New Directions in Linear Acoustics and Vibration: Quantum Chaos, Random Matrix Theory and Complexity PDF
Preview New Directions in Linear Acoustics and Vibration: Quantum Chaos, Random Matrix Theory and Complexity
This page intentionally left blank 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 Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521885089 © 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
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