.18)

Taking the divergence of (1.31) and the time derivative of (1.18) we obtain by subtraction:

t

1

DDt

2 B = (v). (2.73)9This is not a necessary condition for the use of Lighthills analogy. It is the commonly used procedure in which we

derive information on the acoustic eld from data on the ow in the source region.



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2.7 Acoustic energy 25

As the entropy is constant (d s = 0) we have, with (1.12) and (1.27): t

1c2

DiDt

2

B = (v). (2.74)This can be rewritten as

1c2

D20 B Dt

2 B = (v ) + 1c2

D20 B Dt

t

1c2

DiDt

(2.75)

where B = B B0 and D0 Dt = t +U 0 . For the reference ow U 0 we choose a potential ow withstagnation enthalpy B0 .At low Mach number M = v / c0 we have the inhomogeneous wave equation:

1c20

D2

0 B Dt 2 2 B = (v ) (2.76)which explicitly stresses the fact that the vorticity is responsible for the generation of sound. (Note:i = p / 0 and B = i +v0v .) Some of the implications of (2.76) will be considered in more detailin the next section. The use of a vortex sound formulation is particularly powerful when a simpliedvortex model is available for the ow considered. Examples of such ows are discussed by Howe [81],Disselhorst & van Wijngaarden [49], Peters & Hirschberg [172], and Howe [86].

In free space for a compact source region Powell [177] has derived this analogy directly fromLighthills analogy. The result is that the Coriolis force f c = 0(v) appears to act as an ex-ternal force on the acoustic eld. Considering Croccos equation (1.31) with this interpretation Howe

[82, 85] realized that the natural reference of the analogy is a potential ow rather than the quiescentuid of Lighthills analogy. There is then no need to assume free eld conditions nor a compact sourceregion. Howe [81] therefore proposes to dene the acoustic eld as the unsteady scalar potential owcomponent of the ow:

u a = where = 0 and 0 is the steady scalar potential.At high Mach numbers, when the source is not compact, both Lighthills and Howes analogy becomeless convenient. Alternative formulations have been proposed and are still being studied [150].

2.7 Acoustic energy

2.7.1 Introduction

Acoustic energy is a difcult concept because it involves second order terms in the perturbations likethe kinetic energy density 12 0v

2 . Historically an energy conservation law was rst derived by Kirch-hoff for stagnant uniform uids. He started from the linearized conservation laws (2.51a2.51d). Sucha procedure is ad-hoc, and the result, an energy expression of the approximation, is not an approx-imation of the total energy, since a small perturbation expansion of the full non-linear uid energyconservation law (1.6) will contain zeroth and rst order terms and potentially relevant second order

terms O (( / 0)2

) which are dropped with the linearization of the mass and momentum equations.



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However, it appears that for a quiescent uid these zeroth, rst and neglected second order termsare (in a sense) not important and an acoustic energy conservation equation may be derived which is

indeed the same as found by Kirchhoff [175].This approach may be extended to non-uniform ows as long as they are homentropic and irrotational.Things become much less obvious in the presence of a non-uniform mean ow including entropyvariations and vorticity. If required, the zeroth, rst and neglected second order terms of the expansionmay still be ignored, as Myers showed [152], but now at the expense of a resulting energy equationwhich is not a conservation law any more. The only way to obtain some kind of acoustic energyconservation equation (implying denitions for acoustic energy density and ux) is to redirect certainparts to the right hand side to become source or sink terms. In such a case the question of denition,in particular which part of the eld is to be called acoustic, is essential and until now it remains subjectof discussion.

As stated before, we will consider as acoustical only that part of the eld which is related to densityvariations and an unsteady (irrotational) potential ow. Pressure uctuations related to vorticity, whichdo not propagate, are often referred to in the literature as pseudo sound. In contrast to this approachJenvey [96] calls any pressure uctuations acoustic, which of course results in a different denitionof acoustic energy.

The foregoing approach of generalized expressions for acoustic energy for homentropic [152] andmore general nonuniform ows [153, 154] by expanding the energy equation for small perturbationsis due to Myers. We will start our analysis with Kirchhoffs equation for an inviscid non-conductinguid, and extend the results to those obtained by Myers. Finally we will consider a relationship be-tween vorticity and sound generation in a homentropic uniform inviscid non-conducting uid at lowMach numbers, derived by Howe [82].

2.7.2 Kirchhoffs equation for quiescent uids

We start from the linearized mass and momentum conservation laws for a quiescent inviscid andnon-conducting uid:

t + 0 v = m , (2.77a) 0

v t + p = f , (2.77b)

where we assumed that f and m are of acoustic order. Since we assumed the mean ow to be

quiescent and uniform there is no mean mass source ( m0 = 0) or force ( f 0 = 0). From the assumptionof homentropy (d s = 0) we have 10 p = c20 . (2.15)

After multiplying (2.77a) by p / 0 and (2.77b) by v , adding the two equations, and utilizing theforegoing relation (2.15) between density and pressure, we obtain the equation

12 0c20

p 2

t + 12

0v 2

t + ( p v ) = p m

0 +v f (2.78)10 Note that in order to keep equation (2.15) valid we have implicitly assumed that the injected mass corresponding to m

has the same thermodynamic properties as the original uid. The ow would otherwise not be homentropic! In this case

m / 0 corresponds to the injected volume fraction of equation (2.57).



