New Directions in Linear Acoustics and Vibration: Quantum Chaos, Random Matrix Theory and Complexity

286 Pages·5.786 MB·English

by  Matthew Wright, Richard Weaver

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NEW DIRECTIONS IN LINEAR ACOUSTICS
AND VIBRATION
Thefieldofacousticsisofimmenseindustrialandscientificimportance.The
subject is built on the foundations of linear acoustics, which is widely re-
garded as so mature that it is fully encapsulated in the physics texts of the
1950s.Thisviewwaschangedbydevelopmentsinphysicssuchasthestudyof
quantumchaos.Developmentsinphysicsthroughoutthelastfourdecades,
oftenequallyapplicabletobothquantumandlinearacousticproblemsbut
overwhelmingly more often expressed in the language of the former, have
exploredthis.Thereisasignificantnewamountoftheorythatcanbeusedto
addressproblemsinlinearacousticsandvibration,butonlyasmallamount
of reported work does so. This book is an attempt to bridge the gap be-
tween theoreticians and practitioners, as well as the gap between quantum
and acoustic. Tutorial chapters provide introductions to each of the major
aspects of the physical theory and are written using the appropriate termi-
nologyoftheacousticalcommunity.Thebookwillactasaquick-startguide
tothenewmethodswhileprovidingawide-rangingintroductiontothephys-
icalconcepts.
Matthew Wright is a senior lecturer in Acoustics at the Institute of Sound
and Vibration Research (ISVR). His B.Eng. was in engineering acoustics
and vibration, and his Ph.D. was in Volterra series characterization and
identification of nonlinear bioacoustic systems, both from the University
of Southampton. Since then he has worked on flow control for drag and
noisereduction,turbofaninletdesign,aeroacoustictheory,violinacoustics,
andquantumchaosinacoustics,forthestudyofwhichhewasawardedan
EPSRCAdvancedResearchFellowship.Hiscurrentinterestsincludewind
farmnoiseandtheneuroscienceofhearing.HeisaFellowoftheInstitute
ofAcoustics,aFellowoftheInstituteofMathematicsandItsApplications,
a Senior Member of the American Institute of Aeronautics and Astronau-
tics,aMemberoftheAcousticalSocietyofAmerica,andthebookreviews
editoroftheJournalofSoundandVibration.Heteachesmusicalinstrument
acousticsandacousticaldesign.
RichardWeaverreceivedanA.B.degreeinphysicsfromWashingtonUni-
versity in St. Louis in 1971 and a Ph.D. in astrophysics from Cornell
University in 1977. He has been at the University of Illinois since 1981,
after a research associateship in theoretical elastic wave propagation and
ultrasonics at Cornell. He was elected a Fellow of the Acoustical Society
of America in 1996 and received the Hete´nyi Award from the Society for
ExperimentalMechanicsin2004.HeisassociateeditoroftheJournalofthe
AcousticalSocietyofAmerica.

