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Spectrum and Properties of Mesoscopic Surface-Coupled Phonons in Rectangular Wires


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SPECTRUMANDPROPERTIESOFMESOSCOPIC SURFACE-COUPLEDPHONONS INRECTANGULARWIRES By STEVENEUGENEPATAMIA ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2001

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Copyright2001 by StevenE.Patamia

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ACKNOWLEDGMENTS Iwouldliketothankmyadvisor,Prof.PradeepKumar,forindulgingmy interestinpursuingthisvestigialproblem.Iwanttoacknowledgepublicallythat Iunderstandtherisktobothourcareerst hataccompaniedthecontinuedpursuit ofafundamentalandyetelusiveproblemwhich,despiteitssigni“cance,hasbeen declaredwithoutsolution. IwanttothankDr.AlbertMiglioriandtheLosAlamosNationalLaboratories forsupportingthecompletionofthisdi ssertationandforcon“rminginamaterial waythattherewerephysicistsandscienti“centerprisesforwhomtheresultstruly mattered. Iwanttothankmychildren,MandyandSarah,fortheirtoleranceofmy determinationtopursueaPhDlateinlife. Onthechancethatshewillsomedayreadthispage,IwanttoexpressappreciationtoMichelleforreasonsonlyshewillknow. iii

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TABLEOFCONTENTS ACKNOWLEDGMENTS......................iii ABSTRACT............................vi CHAPTERS 1INTRODUCTION........................1 1.1NatureoftheProblem.........................1 1.2History..................................3 1.3AcousticPhononsinQuantumWires.................10 1.3.1Connement ...........................10 1.3.2MesoscopicElectron-PhononInteractions...........12 1.3.3ConnedAcousticPhononsandtheDeformationPotential.15 2ASSUMPTIONSANDCONVENTIONS..............19 2.1PhysicalModelandCoordinateSystem................19 2.2SymbolicConsistenciesandAdoptedTensorNotation........21 2.3LinearElasticityTheory........................22 2.4FreeBoundaryConditions .......................27 2.5NonseparabilityofBoundarySolutions ................30 3THEORETICALASPECTSOFRE CENTNUMERICALMETHODS.33 3.1StationaryLagrangian.........................33 3.2NumericalApproximation.......................39 3.3PartitioningtheProblemintoParityGroups.............42 4MATHEMATICALSTRATEGY..................45 4.1NotationandFunctionExtensionIssues...............45 4.2DeningtheBasisandSuperpositions.................49 4.3DimensionlessRepresentations.....................54 4.4DerivationofRayleigh-LambEquation................55 4.5HowtoTransformSuperpositions...................62 iv

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5DERIVINGNORMA LMODESOFPROPAGATION........70 5.1GeneralConsiderations.........................70 5.2AcousticPoyntingVectorofaNormalMode.............71 5.3PropagatingModesInvolving Hy, HzShear.............77 5.3.1DerivingtheFrequencyEquations...............77 5.3.2Interpretation..........................87 5.3.3ModeDispersions........................91 5.3.4CommentsontheProcessofMappingModeDispersions..98 5.4Disposingofthe Hx=0Possibility.. ................99 6 K =0MODESOFARECTANGULARWIRE..........102 6.1 k =0BoundaryConditions ......................102 6.2Uncoupled(Separable) k =0Modes.................103 6.3UncoupledModesnottheLimitofPropagatingModes.. .....107 6.4Coupled(nonseparable) k =0Modes.................112 6.4.1DerivationAllParityFamilies. ................112 6.4.2ManifestationsByParityFamily...............117 7FRACTALPHASESPACEOFCOUPLEDMODESAT K 0...125 7.1Motivation:LowTemperatureHeatConductance ..........125 7.2EectiveDimension&DensityofPropagatingModesas k 0...129 7.3LowTemperatureHeatConductance. ................135 8CONCLUSION........................140 REFERENCES..........................142 BIOGRAPHICALSKETCH....................146 v

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFul“llmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SPECTRUMANDPROPERTIESOFMESOSCOPIC SURFACE-COUPLEDPHONONS INRECTANGULARWIRES By StevenEugenePatamia December2001 Chairman:PradeepKumar MajorDepartment:Physics Thisdissertationpresentsoriginalanalyticalderivationsofthepropagating modesofcoupledmesoscopicphononsinan isotropicrectangularwirewithstressfreesurfaces.Incidentaltothederivations,novelconsequencesofthederived cutomodesarepresentedastheyaect thelow-energyheatconductanceof suchwires,orindeedanypropertythatdependsuponthedimensionalityofthe phasespacewithinwhichthemodesreside.Owingtononseparabilityofthe free-surfaceboundaryconditions,ananal yticdescriptionofcoupledmesoscopic modeshasheretoforebeenpresumedtobe underivable.Resultspresentedherein showthatthemodespectrumofcoupledmesoscopicphononsisbothsubtleand rich,butconsiderablesuccessintheiranalyticderivationisachieved.Using numericalmethodsdevelopedforresona nceproblems,atleastonecontemporary researcherhaspurportedtoexhibitthelowestdispersionbranchesofpropagating mesoscopicphononmodesinGaAs…which isnotisotropic.Theaccuracyofthese brancheshasnotbeenmeasured,buttheybearaqualitativeconsistencywith isotropicmodesderivedherein.Sincebeforethebeginningofthe20thcentury, analyticalsolutionshavebeenknownforthein“nitethinplateandeventhe vi

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caseofwaveguideswithcircularcrosssections.Solutionsforthesespecialcases taketheformoftranscendentalrelationsamongthewavenumberandboundary parameters,buttheunderlyingwavefunc tionsareseparableinthecoordinates. Theanalyticalresultspresentedhereinforthegeneralrectangularcaseinvolve nonseparablesolutionswhoseseparabl ecomponentsdonotindividuallysatisfy theboundaryconditions.Thesesolutio nsalsotaketheformoftranscendental relations,buttherearesetsoftranscende ntalrelationsforeachfamilyofthe casesthatpartitiontheproblem.Consequently,theeigenspectrum,whilede“ned byexactforms,mustbeenumeratedbyide ntifyingplottedintersectionsofthe rootfamiliesofthesetranscendentalrela tions.Theresultingspectrumsaremore complexandhavelessapparentordert hanthespectrumproducedusingeither periodicboundaryconditionsorrigidbo undaryconditionsforuncoupledphonons. vii

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CHAPTER1 INTRODUCTION 1.1NatureoftheProblem Despitesomeprogressinexhibitingthelow-lyingbandstructureofmesoscopicphononsbynumericalmethods,t hestudyofphenomenainvolvingphonon interactionshasbeenhamperedbyanin abilitytoanalyticallyderivephonon modesinvolvingsurfacecouplingwithi nrectangulargeometries[1].Inunbounded bulkmedia,long-wavelengthphononsareadequatelyrepresentedbyindependent familiesofelasticplanewavesdistinguis hedbypolarization.However,atnonrigid surfaces,elasticboundaryconditionsc annotingeneralbesatisedexceptbya coupledsuperpositionofshearandlongitudinalvibrations.Insomegeometries particularlythoseinvolvingedgesandcor nersthissurfaceconstraintcannoteven besatisedbyseparablewavefunctionsletaloneaderivablesuperpositionofsimple planewaves.Reectingthisdiculty,noge neralanalyticalsolutionhasheretofore beenpublishedfortheeigenvaluesorwavefunctionsofphononsinamesoscopic wireofrectangularcross-section[24]. Ascontemplationshiftsfrombulkmat erialtoboundedsampleswithelongated geometries,themanifestationoflong-wa velengthphononswillincludetorsionand exingofthesample.Bulkphononmodelsb asedonlongitudinalscalarpotentials aloneareintrinsicallyincapableofexhibitinganyexuralortorsionalbehavior sincetheserequirebothsomemechanismforshearandspecickindsofdistortion ofaspeciedsurfacegeometry.Torsionalandexuralbehaviorcanbemodeledto someextentusingbendingmoduliandmini mizingthepotentialenergyassociated 1

