An Introduction to Acoustics

298 Pages·2016·3.08 MB·English

An Introduction to Acoustics
S.W. Rienstra & A. Hirschberg
EindhovenUniversityofTechnology
23Dec 2021
ThisisanextendedandrevisededitionofIWDE92-06.
Commentsandcorrectionsaregratefullyaccepted.
Thisfilemaybeusedandprinted,butforpersonaloreducationalpurposesonly.
© S.W. Rienstra& A.Hirschberg 2004.

Contents
Page
Preface
1 Somefluiddynamics 1
1.1 Conservation lawsandconstitutive equations . . . . . . . . . . . . . . . . . . . . . 1
1.2 Approximations andalternative formsoftheconservation lawsforidealfluids . . . . . 4
2 Waveequation,speedofsound,andacousticenergy 9
2.1 Orderofmagnitude estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Waveequation forauniformstagnant fluidandcompactness . . . . . . . . . . . . . 13
2.2.1 Linearization andwaveequation . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Simplesolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Speedofsound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Idealgas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.3 Bubblyliquidatlowfrequencies . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Influenceoftemperature gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Influenceofmeanflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Sourcesofsound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6.1 Inverse problemanduniqueness ofsources . . . . . . . . . . . . . . . . . . . 22
2.6.2 Massandmomentuminjection . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.3 Lighthill’s analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6.4 Vortexsound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Acousticenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7.2 Kirchhoff’s equation forquiescent fluids . . . . . . . . . . . . . . . . . . . . 30
2.7.3 Acousticenergy inanon-uniform flow . . . . . . . . . . . . . . . . . . . . . 33
2.7.4 Acousticenergy andvortexsound . . . . . . . . . . . . . . . . . . . . . . . . 35
ii Contents
3 Green’sfunctions,impedance,andevanescentwaves 38
3.1 Green’sfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1 Integralrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 RemarksonfindingGreen’sfunctions . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Acousticimpedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.1 Impedance andacousticenergy . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Impedance andreflectioncoefficient . . . . . . . . . . . . . . . . . . . . . . 43
3.2.3 Impedance andcausality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.4 Impedance andsurfacewaves . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.5 Acousticboundary condition inthepresenceofmeanflow . . . . . . . . . . . 48
3.2.6 Surfacewavesalonganimpedancewallwithmeanflow . . . . . . . . . . . . 50
3.2.7 Instability, ill-posedness, andaregularization . . . . . . . . . . . . . . . . . . 52
3.3 Evanescentwavesandrelatedbehaviour . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Animportantcomplexsquareroot . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 TheWalkman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.3 Ill-posed inverseproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.4 Typicalplatepitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.5 Snell’slaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.6 Silentvorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Onedimensionalacoustics 63
4.1 Planewaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Basicequations andmethodofcharacteristics . . . . . . . . . . . . . . . . . . . . . 65
4.2.1 Thewaveequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.3 Linearbehaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.4 Non-linear simplewavesandshockwaves . . . . . . . . . . . . . . . . . . . 71
4.3 Sourceterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Reflectionatdiscontinuities andabruptchanges . . . . . . . . . . . . . . . . . . . . 77
4.4.1 Jumpincharacteristic impedanceρc . . . . . . . . . . . . . . . . . . . . . . 77
4.4.2 Smoothchangeinpipecrosssection . . . . . . . . . . . . . . . . . . . . . . 80
4.4.3 Orificeandhighamplitudebehaviour . . . . . . . . . . . . . . . . . . . . . . 80
4.4.4 Multiplejunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.5 Reflectionatasmallairbubbleinapipe . . . . . . . . . . . . . . . . . . . . 85
4.5 Attenuation ofanacousticwavebythermalandviscousdissipation . . . . . . . . . . 89
4.5.1 Reflectionofaplanewaveatarigidwall . . . . . . . . . . . . . . . . . . . . 89
4.5.2 Viscouslaminarboundary layer . . . . . . . . . . . . . . . . . . . . . . . . 93
Contents iii
4.5.3 Dampinginductswithisothermalwalls. . . . . . . . . . . . . . . . . . . . . 94
