Notes on Acoustics

455 Pages·2008·4.24 MB·English

by  Uno Ingard

ON THE CD Notes on
The CD-ROM includes approxiA- COUSTICS
mately 200MB of engineering
Notes on
application software, models,
Uno Ingard
demos, videos, simulations, and
other resources!
Written by a noted authority in the subject area, this book provides an understanding A
COUSTICS
of the fundamentals of acoustics and a solid foundation for problem solving. It may be N
used as a reference by scientists and engineers or as a text in a senior-undergraduate or
o
graduate-level course.
t
e
In addition to the “standard” material on acoustics, several of the chapters contain lesser
s
known areas of the fi eld and hitherto unpublished material. This is the case particularly

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in Chapters 7, 8, 9, and 10 dealing with such topics as fl ow-induced sound and instabili-
n
ties, fl ow interaction with acoustic resonators, atmospheric acoustics, and some aspects
of nonlinear acoustics. The extensive chapter (9) on atmospheric acoustics is a result of
A
the work sponsored by NASA. Chapter 11 is devoted to examples and applications from
Uno Ingard
various parts of the book.
C
KEY FEATURES
■ Written by an international authority and winner of the Rayleigh Medal (Institute of Acoustics)
■ Emphasizes a descriptive presentation to provide “physical insight” and aspects of “problem-
O
solving” without sacrifi cing the importance of mathematical analysis
■ Includes the author’s unpublished material on fl ow-induced sound and instabilities, fl ow
interaction with acoustic resonators, atmospheric acoustics, and some aspects of nonlinear
acoustics U
BRIEF TABLE OF CONTENTS
1. Introduction. 2. Oscillations. 3. Sound Waves. 4. Sound Refl ection, Absorption, and Transmission.
S
5. The Wave Equation. 6. Room and Duct Acoustics. 7. Flow Induced Sound and Instabilities.
8. Sound Generation by Fans. 9. Atmospheric Acoustics. 10. Mean-Flow Effects & Non-Linear Acoustics.
11. Examples. Appendixes. Index. T
ABOUT THE AUTHOR
Uno Ingard is the internationally recognized professor emeritus of physics (MIT) who was awarded the I
Gold Medal of the Acoustical Society of America, the Rayleigh Medal by the Institute of Acoustics (UK), the
C
Education Award of the Institute of Noise Control Engineering, the John Ericsson Medal, American Society
of Swedish Engineers and the Per Bruel Gold Medal of the American Society of Mechanical Engineers.
He has written numerous books, as well as journal articles in the fi elds of acoustics and fl uid physics
that have appeared in the Journal of the Acoustical Society of America, Physics of Fluids, Physical Review, S
Physical Review Letters, Transactions of ASME, and Acustica (Germany); he is a fellow of the American
Physical Society and was awarded membership in the prestigious National Academy of Engineering.
ISBN: 978-1-934015-08-7
U.S. $69.95 / Canada $72.50
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Notes On
ACOUSTICS
May6,2008 15:26 ISP acoustics_00
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Notes On
ACOUSTICS
By
Uno Ingard
INFINITY SCIENCE PRESS LLC
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Preface
Having been involved periodically for many years in both teaching and research in
acousticshasresultedinnumeroussetsofinformalnotes. Theinitialimpetusforthis
bookwasasuggestionthatthesenotesbeputtogetherintoabook. However, new
personalcommitmentsofminecausedtheprojecttobeputonholdforseveralyears
anditwasonlyaftermyretirementin1991thatitwastakenupseriouslyagainfora
coupleofyears.
In order for the book to be useful as a general text, rather than a collection of
research reports, new material had to be added including examples and problems,
etc. Theresultisthepresentbook,which,withappropriatechoiceofthematerial,can
beusedasatextingeneralacoustics. Takenassuch,itisontheseniorundergraduate
or first year graduate level in a typical science or engineering curriculum. There
shouldbeenoughmaterialinthebooktocoveratwosemestercourse.
Muchofthebookincludesnotesandnumericalresultsresultingtoalargeextent
frommyinvolvementinspecificprojectsinareaswhichbecameofparticularimpor-
tanceattheearlypartofthejetaircraftera. Insubsequentyears,inthe1950’sand
1960’s,muchofourworkwassponsoredbyNACAandlaterbyNASA.
Afterseveralchaptersdealingwithbasicconceptsandphenomenafollowdiscus-
sionsofspecifictopicssuchasflow-inducedsoundandinstabilities,floweffectsand
nonlinearacoustics,roomandductacoustics,soundpropagationintheatmosphere,
andsoundgenerationbyfans. Thesechapterscontainhithertounpublishedmaterial.
TheintroductorymaterialinChapter2ontheoscillatorisfundamental,butmay
appeartoolongasitcontainssummariesofwellknownresultsfromspectrumanalysis
whichisusedthroughoutthebook. Asexamplesinthischaptercanbementionedan
analysisofanoscillator,subjecttoboth‘dynamic’and‘dry’friction,andananalysisof
thefrequencyresponseofamodeloftheeardrum.