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2 withthedistortionofniteobjects(See,e.g.,[5] 4.12).Yet,correcttorsional andexuralmodesoughttobeautomati callyincludedwithinacomprehensive phononmodelwhichcorrectlyidenties phononsasmixedlongitudinalandshear vibrationswhichsatisfynonrigidbounda ryconditionswithinaspecicgeometry. Inmodernterms,thenatureoftheproblemcanbesuccinctlydescribedin thefollowingusefulway.Withinthelim itsoflinearelasticity,phononsinbulk aredistinguishedbypolarizationdirectionandpropagationvelocityandtheydo notinteract.Atanynon-rigidsurface, thereductionofexternalstresstozero isaboundaryconditionthatcanbesatisedingeneralonlyifthetwospecies ofmodesbecomecoupledatthesurface.Ineect,eachspeciesscattersintothe otheruntilanenergytransferbalanceisachieved.Adetailedexaminationofthe boundaryconditionsclariesanimportant featureofthiscoupling.Namely,the nondegeneratelongitudinalphononspola rizedalongtheirpropagationdirection (Iamignoringquasi-shearandquasi-lo ngitudinalsituationswhichcanoccurin nonisotropicmaterials)onlycoupletos hearphononswhichhaveapolarization componentnormaltothesurface.Thesurfaceinteractionthatarisesasthesurfaces arepermittedtodistortactsasaperturbat ionthatsplitsthenaturaldegeneracyof theshearphonons.Shearphononspolarized paralleltothesurfacearenotaected andreectspecularlysubjecttotheirdisplacementsvanishingatthesurface.With respecttothephononbandstructure,thisde generacybreakingismanifestaslevel repulsionsamongphonondispersionsubbands. Itshouldimmediatelybenotedthatdegenerateperturbationtechniquesare notfruitfulinderivingsolutionsfort hecoupledphononproblem.Thereasonfor thisisthatsuchanapproachdependsuponanabilitytondeigenfunctionsofthe perturbationitself.Pursuingthisinevit ablyencountersthefundamentalsourceof mathematicaldicultywhichplaguestheproblemgenerally.Namely,solutions

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3 thatincorporatethefullboundaryconditi onsatadjacentsurfacesinrectangular geometriesareeasilyshowntobenon-separable.Deningaperturbationthat reectscouplingofadjacentsidesandtheni dentifyingabasisforitseigenfunctions becomeintractable. 1.2History Earlyinthehistoryofelasticitytheor y,vibrationalmodeswhichaccompany fullrelaxationofappliedsurfaceforces (formally,astheprojectionofthestress tensorontoavectornormaltothesurfacevanishes)weresuccessfullyderivedfor certaingeometries.Asolutionwasrstobt ainedforcircular,innite-lengthbars byPochammer[6]in1875andindependentlybyChree[7]in1886.In1889Lord Rayleigh[8]publishedtheanalogoussolutionforaninniteplatewithstressfree surfaces. Intheforegoingcases,thesolutionsobtainedtooktheformofso-called frequencyequationswhichdeneatranscendentalrelationshipamongshear andlongitudinalwavevectorcomponents.Therootsoftheseequations,foreach propagationwavenumber,constitutetheeigenspectrum.Reectingthediculties inherentinanalyticallyexposingtheroo tsofsuchtranscendentalrelations,only someasymptoticrootsofthePochammer-Chreefrequencyequationsaccompanied themimmediately,andasymptoticsolutionstoRayleighsplatesolutiondidnot appearuntilitwasrevisitedbyLamb[9]in1917.AsLambexploresasymptotic rootsandthedisplacementfunctions,hemakesperhapstherstobservationthat, atincreasingpropagationwavenumbers,twoofthefundamentalmodesconverge toformRayleighSurfaceWaves.Oversucceedingdecades,frequencyequations forplatesandrodswereextensivelystudiedandnumericallyderivedrootsofthe

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4 underlyingtranscendentalrelationsareex tensivelycharacterizedandwellknown inengineeringapplications[10]. Itmightseemsurprisingthatdespitedicultiesidentiedabove,thatthe frequencyequationforcircularwaveguideswasfoundatall,letalonebefore, thepublishedanalogoussolutionfortheinniteplate.Thereasonisthata cylindricalcoordinatesystemreadilyacco mmodatesthefactthatsurfacecoupling onlyinvolvesshearwaveswhosepolarizationsresultindisplacementsnormalto thesurface.Forcylindricalenvironments,thesheardisplacementeldisnaturally decomposableintoaradialcomponentcoupledtothelongitudinaleldanda componentparalleltothesurfacewhich isnot.Bycontrast,forsurfaceswith sharpcorners,onlyanunobvioussuperp ositionofwavesforms(notnecessarily plane)willaccommodatecouplingatadjacentfaces.Makingthegeometrynite whetherwithatorroundedendsfurtherco mplicatesthisdiculty,aswasnoted evenbyChree. Oncethewaveguidetakesonarectangu larcrosssection,theresultantboundaryvalueproblemdevelopspathologieslinkedtothefactthatsolutionsaregenerallynonseparableinthecoordinates.Thelackofananalyticalsolutiontothe rectangularcrosssectionproblemhasconsequentlybeenwidelyconcededinthe literature(seee.g.,[24])Inviewofthispe rceivedlimitation,approximatesolutions forvarioussubsetsofthemodesforabar havebeendevelopedTimoshenko[15] beamtheorybeinganimportantexampleintherealmofengineeringapplications. Intheabsenceofdirectanalyticalsolutionsfortherectangularcrosssection,it remainedpossibletoapplyavariationala pproachtogeneratesolutionsthatcoincidentallysatisfytheboundarycondition s.Thepossibilityofthistacticemerges intheliteratureatleastasearlyas1966whenMedick[16]attemptedtodevelopa

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5 1-dimensionalwavepropagationtheoryforrectangularbars.Medickreviewedthe importantresult[17,18],alreadythenadecadeold,thatsolutionswhichrender theLagrangianstationaryarepreciselythosewhichsimultaneouslysatisfythe bulkwaveandnaturalboundarycondition s.Medick,however,didnotapplythe variationalprincipledirectlytonumericalcomputation. Thebasicobservationthattheeigenfrequenciesofafreelyvibratingelasticbody arestationaryinthespaceoffunctionswasdeducedearlyonbyLordRayleigh whoincorporatedi ti nt ohistreatise The T heory of Sound [19]in1877.Thisresult isusuallyreferredtoasRayleighsPrinciple[20].NotethatusingRayleighs Principletosupportavariationalsolutioncanbedistinguishedfromapproaches discussedwithinthatrelyexplicitlyuponstationarityofthetime-independent LagrangianasamanifestationofHamiltonsprinciple. Earlydirectapplicationofavariationa lprincipletocomputethelow-order modesofparallelepipedswaspresent edbyHolland[21]in1968.Beginningwith stationarityofthetime-independentLagrangian,HollandarticulatesanessentiallyRayleigh-Ritzapproachwhereinnitecombinationsoftrialfunctionsare minimized.Hollandsparticularinterestisinpiezoelectricmaterials,buthis computationsarenotinanywayrestrictedbythis.Hollandexaminestheresulting eigenfunctionsinanattempttodescribethedisplacementpatternofthemodes themselvesalongwiththespectrum.Hollandstrialfunctionsforcomponents ofthedisplacementeldareproductsofsinesandcosinesofthecoordinates. Later,Demarest(below)believedthatsuchchoicescouldprecipitateinaccuracy intheresultsofavariationalcalculatio n.Inrecognitionoftheimportanceofthis convergenceissue,HollandandEerNisseexaminethisfactorcriticallyinalater monograph[22].