4.6 Onedimensional Green’sfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.6.1 Infiniteuniformtube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.6.2 Finiteuniform tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.7 Aero-acoustical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.7.1 Soundproduced byturbulence . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.7.2 Anisolated bubbleinaturbulent pipeflow . . . . . . . . . . . . . . . . . . . 101
4.7.3 Reflectionofawaveatatemperature inhomogeneity . . . . . . . . . . . . . . 102
5 Resonatorsandself-sustained oscillations 108
5.1 Self-sustained oscillations, shearlayersandjets . . . . . . . . . . . . . . . . . . . . 108
5.2 Someresonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.2 Resonance inductsegment . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.3 TheHelmholtzresonator (quiescent fluid) . . . . . . . . . . . . . . . . . . . 122
5.2.4 Non-linear lossesinaHelmholtzresonator . . . . . . . . . . . . . . . . . . . 124
5.2.5 TheHelmholtzresonator inthepresence ofameanflow . . . . . . . . . . . . 125
5.3 Green’sfunction ofafiniteduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 Self-sustained oscillations ofaclarinet . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.2 Linearstability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.3 Rayleigh’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.4 Timedomainsimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.5 Somethermo-acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.5.2 Modulated heattransferbyacoustic flowandRijketube . . . . . . . . . . . . 135
5.6 Flowinduced oscillations ofaHelmholtzresonator . . . . . . . . . . . . . . . . . . 139
6 Sphericalwaves 148
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2 Pulsatingandtranslating sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.3 Multipoleexpansion andfarfieldapproximation . . . . . . . . . . . . . . . . . . . . 154
6.4 Methodofimagesandinfluenceofwallsonradiation . . . . . . . . . . . . . . . . . 159
6.5 Lighthill’stheoryofjetnoise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.6 Soundradiation bycompactbodiesinfreespace . . . . . . . . . . . . . . . . . . . . 165
6.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.6.2 TailoredGreen’sfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.6.3 Curle’smethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.7 Soundradiation fromanopenpipetermination . . . . . . . . . . . . . . . . . . . . 170
iv Contents
7 Ductacoustics 177
7.1 Generalformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.2 Cylindrical ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.3 Rectangular ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.4 Impedancewall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.4.1 Behaviour ofcomplexmodes . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.4.2 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.5 Annularhard-walled ductmodesinuniform meanflow. . . . . . . . . . . . . . . . . 189
7.6 Behaviourofsoft-wallmodesandmeanflow . . . . . . . . . . . . . . . . . . . . . . 193
7.7 Sourceexpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.7.1 Modalamplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.7.2 Rotatingfan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.7.3 TylerandSofrinruleforrotor-stator interaction . . . . . . . . . . . . . . . . . 196
7.7.4 Pointsourceinalinedflowduct. . . . . . . . . . . . . . . . . . . . . . . . . 198
7.7.5 Pointsourceinaductwall . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.7.6 Vibratingductwall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.8 Reflectionandtransmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.8.1 Adiscontinuity indiameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.8.2 Theirisproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.8.3 Theedgecondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.9 Reflectionatanunflangedopenend . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8 Approximationmethods 217
8.1 Webster’shornequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.2 Multiplescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.3 Helmholtzresonator withnon-linear dissipation . . . . . . . . . . . . . . . . . . . . 225
8.4 Slowlyvaryingducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.5 Reflectionatanisolated turningpoint . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.6 Rayacoustics intemperature gradient . . . . . . . . . . . . . . . . . . . . . . . . . 237
8.7 Refractioninshearflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
8.8 Matchedasymptoticexpansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.9 Ductjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
8.10 Co-rotating line-vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Contents v