In hindsight, I believe that parts of the book, particularly the chapters on sound
generationbyfansprobablywillberegardedbymanyastoodetailedforanintroduc-
torycourseanditshouldbeapparentthatinteachingacoursebasedonthisbook,
appropriatefilteringofthematerialbytheinstructoriscalledfor.
As some liberties have been taken in regard to choice of material, organization,
notation, and references (or lack thereof) it is perhaps a fair assessment to say that
the‘Notes’inthetitleshouldbetakentoimplythatthebookinsomerespectsisless
formalthanmanytexts.
In any event, the aim of the book is to provide a thorough understanding of the
fundamentalsofacousticsandafoundationforproblemsolvingonalevelcompatible
v
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vi
with the mathematics (including the use of complex variables) that is required in a
typicalscience-engineeringundergraduatecurriculum. Eachchaptercontainsexam-
ples and problems and the entire chapter 11 is devoted to examples with solutions
anddiscussions.
Although great emphasis is placed on a descriptive presentation in hope of pro-
viding‘physicalinsight’itisnotattheexpenseofmathematicalanalysis. Admittedly,
inclusion of all algebraic steps in many derivations can easily interrupt the train of
thought,andinthechapterofsoundradiationbyfans,muchofthisalgebrahasbeen
omitted,hopefullywithoutaffectingthepresentationofthebasicideasinvolved.
Appendix A contains supplementary notes and Appendix B a brief review of the
algebraofcomplexnumbers.
Acknowledgment.
I wish to thank colleagues and former students at M.I.T. as well as engineers and
scientistsinindustrywhohaveprovidedmuchofthestimulationandmotivationfor
thepreparationofthisbook. Specialthanksgotoseveralindividualswhoparticipated
in some of the experiments described in the book, in particular to Stanley Oleson,
DavidPridmoreBrown,GeorgeMaling,DanielGalehouse,LeeW.Dean,J.A.Ross,
Michael Mintz, Charles McMillan, and Vijay Singhal. At the time, they were all
studentsatM.I.T.
AgrantfromtheDuPontCompanytotheMassachusettsInstituteofTechnology
forstudiesinacousticsisgratefullyacknowledged.
UnoIngard,ProfessorEmeritus,M.I.T.
KitteryPoint,May,2008
May6,2008 15:26 ISP acoustics_00
Contents
1 Introduction 1
1.1 SoundandAcousticsDefined . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 FrequencyIntervals. MusicalScale . . . . . . . . . . . . . . 3
1.1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 AnOverviewofSomeSpecialtiesinAcoustics . . . . . . . . . . . . 3
1.2.1 MathematicalAcoustics . . . . . . . . . . . . . . . . . . . . 4
1.2.2 ArchitecturalAcoustics . . . . . . . . . . . . . . . . . . . . 4
1.2.3 SoundPropagationintheAtmosphere . . . . . . . . . . . . 5
1.2.4 UnderwaterSound,Geo-acoustics,andSeismology . . . . . 6
1.2.5 Infrasound. ExplosionsandShockWaves. . . . . . . . . . . 7
1.2.6 NoiseControl . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.7 Aero-acoustics . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.8 Ultrasonics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.9 Non-linearAcoustics . . . . . . . . . . . . . . . . . . . . . . 9
1.2.10 AcousticInstrumentation . . . . . . . . . . . . . . . . . . . 9
1.2.11 SpeechandHearing . . . . . . . . . . . . . . . . . . . . . . 10
1.2.12 MusicalAcoustics . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.13 PhononsandLaserLightSpectroscopy . . . . . . . . . . . . 10
1.2.14 Flow-inducedInstabilities . . . . . . . . . . . . . . . . . . . 11
1.2.15 Aero-thermoAcoustics. CombustionInstability . . . . . . . 11
1.2.16 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Oscillations 13
2.1 HarmonicMotions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 TheComplexAmplitude. . . . . . . . . . . . . . . . . . . . 15
2.1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 SumsofHarmonicFunctions. Beats . . . . . . . . . . . . . 18
2.1.4 Heterodyning . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 TheLinearOscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 EquationofMotion . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 The‘Real’Spring. Compliance . . . . . . . . . . . . . . . . 23
2.2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 FreeDampedMotionofaLinearOscillator . . . . . . . . . . . . . 26
2.3.1 EnergyConsiderations. . . . . . . . . . . . . . . . . . . . . 26
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2.3.2 OscillatoryDecay . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.3 UseofComplexVariables. ComplexFrequency . . . . . . . 28
2.3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 ForcedHarmonicMotion . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 WithoutComplexAmplitudes . . . . . . . . . . . . . . . . . 31
2.4.2 WithComplexAmplitudes. . . . . . . . . . . . . . . . . . . 32
2.4.3 ImpedanceandAdmittance . . . . . . . . . . . . . . . . . . 33
2.4.4 PowerTransfer . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.5 AcousticCavityResonator(HelmholtzResonator) . . . . . . 35
2.4.6 TorsionOscillator . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.7 Electro-mechanicalAnalogs . . . . . . . . . . . . . . . . . . 37