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6 BuildingonHollands1968result,Demarest[23]in1971improvedtheaccuracy ofHollandsmodecomputationsforafreesurfacecubeinpartbychangingthe natureofthetrialfunctions.Thetrial functionsofDemarestsownRayleighRitzexpansionareproductsofLegendreP olynomialsofthecoordinates.Along withidentifyinganinadequacyinHolla ndschoiceoftrialfunction,Demarest makestheobservationthatwhileLegendr ePolynomialsexhibitacomputationally usefulorthogonalityintherelevantspace,inprincipletheycouldbereplacedby apowerseriesinthecoordinatesofwhichLegendrePolynomialsarejustalinear combination. NeitherHollands1968papernorDemarests1971paperderivethegeneralrelationshipbetweensatisfactionoftheboundary-valueconditionandtheecacyof thevariationalprincipleutilized.Sincetheyareabletodemonstrateasatisfactory correspondencebetweenactualmeasurementresultsandthenumericalpredictions, theyappeartoconsiderthetheoretical ,albeitimportant,questionofboundary conditionstobemoot.Demarestinsteadfocusesonthefactthatderivativesofhis trialfunctionsconstituteamorerelativelycompletesetofbasisfunctionsthan Hollands.Bymorerelativelycomplete,hemeansthatlinearcombinationsof selectionsfromthisnitebasisareabletoconvergetoabroaderrangeofpossible functions.Holland,asDemarestpointsout,manipulateshistrialfunctionchoices tofavorconvergenceoftheirderivativestozeroattheboundary.Demarests choiceofLegendrePolynomialsenrichesthebasissothatconvergencebothto zerodisplacementsandzeroderivativesofdisplacementarepossible.Whilethis issucienttoimprovetheaccuracyofnumericalresultsascomparedagainst measurements,itcannotbedecisive.Theactualboundaryconditionswhich willbeexploredhereincannotbesimplypartitionedintocasesinvolvingzero displacementsversuszeroderivativesofdisplacements.Moreover,inmostcases

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7 unobvioussuperpositionsareneededto assembleboundarysolutionswhichare provablynonseparableandthenatureoftheirboundarybehaviorisimpossible todiscerninadvanceofactuallyexhibitingthem.Whateveritsincremental improvement,Demarestschoiceofbasisalsothwartstheexpectationthatthe nalsolution,andpresumablyitscomp onents,mustbeconstitutedofnomore thantwopartswhichseparatelysatisfyabasicwaveequationasisprovablefrom ElasticityTheory. BothHollandandDemarestutilizeaRayleigh-Ritzkindofprocedurewhich approximatesthesolutiontoaneigenva lueproblembyndingstationarypoints amongacombinationoftrialfunctions.TheconventionalRayleigh-Ritzprocedure isimplementedusingtrialfunctionswhichindividuallysatisfyapplicableboundary conditions(seeanytextonmathematicalphysics,e.g.,[24] 17.8).This,however, isnotpossibleundercircumstanceswher etherearenoavailablebasicsolutions forthegeometrythatareknowntosati sfythoseboundaryconditionsintherst place.ItthuswasopenlyexpressedbyVisscher,MiglioriandB ell[25]in1991to befortuitousthatdisplacementfunctionswhichcouldbefoundtominimizethe time-independentLagrangianwouldautomaticallysatisfythebulk-waveandfreeboundaryconditionssimultaneously.AsIwillpointoutmoreparticularlywithin, thisfortuityislessdramaticallyusefulthanitsdiscoveryinfers.Thereasonisthat itisonlyashelpfulasthesetoftrialfunctionsisversatileandisliterallytrueonly ifthesetoftrialfunctionshappenstoincludeacombinationthatisasolution. Again,thecentralimpedimentisthatbasisfunctionswhichatleastsatisfythe boundaryconditionsdonotevenexist.Vi sscheretal.,therefore,concentrated uponassemblingabasischosenforitsversatilityandtheconveniencewithwhich numericalcomputationscouldincorporate itselements.Infact,thebasiselements theychoose,whilelinearlyindependent,arenotorthogonal,anddonotevensatisfy

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8 awaveequationletaloneboundaryconditi ons.Remarkably,atleastintermsof resonancesofclosedobjects,thebasisisneverthelesssucienttoproduceuseful resultstoasometimeshighdegreeofaccuracy[26]. Possiblybecauseengineeringandphysicsc ommunitiesdonotconsistentlyoverlap,Visscheretal.wereprobablynotawarethatMedickhadpreviouslynoticed thesamefortuityaquarter-centurybeforeandhadbeenreviewingresultsavailable adecadebeforethat.Nevertheless,Visscher,MiglioriandBelloeranextremely elegantderivationoftheprincipleandclarifygreatlythetheoreticalunderpinnings ofthealgorithmdevelopedbyHollandandDemarest.Moreover,equippedwith accesstosomeoftheearliesttoolstobearthedesignationsupercomputer, Visscheretal.testeddirectlytheecacyofaparticularlyusefulandelegant choiceofbasisfunctions.Specically,the yimplementDemarestssuggestionthat productsofpowersofthecoordinatesshouldconstituteasetofbasisfunctions thatwouldbesucientlycompletetoc onvergetoaccuratesolutionsatleast forthefrequencyeigenvalu esinresonanceproblems. LikeHollandandDemarest,Visscheretal.weremotivatedbytheneedtond accuratelow-ordereigenmodeswhichcouldbeveriedbycomparisontoactual measuredresonancesofsamples.Theyshowedthatanalgorithmemployinga basisbuiltfromsimpleproductsofpowersofthecoordinateswouldaccomplish thisevenforgeometricshapesfarmorech allengingthentheparallelepipedsof DemarestandHolland.Subsequently,Miglio rihasperfectedthistechniqueinto anaccuratestandardmeansofndingthemic roscopicelasticconstantsofmaterials havingarbitrarycrystalorcompositestructure[26]. Thedirectnessofcomputabilityandbroad applicabilityofthecomputational algorithmrenedbyVisscher,Migliori,andBellhasrecentlybeenadaptedand appliedtoexhibitthemodesofaninniterectangularwire.Throughthesimple