New Directions in Linear Acoustics
and Vibration
QUANTUM CHAOS, RANDOM MATRIX
THEORY, AND COMPLEXITY
Editedby
Matthew Wright
UniversityofSouthampton
Richard Weaver
UniversityofIllinois
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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© Cambridge University Press 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2010
ISBN-13 978-0-511-78996-0 eBook (NetLibrary)
ISBN-13 978-0-521-88508-9 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents
ForewordbyMichaelBerry pagevii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
MatthewWrightandRichardWeaver
1 TheSemiclassicalTraceFormula . . . . . . . . . . . . . . . . . . . . . . . . . . 5
MatthewWright
2 WaveChaosfortheHelmholtzEquation . . . . . . . . . . . . . . . . . . . . 24
OlivierLegrandandFabriceMortessagne
3 TheUnreasonableEffectivenessofRandomMatrixTheoryfor
theVibrationsandAcousticsofComplexStructures . . . . . . . . . . . . . 42
RichardWeaver
4 GaussianRandomWavefieldsandtheErgodicModeHypothesis . . . . 59
MarkR.Dennis
5 ShortPeriodicOrbitTheoryofEigenfunctions. . . . . . . . . . . . . . . . .77
EduardoG.VerginiandGabrielG.Carlo
6 ChaoticWaveScattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
JonathanP.KeatingandMarcelNovaes
7 TransferOperatorsAppliedtoElasticPlateVibrations . . . . . . . . . . 110
NielsSøndergaard
8 MesoscopicsinAcoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
RichardWeaver
9 DiagrammaticMethodsinMultipleScattering . . . . . . . . . . . . . . . . 131
JosephA.TurnerandGoutamGhoshal
10 Time-ReversedWavesinComplexMedia . . . . . . . . . . . . . . . . . . . 146
MathiasFink
v
vi Contents
11 OceanAcoustics:ANovelLaboratoryforWaveChaos . . . . . . . . . . 169
StevenTomsovicandMichaelBrown
12 MesoscopicSeismicWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
MichelCampilloandLudovicMargerin
13 RandomMatricesinStructuralAcoustics. . . . . . . . . . . . . . . . . . . .206
ChristianSoize
14 TheAnalysisofRandomBuilt-UpEngineeringSystems. . . . . . . . . .231
RobinLangley
References 251
Index 271
Foreword
MichaelBerry
H.H.WillsPhysicsLaboratory,UniversityofBristol,Bristol,UK
Intheearly1970s,MartinGutzwillerandRogerBalianandClaudeBlochdescribed
quantumspectraintermsofclassicalperiodicorbits,andinthemid1970sitbecame
clearthattherandommatrixtheorydevisedfornuclearphysicswouldalsodescribe
thestatisticsofquantumenergylevelsinclassicallychaoticsystems.Itseemedob-
vious even then that these two great ideas would find application in acoustics, but
it has taken more than three decades for this insight to be fully implemented. The
chapters in this fine collection provide abundant demonstration of the continuing
fertility, in the understanding of acoustic spectra, of periodic orbit theory and the
statistical approach. The editors’ kind invitation to me to write this foreword pro-
videsanopportunitytomakearemarkabouteachofthesetwothemes.
First, here is a simple argument for periodic orbit theory being the uniquely
appropriatetoolfordescribingtheacousticsofrooms.Thereasonforconfiningmu-
sic and speech within auditoriums – at least in climates where there is no need to
protectlistenersfromtheweather–istopreventsoundfrombeingattentuatedby
radiatingintotheopenair.Butiftheconfinementwereperfect,thatis,ifthewallsof
theroomwerecompletelyreflecting,soundswouldreverberateforeverandgetcon-
fused.Toavoidtheseextremes,thewallsinarealroommustbepartiallyabsorbing.
This has the effect of converting the discrete eigenvalues with perfectly reflecting
wallsintoresonances.Iwillarguethatforrealroomsthewidthofresonancesusu-
allyexceedstheirspacing.Thisisimportantbecauseitcastsdoubtontheusefulness
oftheconceptofanindividualmodeinassessingtheacousticresponseofrooms;a
smootheddescriptionofthespectrumseemspreferable.Butsmoothingisprecisely
what periodic orbit theory naturally describes. When there is no absorption, the
contributionsfromthelongperiodicorbitsmaketheconvergenceofthesumprob-
lematic,frustratingthedirectcalculationofindividualeigenvalues,forexample,in
quantumchaology.Absorptionattentuatesthelongorbits,andtheoscillatorycon-
tributions from few shortest orbits are sufficient to describe the acoustic response.
Butthesefeworbitsareimportant:thecrudestsmoothing,basedsimplyontheav-
eragespectraldensity,obliteratesallthespectraloscillationsandfailstocapturethe
characteristicsofmostrealrooms.
Toassessthesignificanceofabsorption,startfromtheWeylcountingformula
forthenumber Nofmodeswithfrequencieslessthan f,foraroomofvolume L3:
vii
viii Foreword
1.0
0.8
0.6 Poisson
C(S)
0.4
0.2
Wigner
0 0.5 1.0 1.5 2.0
S
ifthespeedofsoundisc=330ms−1,
4πL3f3
N= .
3c3
Inthepresenceofabsorption,modeledapproximatelybyanexponentialamplitude
decaytime T,thatis,intensity∼exp(−2t/T),theresonancewidthcorrespondsto
afrequencybroadening,
1
(cid:3)f = .
2πT
Thus, incorporating the reverberation time T , corresponding to 60-dB intensity
60
reduction, that is, T =T /3log 10, the number (cid:3)N of modes smoothed over by
60 e
thebroadeningis
L3f2
(cid:3)N=6log 10 .
e c3T
60
For estimates, we can choose the frequency middle A (f =440Hz). Then, for
a small auditorium with L=6m, and a reverberation time T =0.7s, (cid:3)N∼23,
60
whichisunexpectedlylargeforsuchasmallroom.FortheAlbertHallinLondon,
where the effective L∼60m, and taking T =2s, (cid:3)N∼ 8,200. These estimates
60
stronglysuggestthatthereislittlesenseinstudyingindividualmodes.
Second,hereisanunusualapplicationofspectralstatisticsfrom1993,inspired
by a visit to Loughborough University, where I talked about quantum chaos and
mentionedthattheideascouldbeusefullyappliedinacoustics.Afterward,Robert
Perrinshowedmehismeasurements(Perrinetal.1983)ofeigenfrequenciesofone
English church bell, ranging from 292.72Hz – the lowest mode, called the hum,
throughthefirstfewharmonics,withtheirtraditionalnamesFundamental,Tierce,
Quint,Nominal,Twister,Superquint–uptothe134thfrequencyof9,285Hz.This