2.4.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 ImpulseResponseandApplications . . . . . . . . . . . . . . . . . . 38
2.5.1 GeneralForcedMotionofanOscillator . . . . . . . . . . . 39
2.5.2 TransitiontoSteadyState . . . . . . . . . . . . . . . . . . . 39
2.5.3 SecularGrowth. . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.4 BeatsBetweenSteadyStateandTransientMotions . . . . . 40
2.5.5 PulseExcitationofanAcousticResonator . . . . . . . . . . 41
2.5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 FourierSeriesandFourierTransform . . . . . . . . . . . . . . . . . 42
2.6.1 FourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.2 FourierTransform . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.3 SpectrumDensities;Two-sidedandOne-sided. . . . . . . . 45
2.6.4 RandomFunction. EnergySpectraandCorrelation
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6.5 RandomExcitationoftheLinearOscillator . . . . . . . . . 49
2.6.6 ImpulseandFrequencyResponseFunctions;
GeneralizationandSummary . . . . . . . . . . . . . . . . . 50
2.6.7 CrossCorrelation,CrossSpectrumDensity,
andCoherenceFunction . . . . . . . . . . . . . . . . . . . 52
2.6.8 SpectrumAnalysis . . . . . . . . . . . . . . . . . . . . . . . 53
2.6.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7 ThePotentialWellandNonlinearOscillators . . . . . . . . . . . . . 55
2.7.1 PeriodofOscillation,LargeAmplitudes . . . . . . . . . . . 57
2.7.2 Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.7.3 Oscillatorwith‘Static’and‘Dynamic’ContactFriction . . . 58
2.7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 SoundWaves 63
3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.1 TravelingWaves . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.2 TheComplexWaveAmplitude . . . . . . . . . . . . . . . . 65
3.1.3 StandingWave . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1.4 TheWaveEquation . . . . . . . . . . . . . . . . . . . . . . 66
3.1.5 WaveLines . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1.6 TheDopplerEffect . . . . . . . . . . . . . . . . . . . . . . 67
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ix
3.1.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 SoundWaveinaFluid . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.1 Compressibility. . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.2 PistonSourceofSound . . . . . . . . . . . . . . . . . . . . 73
3.2.3 SoundSpeedandWaveImpedance . . . . . . . . . . . . . . 74
3.2.4 AcousticLevels. Loudness . . . . . . . . . . . . . . . . . . 79
3.2.5 HearingSensitivityandEarDrumResponse . . . . . . . . . 82
3.2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3 WavesonBars,Springs,andStrings . . . . . . . . . . . . . . . . . . 85
3.3.1 LongitudinalWaveonaBarorSpring . . . . . . . . . . . . 85
3.3.2 TorsionalWaves . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.3 TransverseWaveonaString. Polarization . . . . . . . . . . 87
3.3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4 NormalModesandResonances . . . . . . . . . . . . . . . . . . . . 90
3.4.1 NormalModesandFourierSeries . . . . . . . . . . . . . . 90
3.4.2 The‘Real’Mass-SpringOscillator . . . . . . . . . . . . . . . 92
3.4.3 EffectofSourceImpedance . . . . . . . . . . . . . . . . . . 94
3.4.4 FreeMotionofaString. NormalModes . . . . . . . . . . . 95
3.4.5 ForcedHarmonicMotionofaString . . . . . . . . . . . . . 95
3.4.6 RectangularMembrane . . . . . . . . . . . . . . . . . . . . 97
3.4.7 RectangularCavity . . . . . . . . . . . . . . . . . . . . . . . 98
3.4.8 ModalDensities . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.5 TheFlowStrengthofaSoundSource . . . . . . . . . . . . . . . . . 100
3.5.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.6 SoundontheMolecularLevel . . . . . . . . . . . . . . . . . . . . . 102
4 SoundReflection,Absorption,andTransmission 105
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.1 Reflection,anElasticParticleCollisionAnalogy . . . . . . . 105
4.1.2 GaseousInterface . . . . . . . . . . . . . . . . . . . . . . . 106
4.1.3 ReflectionfromanAreaDiscontinuityinaDuct . . . . . . . 107
4.1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2 SoundAbsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.1 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.2 TheViscousBoundaryLayer . . . . . . . . . . . . . . . . . 111
4.2.3 TheThermalBoundaryLayer . . . . . . . . . . . . . . . . . 114
4.2.4 PowerDissipationintheAcousticBoundaryLayer . . . . . 118
4.2.5 ResonatorAbsorber . . . . . . . . . . . . . . . . . . . . . . 119
4.2.6 Generalization;ImpedanceBoundaryCondition . . . . . . . 120
4.2.7 MeasurementofNormalIncidenceImpedanceand
AbsorptionCoefficient . . . . . . . . . . . . . . . . . . . . . 125
4.2.8 UniformPorousAbsorber . . . . . . . . . . . . . . . . . . . 126
4.2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.3 SoundTransmissionThroughaWall. . . . . . . . . . . . . . . . . . 132
4.3.1 LimpWallApproximation . . . . . . . . . . . . . . . . . . . 132