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9 expedientofreplacingonecoordinatef actorofthebasisfunctionswithacommon periodicfunctionofthelongcoordinatealone, i.e., xpyrzq xpyreiqz, Nishiguchi,AndoandWybourne[1]resolvedthetransversecrosssectioncomponentsusingthesamevariationalmethodsasVisscheretal.whileretaining translationalinvariancealongthelength oftheinnitewire.Insteadofobtaining anitesetofresonanteigensolutionsforaresonator,heobtainsanitefamilyof solutionsforeachvalueof q inawaveguide.theythengeneratethedispersionrelationsalongthe z -axisbyplottingtheresultingsolutionswhichremaincontinuous intheparameter q .Sincethewiresareinnitelength,therearenoconsiderations ofmixingfromend-eects.Thereisalsononeedforthevariationalalgorithmto resolvethemodalpatternalongthelongdi mension.Asaconsequence,convergence islimitedonlybythealgorithmsabilityt oresolvethetransversedisplacements whosemodestperiodicityoveralimitedrangeismorereadilyapproximatedby anitepowerexpansion.TheirspecicresultsarecomputedforGaAs,butas Visscheretal.hademphasized,thereisnothinginthenatureofthenumerical algorithmwhichinherentlylimitsittoanymaterialorcrystalstructure. ThemodedispersionsdevelopedbyNishiguchietal.lackcomparisonwith measurement.Directlymeasuringtheeigenfrequenciesofaresonatorisstraightforward,butdirectlymeasuringthelong-w avelengthsubbanddispersionsofphonons subjecttoaspecicgeometricboundarymaynotbe.Thetechniqueofnding bulkphonondispersionsusingneutrons catteringcannotbeeasilyappliedtoa geometricallyconnedlow-wavelengthenvironmentparticularlyifthesampleis microscopic.Also,fullresonancesarepr ecisebydenitionandrepresentaseverely constrainedcomputationalproblem.Itisnotobviouswhetherasetoflinearly

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10 independent,butnotinherentlyorthogonal,trialfunctionsremainsimilarlyable toconvergestronglytoasolutionwhereth ebasisisconstrainedconceptually,but notactually,alongonedirection.Thesec oncernswillbefurtheraddressedinthe resultspresented. Ingeneralcontemporaryresearch,mesoscopicacousticphononshavebeen investigatedtheoretically,andtosomeextentexperimentally,inprimarilytwo kindsofsituations.The“rstisinrespectoftheirinteractionswithphononsin quantumwiresandthesecondistheirroleascarriersofthermalenergy.The twotopicshavesomeoverlap.Inthenextsection,Iwillsummarizethe“ndings withrespecttoelectronicinteractionsinquantumwiresandinCh.7Iwilltake upthemoreemergenttopicofkineticheattransportbyacousticphononsinthe low-temperaturehigh-con“nementregime. 1.3AcousticPhononsinQuantumWires 1.3.1Con“nement Fromtheendofthe1980sandcontinuingintothepresent,mesoscopicphysics hasenjoyedanobviousprominencedrivenlargelybytherelentlessdownscalingof electronicdevices.Commercialapplicat ionsaside,theemergenceofanability tofabricatedeviceswithinwhichquantumbehaviorbecomesobservableasa consequenceofdimensionalcon“nementitse lfstimulatespersistentandwidespread interestsintheeectsofsmallsizeandlowtemperatures.Certainly,today,there isnomoreubiquitousdeviceinthetheore ticalandexperimentalcondensedmatter inventoryasthequantumwire.Ž Inthiscontext,so-calledphononco n“nementŽhasalsobecomeaphenomenon ofspeci“cinterest.Phonons,beingacla ssicalratherthanq uantumphenomenon

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11 perse,arenotsubjecttothesharpdimensionalcon“nementspossibleforquantum particles,likeelectrons.Whencharacter izingphononsascon“ned,Žauthorsare onlydistinguishingphononmodessubjecttoquantizationbyphysicalboundaries frombulkphononsquantizedusingperio dicboundaryconditions.Thereisno inferredlossofdimensionalityofthepho nons,though,ofcourse,thelowestsubbandstypicallyhaveimaginarytransversewavenumbersandhighersubbandsmay sometimesbecomeenergeticallyinaccessible. Whilecon“nementŽatdrasticallysm allscalescanleadtoactualdimensionalreductionofquantumparticles,i tsgenericeectistorenderthephysical boundariesrelevant.ThesignaturefeatureofwhatiscalledmesoscopicŽremains thecommensuratenessofthecharacteristicwavelengthwiththedistancebetween boundaries.Forphonons,however,actuals mallscalesandlowtemperaturestend tomagnifythedeparturesfrombulkbehav iorprovokedbymesoscopicconsiderations.Certainly,atanyscale,mesoscopicphononsexistsincesomesubsetofthe phononspectrumconsistsofphononsatwa velengthscomparabletothesizeofthe physicalenvironment.Atsmallscalesandlowtemperatures,however,mesoscopic phononsarethedominantonesandtheiractu alfrequencies,whichscaleinversely withoverallsize,becomelargeenought oaugmenttheprobabilityofelectronic interactions. Mesoscopicphononsare,byde“nition ,ofawavelengthlargeenoughthat surfaceeectsandgeometrycontroltheirdispersionandwavestructure.Periodic boundaryconditionsandplanewavemodelinggivewaytoexplicitboundary solutionsforspeci“cgeometries.Surfa cecouplingendemictothenon-rigidboundaryconditionsdeterminestheeigensp ectrumandmixeslongitudinalandshear contributions.

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12 1.3.2MesoscopicElectro n-PhononInteractions Absentananalyticalsolutionforsurfa ce-coupledphonons,researchintocon- nedphononinteractionsprogressedbyu singapproximationsorroughmodels whichwerebelievedtomimicthemostimportantfeaturesofactualphonons. Abriefsurveyoftheresultsproducedinthismannerthusservestoidentify conclusionsandcalculationswhichshoul dberevisitedinthelightofanalytical resultsfromthisinvestigation. Bythe1990s,papersexplicitlycalculatingtheinteractionofelectronsin aquasi-one-dimensionalgas(i.e.,in aquantumwire)withconnedphonons wereappearingregularly[27,28].Initialinvestigationsfocusedprimarilyonhotelectronrelaxationviaemissionofco nnedopticalphonons.Thephononmodels utilizedduringthistimeweretypicallyuncoupledsymmetricandantisymmetric transverseresonanceswherethetransversecomponentsweretakeneithertobe zero(denominatedguidedmodes)ora maximaattheboundaries(denominated slabmodes).Thelatterconditionisactu allysimilartostress-freeconditionsfor elasticphonons.Resultsfromthetwoa vorsofboundaryconditionsarecompared. Amongtheinterestingconclusionsisthatboundaryconditionsresemblingstressfreesurfacesleadtoanorderofmagnitudefasterrelaxationratethanobtainedby assumingzerodisplacem entboundaries[27]. Longitudinalopticalphononsinpolars emiconductorswilldecaytoacoustic phonons(seeRef.[29] 6. 2)withinsomeaverageLOlifetimewhichcanactuallyaugmentdissipationbyvirtueofprecludingLOphononsintheprocessof decayfrombeingabsorbedbackbythecarriers.However,atlowtemperatures inquantumwires(e.g.,below30Kfora100 AwideGaAswire)hot-electron scatteringtoLOphononsbecomesexponen tiallyweakanddirectacousticphonon