9 Effectsofflowandmotion 260
9.1 Uniformmeanflow,planewavesandedgediffraction . . . . . . . . . . . . . . . . . 260
9.1.1 LorentzorPrandtl-Glauert transformation . . . . . . . . . . . . . . . . . . . 260
9.1.2 Planewaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9.1.3 Half-plane diffraction problem . . . . . . . . . . . . . . . . . . . . . . . . . 262
9.2 MovingpointsourceandDopplershift . . . . . . . . . . . . . . . . . . . . . . . . . 264
9.3 Rotatingmonopoleanddipolewithmovingobserver . . . . . . . . . . . . . . . . . 266
9.4 FfowcsWilliams&Hawkingsequation formovingbodies . . . . . . . . . . . . . . . 269
Appendix 274
A Integrallawsandrelated results 274
A.1 Reynolds’transport theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
A.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
A.3 Normalvectorsoflevelsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
A.4 Vectoridentities andtheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
B Ordersofmagnitude: O ando. 278
C Fouriertransformsandgeneralized functions 279
C.1 Fouriertransforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
C.1.1 Causalitycondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
C.1.2 Phaseandgroupvelocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
C.2 Generalized functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
C.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
C.2.2 Formaldefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
C.2.3 Thedeltafunction andotherexamples . . . . . . . . . . . . . . . . . . . . . 288
C.2.4 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
C.2.5 Fouriertransforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
C.2.6 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
C.2.7 Higherdimensions andGreen’sfunctions . . . . . . . . . . . . . . . . . . . . 291
C.2.8 Surfacedistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
C.3 Fourierseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
C.3.1 TheFastFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
D Besselfunctions 299
E FreefieldGreen’sfunctions 307
F Summaryofequationsforfluidmotion 308
F.1 Conservation lawsandconstitutive equations . . . . . . . . . . . . . . . . . . . . . . 308
F.2 Acousticapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
F.2.1 Inviscid andisentropic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
F.2.2 Perturbations ofaninviscid non-heat conducting meanflow . . . . . . . . . . 311
F.2.3 Myers’EnergyCorollary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
F.2.4 Zeromeanflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
F.2.5 Timeharmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
F.2.6 Irrotational isentropic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
F.2.7 Uniformmeanflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
F.2.8 Parallelmeanflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
G Answerstoexercises. 316
Bibliography 327
Index 339
Preface
Acousticswasoriginallythestudyofsmallpressurewavesinairwhichcanbedetectedbythehuman
ear:sound.Thescopeofacousticshasbeenextendedtohigherandlowerfrequencies:ultrasound and
infrasound. Structural vibrations are now often included in acoustics. Also the perception of sound
is an area of acoustical research. In our present introduction we will limit ourselves to the original
definition andtothepropagation influidslikeairandwater. Insuch acaseacoustics isapartoffluid
dynamics.
Amajorproblemoffluiddynamicsisthattheequationsofmotionarenon-linear. Thisimpliesthatan
exact general solution of these equations is not available. Acoustics is a first order approximation in
whichnon-linear effectsareneglected. Inclassical acoustics thegeneration ofsound isconsidered to
beaboundary condition problem. Thesound generated byaloudspeaker or anyunsteady movement
of a solid boundary are examples of the sound generation mechanism in classical acoustics. In the
present course we will also include some aero-acoustic processes of sound generation: heat transfer
andturbulence. Turbulence isachaotic motiondominated bynon-linear convectiveforces. Anaccur-
atedeterministicdescriptionofturbulentflowsisnotavailable.ThekeyofthefamousLighthilltheory
of sound generation by turbulence is the use of an integral equation which is much more suitable to
introducing approximations than a differential equation. We therefore discuss in some detail the use
ofGreen’sfunctions toderiveintegralequations.
Next to Lighthill’s approach which leads to order of magnitude estimate of sound production by
complex flowswealso describe briefly the theory of vortex sound which can be used when a simple
deterministic description is available fora flowatlow Mach numbers (for velocities small compared
tothespeedofsound).