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which can be interpreted as a conservation law for the acoustic energy

E

t + I = D (2.79)if we DEFINE the acoustic energy density E , the energy ux or intensity 11 I and the dissipation D as:

E = p 2

2 0c20 + 0v 2

2, (2.80a)

I = p v , (2.80b)D =

p m 0 v f . (2.80c)

In integral form this conservation law (2.79) can be written for a xed control volume V enclosed bya surface S with outer normal n as

ddt

V

E d x + S

I n d = V

D d x , (2.81)

where we have used the theorem of Gauss to transform I d x into a surface integral. For aperiodic acoustic eld the average E of the acoustic energy over a period is constant. Hence we ndP =

S

I n d = V

D d x , (2.82)

where P is the acoustic power ow across the volume surface S . The left-hand side of (2.82) simplycorresponds with the mechanical work performed by the volume injection (m / 0) and the externalforce eld f on the acoustic eld. This formula is useful because we can consider the effect of themovement of solid boundaries like a piston or a propeller represented by source terms m and f .We will at the end of this chapter use formula (2.82) to calculate the acoustic power generated by acompact vorticity eld.

We will now derive the acoustic energy equation starting from the original nonlinear energy conser-vation law (1.6). We consider the perturbation of a uniform quiescent uid without mass source term(v0 = 0, m = 0, f 0 = 0, p0 and 0 constant). We start with equation (1.6) in standard conservationform:

t e + 12 v 2 + v e + 12 v 2 + p = q + ( v) + f v, (2.83)where we note that the total uid energy density is

E tot = e + 12

v 2 , (2.84a)

and the total uid energy ux is

I tot = v( e + 12

v 2 + p). (2.84b)11 There is no uniformity in the nomenclature. Some authors dene the acoustic intensity as the acoustic energy ux,

others as the time-averaged acoustic energy ux.



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We have dropped here the mass source term m because, in contrast to the force density f , it does notcorrespond to any physical process.

For future reference we state here some related forms, a.o. related to the entropy variation of the uid.Using the continuity equation we obtain

DDt

e + v2

2 = ( pv) q + ( v) + f v , (2.85)which by using the fundamental law of thermodynamics (1.8) may yield an equation for the changein entropy s of the uid:

T DsDt

p

DDt +

2

Dv2

Dt = ( pv) q + ( v) + f v. (2.86)By subtraction of the inner product of the momentum conservation equation with the velocity, thismay be further recast into

T DsDt = q + : v . (2.87)

In the absence of friction ( = 0) and heat conduction ( q = 0) we have the following equations forenergy and entropy:

DDt

e + 12

v2 = ( pv ) + f v (2.88)DsDt = 0 . (2.89)

We return to the energy equation in standard conservation form, without friction and heat conduction:

t

e + 12

v 2 + v( e + 12

v 2 + p) = v f . (2.90)From the fundamental law of thermodynamics (1.8):

T ds = de + p d( 1) (1.8)we have for isentropic perturbations:

e s =

p 2

, and so

e

s = e

+ p

= i,

2 e

2 s = 1

p

s = c2

,

where i is the enthalpy (1.26) or heat function. We can now expand the total energy density, energyux and source for acoustic ( i.e. isentropic) perturbations up to second order, to nd ( v0 = 0):

e + 12 v 2 = 0e0 +i0 + 12 0c02 0

2

+ 12 0v 2 , (2.91a)v( e + 12 v 2 + p) = v (i0 0 +i0 + p ), (2.91b)

v f = v f . (2.91c)Noting that the steady state is constant, and using the equation of mass conservation

t + ( 0v + v ) = 0



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in (2.90), with (2.91a2.91c) substituted in it, we nd that the zeroth and rst order terms in / 0vanish so that (2.90) becomes within an accuracy of O (( / 0)3) :

t

p 2

2 0c20 + 0v 2

2 + ( p v ) = v f , (2.92)which demonstrates that Kirchhoffs acoustic energy conservation law (2.79) is not only an energy-like relation of the approximate equations, but indeed also the consistent acoustic approximation of the energy equation of the full uid mechanical problem.

2.7.3 Acoustic energy in a non-uniform ow

The method of Myers [152] to develop a more general acoustic energy conservation law followssimilar lines as the discussion of the previous section. We consider a homentropic ow (d s

= 0, so

that d e = ( p/ 2)d ) with v0 = 0. In this case the total enthalpy B = e + p/ + 12 v2 appears to be aconvenient variable. In terms of B the energy conservation law (2.90) becomes:

t

( B p) + ( Bv) = v f . (2.93)The momentum conservation law in Croccos form (1.31) also involves B:

v t + B +v = f / . (2.94)

By subtracting 0v0 times the momentum conservation law (2.94) plus B0 times the continuity equa-

tion (1.18) from the energy conservation law (2.93), substituting the steady state momentum conser-vation law:

B0 +0v0 = f 0/ 0 , (2.95)subtracting the steady state limit of the resulting equation, and using the vector identity v(v) = 0,Myers obtained the following energy corollary:

t

E exact + I exact = D exact (2.96)where E exact , I exact and D exact are dened by:

E exact = ( B B0) ( p p0) 0v0(v

9. 2004 s.w. rienstra & a. hirschberg-acoustics

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