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13 emissionistheonlyimportantdissipativeprocess[30]thoughforquantumwire wells,LOscatteringmaycontinuetodominateintolowertemperatureregimes[31]. Earlyinthisperiod,theoreticalinterestinhotelectronrelaxationtoacoustic phononsinwireswasencouragedbyexperimentaldevelopments.In1992Seyler andWybournepublishedaPRL[32]reportingonthedet ectionofr esonanceswith presumablyacousticmodesinsmall(approximately20nmthickandbetween 30and90nmwide)Au-Pdwiresoverabroadrangeoflowtemperatures(120K).Forthebalanceofthedecadethroughtothepresent,thisparticular experimentalobservationisamongthemostconsistentlycitedasjusticationfor furthertheoreticalexplorationofthe importanceofacoustic-phononconnement inreduceddimensionalelectronicstructures[4]. Theessentialroleofacousticphononsinenergydissipationwithinquantum wireshasbeenscrutinizedinvariousways.SennaandDasSarmahaveinvestigated aGiantMany-BodyEnhancementofelectron-acousticphononcouplingatlow temperaturesbyrenormalizingthephonons inthepresenceoftheelectrongas[33]. Theyndthatatlowtemperature(viz.1K),belowwhichdirectplasmaresonance cannotbesignicant,thequantum-mechani cal(whichtheydistinguishfromthermal)uncertaintyinthephononmodescreatesaninteractionenhancementorders ofmagnitudegreaterthanforbarephonons.Mickevi cius,Mitin,andKochelap usedMonteCarlocalculationstoinvestigatephononradiationfromaQuasi-1D electrongasinrectangularGaAsquantumwiresof80x80 Aintheneighborhood of4K[34].Theyreportthatvirtuallytheentirepowerdissipationisduetothe transverseradiationofba llisticacousticphonons. AlloftheabovecitedtheoreticalexplorationsinvolveGaAsand/orAlAsas thespecicmaterialofchoice.WhileGaAsremainedanalmostpredominant materialchosenfortheoreticalstudies duringandsincethistime,Sihasalsobeen

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14 explored.Inacomprehensivesetofcalcul ations,Sanders,Stanton,andChangcalculated[35]arangeoftransportpropertiesforaSiquantumwireincludingdeformationpotentialscatteringbyquantum connedphonons.Althoughtheauthors explicitlyreservethetermacoustic forthelowestsubband,thephononmodel usedcorrespondsexclusivelytolongitudi nalelasticwavescoupledtoelectronsvia adeformationpotential.Possiblytoavoidintimatingthathighersubbandsof theseconnedelasticmodesshouldbeca lledopticaltheauthorsdescribethem asbeingexcitedquantum-connedphonons.Inanycase,thefocusinthispaper isonasecond-quantizationrepresentationofthephononswithoutattemptingto mimicspecicsurfaceboundaryconditions. Ina1994study,Yu,Kim,Stroscio,Iafrate,andBallato[4](liberallycitedelsewhereherein)builduponthe1949Ph.D.thesisworkofMorse[36,3]toimplement amorerealisticconnedacousticphononm odelthatincorporatessomesurface couplinginarectangularquantumwireenvironment.UponrecitingthatAsis wellknown,therearenoexactsolutio nsforthecompletesetofphononmodes forarectangularwireYuetal.adoptMorsescompromisestrategywhichwas expectedtobeadequatewhenthecrosssectionalaspectratiowasgreaterthan two.Morsehadmadeprogresssolvingtherectangularproblem(whichIinvestigate moresuccessfullyherein)bytreatingthecl osestparallelsurfacesasanindependent platescenario(seesection4.4herein)andndingwhatamountstoRayleigh-Lamb modesrelativetothosesurfaces.Hethe nobservedthattwoofthethreesurfacenormalstresscomponentsbecomesmallattheadjacentsurfacesastheaspect ratiogrowsandtunesafreeparameterinhisplatesolutiontoforceoneofthe twodiminishedstressestobezero.Morsestranscendentalequationderivedfrom consideringthecloserofthesurfaces(e quation(14)inRef.[4]underdiscussion) isverysimilarto,thoughlessgeneralthan,myownintermediateequation(5.27)

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15 whichappearswithinmyderivationofpropagatingcoupledmodes.Hisstarting assumptionsarefarlessgeneral,butatthatcorrespondingpointofthederivations thesituationisarticiallysimilar.Res trictingthemselvestodilatationalmodes (seecommentsbelow),Yuetal.proceedtocomputeanormalizationofMorses separablesolutionandthencomputeelec tron-phononscatteringfromdeformation potentialinteractionforarangeofcrosssections(principally28x57and50x200 A)at77K.Theiressentialndingisthatscatteringratesarenotablyhigherfor suchconnedcoupledacousticmodesthanforbulkacousticmodesandincrease dramaticallyastheoverallscaleisreduced.Infact,theycomputethatthe scatteringratefora28x57 Awireareanorderofmagnitudelargerthanfora 100x200 Acrosssection. 1.3.3ConnedAcousticPhononsandtheDeformationPotential Hereafter,myownuseofthelabelacoust icwill,absentqualication,encompassallphononsthatareamanifestationo felasticdeformation.Consistentwith muchoftheliteraturebeingreviewed(see,e.g.,Ref[4]discussedabove),this meansthatthesubbandsthatarisefro mrealboundaryconnementofelastic waveswillcontinuetobereferredtoasacousticdespitethefactthatinbulk environmentsthistermrefersonlytoadispersionbranchthatoriginateswith zerofrequencyatzero k .Myconventionisappropriatetothisstudywhereonly connedphononsarerelevant,isconsiste ntwithtextsandmonographsdevoted toacousticsofsolids[5,10,37,38]anditappearsconsistentwithemergenttext booksdevotedtophononsinnanostructures [29].Unfortunately,pastpublications arenotalwaysasinclusive,butintheworstcaseitwillsimplybeimportantto noticewhetheracousticmodesunderdiscussionhavebeenrestrictedtoaspecied dispersionbranchorexplicitlystatedtobebulk-like.Asisconsistentinthe

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16 historicalliteratureofacousticsperse, Iwilldistinguishmodeswhosedispersion goestozeroas k 0bycallingthemfundamental. Withinthemostinclusiveuseofthete rm,acousticphononsarecoupledto carrierssolelybythedeformationpotentialwhichtrackstheiondensityuctuations.Ineachoftheabove-citedarticles,onlylongitudinal(i.e.,compressional) phonons(see,especially,[39])participa teinthiscoupling.Sincebulkshearmode displacementscanbegeneratedbyavectorpotential,thedivergenceoftheir displacementsiszeroandtheycannotaltertheionpotentialeldviauctuations iniondensity. Thereare,nevertheless,circumstanceswhereadeformationpotentialcanbe associatedwithnon-compressionalphonons.Formaterialswithdegeneraciesin bandstructurethatcanbebrokenbyshea rdistortions,itislongbeenknownthat deformationpotentialshouldbegeneralizedtoatensorsuchthatthepotential changeduetodeformationincorporatesshearaswellasdivergenceeects[40,41]. Evenifsuchspecialcircumstancesdonotapply,thereremainsamorefundamental reasontorevisittheexclusivefocuson compressionalmodesoncethephononsare subjecttoboundaryconnement. Inthecontinuumlimit,acousticphonons aremerelyelasticmodesirrespective ofwhethertheenvironmentisconnedo rbulk-like.Thisstudyisconcerned, however,exclusivelywithelasticmode sconnedtorectangularwaveguides.As willbedevelopedindetailinsubsequent chapters,elasticwaveguidemodescan bepartitionedwithrespecttotheparitypatternsoftheirdisplacementsrelative toacoordinatesystemsalignedwiththelongaxisandnormaltothesides. Forthefundamentalmodes(i.e.,thoseconnedbrancheswhichdogotozero as k 0)thesepatternscorrespondtomac roscopicmotionofthewaveguide dilatational,torsional,andexural.Ofthese,thedilatationalissonamedbecause