In contrast to most textbooks we have put more emphasis on duct acoustics, both in relation to its
generationbypipeflows,andwithrespecttomoreadvancedtheoryonmodalexpansionsandapprox-
imation methods. This particular choice is motivated by industrial applications like aircraft engines
andgastransport systems.
ThiscourseisinspiredbythebookofDowlingandFfowcsWilliams:“SoundandSourcesofSound”
[55]. We also used the lecture notes of the course on aero- and hydroacoustics given by Crighton,
Dowling,FfowcsWilliams, HecklandLeppington [45].
Amongtheliteratureonacoustics thebookofPierce[183]isanexcellentintroduction availablefora
lowpricefromtheAcousticalSocietyofAmerica.
Inthepreparationofthelecturenotesweconsultedvariousbookswhichcoverdifferentaspectsofthe
problem [16,18,20,40,51,73,90,96,102,116,126,153,168,176,179,227,241].
1 Some fluid dynamics
1.1 Conservation laws and constitutive equations
Influiddynamics weconsider gasand liquids asacontinuum: weassume thatwecandefinea“fluid
particle” which is large compared to molecular scales but small compared to the other length scales
in our problem. We can describe the fluid motion by using the laws of mass, momentum and energy
conservationappliedtoanelementaryfluidparticle.Theintegralformoftheequationsofconservation
aregiveninAppendixA.Applyingtheselawstoaninfinitesimalvolumeelementyieldstheequations
indifferential form,which assumes that thefluidproperties arecontinuous andthat derivatives exist.
Insomecaseswewilltherefore usethemoregeneralintegrallaws.Aconservation lawindifferential
formmaybewrittenasthetimederivativeofthedensity ofaproperty plusthedivergence oftheflux
ofthispropertybeingequaltothesourceperunitvolumeofthispropertyintheparticle[16,176,183,
227,241].
Indifferential form1 wehaveforthemassconservation:
∂ρ ∂ρ ∂
(ρv) m, or (ρv ) m, (1.1)
i
∂t +∇· = ∂t + ∂x =
i
where ρ is the fluid density and v (v ) is the flow velocity at position x (x ) and time t. In
i i
= =
principle we will consider situations where mass is conserved and so in general m 0. The mass
=
sourcetermm can,however,beusedasarepresentation foracomplexprocesswhichwedonotwant
to describe in detail. For example, the action of a pulsating sphere or of heat injection may be well
approximated bysuchamasssourceterm.
Themomentumconservation lawis2:
∂ ∂ ∂
(ρv) (P ρvv) f mv, or (ρv ) (P ρv v ) f mv , (1.2)
i ji j i i i
∂t +∇· + = + ∂t + ∂x + = +
j
where f (f ) is an external force density (like the gravitational force), P (P ) is minus the
i ij
= =
fluidstress tensor, andtheissuing massaddsmomentum byanamountofmv.Insomecases onecan
represent theeffectofanobject likeapropeller byaforce density f acting onthefluidasasource of
momentum.
Whenweapplyequation (1.1)weobtain3 for(1.2)
∂v ∂v ∂P ∂v
i ji i
ρ P ρv v f, or ρ ρv f . (1.3)
j i
∂t +∇· + ·∇ = ∂t + ∂x + ∂x =
j j
1ForconveniencelaterwepresentthebasicconservationlawsherebothintheGibbsnotationandtheCartesiantensor
notation.Inthelatter,thesummationoverthevalues1,2,3isunderstoodwithrespecttoallsuffixeswhichappeartwiceina
giventerm.Seealsotheappendixof[16].
2Thedyadicproductoftwovectorsvandwisthetensorvw (viwj).
3(ρv)t (ρvv) ρtv ρvt (ρv)v ρ(v )v ρ=t (ρv) v ρ vt (v )v.
+∇· = + +∇· + ·∇ =[ +∇· ] + [ + ·∇ ]