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17 thedisplacementpatternofthefundamentalmodesresemblespropagatingdensity uctuationsonthescaleofthewidthofthewaveguide.Itisnodoubtforthis reasonthatasphononmodelsinquantumwirestakeboundaryconnementinto account,thereisanexplicitassumptionmadethatonlythepartofthemodel correspondingtodilatationalmodesneedbeconsidered.Torsionalandexural modescorrespondtotwistingandbendingandappear,onrstimpression,tobe dominatedbyshearandareassumedtohavenosignicantdeformationpotential (asassumedi nRef.[4]). Transferringallegiancefromlongitudi nalbulkmodestodilatationalconned modespromotesasignicantoversight.Inbulkmedia,torsionandexuraldisplacementpatternsdonotevenexist,but oncethephononsareconnedbyactual surfaces,thesecategoriesarisepreciselybecausesurfacecouplingmixeslongitudinalandshearcontributionsintothedispla cementpattern.Ignoringthetorsionand exuralmodesperhapsforlackofanana lyticalmodelforthesurfacecoupling comesatthecostofignoringdenitesour cesofdensityvariation.Aggravatingthe potentialconsequencesofthisoversight,itwilllaterbeseenthatthebranchshape ofsomefundamentaltorsional,alongwit hthefundamentalexuralmodes,shows ahighdensityofstatesatlowfrequencies. Anyonewhohaseverwrungoutawashragr ealizesthatfundamentaltorsional modesinvolvepatternsofchangeinlocaldensity.Bendingafoamrubberobject createsaclearopportunitytoobservethe compressionsandstretcheswhichaccompanythatmovement.Consistentwiththesereal-lifemacroscopicobservables,the derivationspresentedinsubsequentchapte rsrevealthatthelongitudinalpotential makesanecessarycontributiontovirtuallyeverycoupledmode.Whileitmay turnoutthatsomesubsetofmodeswithinasubbandaredominatedeitherby shearorlongitudinalpatterns,thereisnoapriorijusticationforassumingthat

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18 anycategoryofcoupledmodescanbesafe lyignoredevenifonlythedeformation potentialfromdensityuctuationremainsrelevant.EvenDemaresttookcareto pointoutthatthedilatational,torsional,andexuralnamingattachedtoparity patternsshouldnotbetakentooseriouslyaseachcontainedelementsofshearand dilatation[23].

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CHAPTER2 ASSUMPTIONSANDCONVENTIONS 2.1PhysicalModelandCoordinateSystem Iassumethatthewaveguideiscomposedofanisotropicmaterialformed intoaninnitebarwhoserectangularcross sectionisinvariantalongitslength. Thestinesstensorforanisotropicmaterialhasonlytwoindependentelastic constantsandthese,togetherwiththematerialdensity,determinethebulkshear andlongitudinalvelocities,denoted csand crespectively[5].Tokeeptheresults completelygeneral,Iwillreduceallderi vedobservablestoadimensionlessform rescaledrelativetotheshearvelocity,thesmallerofthehalf-widthsofthecross section, h ,andtheaspectratio a betweenthewidths.Derivedresultswillthusbe intermsofasinglematerialcharac teristictheratioofvelocities R = c/csanda singlegeometricparameterthecrosssectionalaspectratio, a Whencomputingdeniteeigenvaluestodisplayrepresentativequantitative results, R willbearbitrarilysettoavalueof 3.Someresultswillturnoutto beindependentof R .Forperspective,itisawellknownresultoflinearelasticity theory[42]that R cannotbelessthan 2forisotropicmaterials.Commercial aluminum,amodestlysoftandnearlyisotropicmetal,hasan R ofroughly2 whereasGaAs,anon-isotro picsemiconductor,hasan R averagingclosetothe 2 limit. Allphononsarecontemplatedinthem esoscopicregime,bywhichImean thattheirwavelengthsarenotlessthananorderofmagnitudesmallerthanthe smallestcrosssectionalwidth.Ialsoassumethattreatingthematerialasanelastic 19

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20 continuumisjusti“edby“rstassumingth ewaveguidematerialhasatypicalinter latticespacingmuchsmallerthanthesma llestphononwavelengthconsidered.As apracticalmatter,thiswouldstillpermitimportantresultstoapplytoquantum wireswithwidthsontheorderofafewhundredatoms.Thedisplacementswillbe assumedsucientlysmallthatappliedela sticitytheoryiswellwithinthelinear regime.Forphononsthatarethermallyexci tedorscatteredfrominteractionswith itinerateelectrons,magnons,andsimilarparticles,thisisareasonablephysical assumptionconsistentwithremainingwit hinthephononsownmesoscopicregime. Calculationsandrepresentationswillbe renderedwithinaright-handedCartesian Figure2.1:NomenclatureofElasticWaveguide.Frequencyandwavenumberwill berescaledrelativeto h (and cs).Thesmallerhalfwidthwillbedenoted hz= h Theonlygeometricfactorin“nalresultswillbe a ,thecross-sectionaspectratio. coordinatesystem.Thelongaxisofthebarwillbeconsideredthe x axisto facilitateeasycomparisonwithpublicatio nsofhistoricalsigni“canceinwhichthis isthemorecommonconvention.Thelongcoordinatewillbeembeddedalongthe geometricalcenterofthebar.Thisplacem entofthelongaxissymmetricallydivides thebar.Accordingly,thebarwillbetransverselyboundedby Š hy y hyand Š hz z hz.Forconvenience, hzwillbeconsistentlytakenasthesmallerhalf-

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21 widthshould hy = hzand h withoutsubscriptwillrefertothissmallerquantity.As alreadyindicated, a willdenotethecrosssectionalaspectratiosothat hy= ahz. 2.2SymbolicConsistenciesandAdoptedTensorNotation Inadditiontoforegoingnomenclature,the basesymbolforallquantitiesrelated toelasticdisplacementswillbetheletter u .Vectors,asopposedtotheircomponents,willbebolded.Whennecessarytodi stinguishshearversuslongitudinal displacementcontributions,aparenthesizedsuperscriptwillbeused,asin u( )and u( s ).Thedisplacement“eldwillbedecomp osedintolongitudinalandshear partsgeneratedbyascalarpotential andavectorpotential H respectively.The Greekletter willbeconsistentlychosenasab asesymbolforlongitudinalwave numbersand forshearwavenumberssothattheseassociationscanbeperceived at“rstglance. Einsteinnotationwillbeusedintens orequations.Repeatedindiceswill, unlessotherwisenoted,implyasummationover { x,y,z } coordinates.Commas precedingoneormoreindiceswillabbreviatederivativestakenwithrespectto them.However,theproblemwillbeinvestigatedwithinaCartesianmetricand sotherewillbenodistinctionbetweencovariantandcontravariantvectorsand correspondinglytherewillbenomethodicaluseofraisedversusloweredindices. Absenttheabilitytorestrictsummationstomatchingupperandlowerindices, summationmaybeassumedforallmatchingpairsofindicesonanyonesideofan equation„subjecttocontrarycommentary. Thesymmetricandantisymmetricpartsoftensorswithrespecttoanysubsetof indiceswillsometimesbeexplicitlyindica tedusingthestandardnotationaldevices ofenclosingthelistof p indicesthataresymmetrizedinparenthesisandanythat

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22 aretobeantisymmetrizedinsquarebrackets. A... ( ij... )=1 p ![ A...ij...+ A...ji...+ ... ] A... [ ij... ]=1 p ![ A...ij... A...ji...+ ... ] Hz,y Hy,z=2 H[ z,y ]= 2 H[ y,z ]Thetotallyantisymmetrictensorofanyappropriaterankwillbeindicatedusing an (asopposedto ),asin [ H ]i= ijkHj,k2.3LinearElasticityTheory Thephononsinthisstudyaremodeledaselasticvibrations.Itishelpful andimportanttoidentifyandoutlinetheoriginofspecicelementsofelasticity theoryessentialtothederivationsdevelopedherein.Besidesclarifyingnotation andconnectingittostandardliterature, thiswillservetheimportantpurposeof exposingimpliedandexplicitassumptionswhichunderliemyownderivationsand whichcouldlimitapplicabilityoftheresults. Thebasicresultsofthelineartheoryarestraightforward,butelasticitytheoryhasmanysubtletiesandcomplexitieswhichwillnotbeneededinwhatfollows.Forfulldevelopmentofthetopic, monographsandtextsrangefromcomprehensiveclassics[43],torelativelyrecentstandardtextswhicharecitedby mostcontemporaryresearcherswhoutilizea nelasticitymodeloflong-wavelength phonons[37,38].Asarecommendedsupplement,asuccinctdevelopmentofelementarylinearelasticitytheoryisprovide dwithinthetheoreticalphysicsseriesof

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23 LandauandLifshitz[42],butsomelesse rknowntextsandmonographsaremore relevanttothestudyofelasticwaveguides[5,10]. Linearelasticitycanbeviewedasage neralizationofHookesLawapplied toacontinuumwhoseelasticresponseneednotbeisotropic.Hence,thebasic stress-strainrelationlawtakesthefollowinggeneralform: ij= cijkuk(2.1) Thetensor cijkissometimescalledthestinesstensorincontradistinctionto thecompliancetensorwhoseelementsarethoseoftheinverseofthestiness tensorsmatrixrepresentation.Itisnot uncommontorefertothestinesstensoras theelastictensor.Thetensor ukisadimensionlessstraindenedtoencapsulate deformationinaforminvariantunderpurerotationwhileomittingnon-linear terms.Thisisaccomplishedbydeningittobethesymmetricpartofthegradient ofthedisplacementvector(sometimesdenoted su [38]). [ u ]( k )= uk= u( i,j )=1 2( uk,+ u,k)(2.2) Thepotentialenergydensitythenhasthefollowingform: V =1 2ukcijkuk(2.3) Since uijissymmetric,directionalinvarianceof V leadsultimatelytoessential symmetriesoftheelastictensorasfollows: cijk= cijk= cjik= cijk= cjikand cijk= ckij(2.4)

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24 Theseintrinsicsymmetriessubstantially reducethenumberofuniquecomponents (81,beforeapplyingthesymmetries)andgiverisetoanabbreviatednotation, ubiquitousinmaterialsscience,forden otingthe,atmost21,uniqueelasticconstantswhichremain.Abbreviatedinde xingre”ectsthepatternofsymmetriesby denotingpairswithsingledigits.Thestra in,stress,andelastictensorcomponents arethenexpressedintermsofthecorrespondinglyreducednumberofindices.The abbreviationschemeissimply 11 122 233 3 23 32 413 31 512 21 6 Thisallowsequation(2.1)tobeexpresseda sthefollowingtwo-dimensionalmatrixvectorproductrelation: 123456 = c11c12c13c14c15c16c22c23c24c25c26c33c34c35c36c44c45c46c55c56c66 u1u2u3u4u5u6 (2.5) whereomittedmatrixelementsaresymmetri cre”ections.Variouscrystalsymmetrieswillreducefurtherthenumberofuniqueelementsof cpq.AlthoughIwill sometimesrelyuponandrefertogeneralpropertiesof cpq,theimportantnovel resultsdependuponthematerialbeingisotropic.Isotropycanberepresentedby imposingrotationalinvarianceonanexistingcubicsymmetry.Thefullelastic

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25 matrixforacubicmaterial(inabbreviatednotation)is c11c12c12000 c12c11c12000 c12c12c11000 000 c4400 0000 c440 00000 c44 (2.6) whichcanberenderedrotationallyinvariantbyrequiringthat c44=1 2( c11 c12)(2.7) Theresultisthatthethreecomponentscanbeparameterizedusingonlytwo constants(ormoduli),andthesecanbechosentobeequivalenttotheso-called Lam econstantswhichemergenaturallywhenthetheoryisderivedstartingfrom anassumptionofisotropy(see,e.g.,[42]). = c12and = c44with c11= +2 (2.8) Foranisotropicmaterial,thestress-strainrelationship(2.1)canthenbesummarizedas ij= ijrurr+2 uij(2.9) NoticethatIhavenotbeencontenttoallow urrtoimplysummationbecauseone ofthepenaltiesofavoidingtheuseofupperindicesisthat,underthedenition (2.2), uiiispropernotationforanindividualdiagonalcomponentof u .Having calledattentiontothisproblem,itcanbemitigatedbyadoptingaconvention

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26 that,absentclarifyingcommentsorexplicitsummation,repeatedindicesinarank twotensorwillimplysummation(i.e.,contraction)onlywhentheusualletters setasideascoordinatevariablesarenotused.Underthisrule,theuseof r asa subscriptaboveinlieuof i j ,or k wouldhavesignaledthattheambiguityshould beresolvedinfavorofimplyingsummation. Thefree-bodyequationofmotionwithinthemediafollowsimmediatelyfrom Newtonslawandsubstitutingequation(2.1)into(2.3). ui,tt= V,i= ij,j(2.10) Iftherightsideofequation(2.10)isexpandedusingequations(2.1),(2.2),and thesymmetriesidenti“edin(2.4),thentheequationofmotioncanbeexpressed intermsofdisplacementsas ui,tt= cijkuk,j(2.11) which,inthecaseofanormalmode(i.e., eittimedependency),immediately yieldsaformofvectorwaveequationwithpossiblynon-factorablevelocitiesŽ. 2ui+ cijkuk,j=0(2.12) Toexhibitthewavepictureinthecaseof isotropy,Icanuse,instead,equation (2.9)inexpandingequation(2.10).Thislea dsultimatelytoausefulcoordinate-free equationofmotion u,tt= + ( u )+ 2u (2.13) ThisexpressionisusefulŽinthesensethatanassumptionthatthedisplacement “eldvanishesatin“nitypermits,viaawell-knowntheoremofvectoralgebra, u

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27 tobeseparatedintodivergencelessandsolenoidalpartswhich,inturn,produces separatedwaveequationswithdistinctvelocitiesasfollows: u( ) ,tt= +2 2( ) u( s ) ,tt= 2( H )(2.14) HereIhavebeguntoutilize asthescalarpotentialgeneratingthelongitudinal displacementand H asthevectorpotentialgeneratingthesheardisplacement.In general,thesepotentialsymbolsmaythemselvesrepresentsuperpositions. Theforegoingclearlyidenti“esthetwoisotropicbulkvelocities: cs= = c44 (2.15) c= +2 = c11 (2.16) Thesecanthenbesubstitutedbackintoe quation(2.9)tobeginutilizingthebulk velocitiesastheprimarycharacteristicsofanisotropicmaterial. ij= ( c2 Š 2 c2 s) iju+2 c2 suij(2.17) 2.4FreeBoundaryConditions Theprobleminvestigatedhereinis de“nedbytakingthestressnormalto thesurfacestobezero.Ibegintotransit iontowardadimensionlessformof thissituationby“rstdividingequation(2.17)by c2 s.Recallthat R hasbeen designatedtobetheisotropicmaterialpropertyde“nedby c/cs.Then,inthe coordinatesystemoftheproblem,theassertionthatstressnormaltothe i th

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28 surfaceat xi= hxiiszerocanbeexpressedinaformindependentof and csas ( R2Š 2) ijurr+2 uij=0at xi= hxi(2.18) Since ui,i= uii,thecontraction urrinequation(2.18)isjust u .Sincethe divergenceofacurliszero,thereisnoshearcontributiontothisterm.Then,since thelongitudinaldisplacementisthegradientofthescalarpotential ,Icanreplace u with 2 .Since issolutionofawaveequation,wecan“nallyreplace 2 with Š k2 where k= /cattheeigenfrequency Finally,sinceeach u( ) i= ,ilongitudinalstrainsaresuccinctlyexpressiblein termsofthescalarpotentialintheform u( ) ij= ,ij. Separatingtheshearandlongitudina lpartsandutilizingtheforegoing,the boundaryconditionsateitherofadjacen tpairsofparallelsurfacescannowbe restatedasasetofequalitiesrelatingshearandlongitudinalcontributions.To emphasizetheessentialstructureindepe ndentofchosensurface,Iwillhereusea surface-orientednotationalschemeasfollows:Denotewith s thecoordinateaxes (i.e., y ,or z )normaltothesesurfaces.With s asthestartingcoordinate,denote thenextcycliccoordinate(right-handedprogression)as p andthenextafterthat as p .Intermsofthisnotation,theboundaryconditionsforanytransversesurface canbewritteninthefollowingform: At s = hsforall p [ Š hp,hp]andall p [ Š h p,h p]: Š k2 +2 ,ss= Š 2 u( s ) ss,sp= Š u( s ) sp,s p= Š u( s ) s p(2.19)

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29 where n = R2 2. Thesheardisplacementvectorcomponentsintermsof H componentsare explicitly: u( s )= HxHyHz = Hz,y Hy,zHx,z Hz,xHy,x Hx,y (2.20) Theshearstraintensorsarethen: u( s ) xx= Hz,xy Hy,zxu( s ) yy= Hx,yz Hz,xyu( s ) zz= Hy,zx Hx,yzu( s ) xy=1 2( Hz,yy Hy,yz+ Hx,xz Hz,xx) u( s ) yz=1 2( Hx,zz Hz,zx+ Hy,yx Hx,yy) u( s ) zx=1 2( Hz,zy Hy,zz+ Hy,xx Hx,xy)(2.21) withtheorderofdierentiationhavingbeenarrangedtohighlightpatternsand symmetries. Whentensorsfromequations(2.21)ares ubstitutedintosurfacemanifestations ofequation(2.19),theoverallcyclicpatternofthesubscriptsisfullyapparentand theboundaryconditionscanberesummarizedcompletelyintermsofpotentials asfollows: At s = hsforall p [ hp,hp]andall p [ h p,h p]: nk2 +2 ,ss= 2( H p,p Hp, p),s,sp= 1 2[( H p,pp H p,ss)+( Hs,s Hp,p), p] ,s p=+1 2[( Hp, p p Hp,ss)+( Hs,s H p, p),p](2.22)

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30 2.5NonseparabilityofBoundarySolutions Utilizingequation(2.19),Iwilldirect lydemonstratethatassumingtheexistenceofseparablepotentialfunctionsinthisscenarioleadstoanessentialcontradiction.Thez-boundaryandy-bo undaryconditionsareexplicitly at z = h forall y [ h,h ]: nk2 +2 ,zz= 2 u( s ) zz,yz= u( s ) yz,xz= u( s ) xz(2.23) andat y = h forall z [ h,h ]: nk2 +2 ,yy= 2 u( s ) yy,yz= u( s ) yz,xy= u( s ) xy(2.24) Supposethat isseparablewavefunctionrepresentableastheproductofwave functionssuchas p ( kx ) y( yy ) z( zz ).Sincelongitudinalstrainsaresymmetrical inthederivativestakentoformthemfrom ,separabilityof willperpetuateto thelongitudinalstrains.The x dependenciesofthepotentialswillbeassumedto becommonbecausethatisthebasischoiceappropriatetopropagatingmodes.I canthusanticipatethatthe x -dependenciesofeachboundaryconditionwillcancel andwillhenceforthomitthemfromtheremainderofthisproof. Assumealsothat H generatesa u( s )suchthatshearstrainsformedbycombiningtheirderivativescanberepresentedasseparableproductssuchas u( s ) ij=

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31 uij y( yy ) uij z( zz ).Again,since H mustbeawavefunctionsatisfyingtheboundaryconditions,theseparableproductsmakingupthestrainsitgenerateswill ultimatelyalsobewaveequationsforeachdirection. Iftheshearcomponentswereindividua llyseparable,thebestpossiblecase wouldbethattheshearstrainswhichcombinederivativesofthese,nevertheless produceindividuallyseparablefunction s.Iftheydonot,theshearstrainswill immediatelybenonseparableandtherewillbenopointingoingfurthersincethe casewillhavebeenmade.Itissucient,therfore,toassumethisbestpossible case. Substitutingtheseassumptionsintoequation(2.23)at+ hz: ( nk2 + 2 z) y( yy ) z( zhz)= 2 kuzz y( yy ) uzz z( zhz) yz y( yy )z( zhz)= yuyz y( yy ) uyz z( zhz) kzy( yy )z( zhz)= uyz y( yy ) uyzz( zhz)(2.25) wherethetildesoverfunctionsconveysthattheyhavebeendierentiated.Itwill notmattertotheproofpreciselywhattheseparableproductoritsderivativeis becausetheonlyfactsessentialtotheresu ltarethattheyarewavefunctionswith denitewavenumbersxingtheirdirectionaldependencies. Fortheforegoingequationstobevalid,theymusteachbeinvariantiny.By inspection,thiswillonlybetrueif y= yincludingthepossibilitythattheyare bothzero. Giventhecyclicsymmetryconnectingt heboundaryconditionsatadjacent sides,substitutingthesameassumptionsintoequation(2.24)willsimilarlyresult inthesymmetricalrequirementthat z= z.

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32 Separabilityofthepotentialstogetherwi ththeirsatisfactionofwaveequations impliesthat 2 y+ 2 z+ k2= t2 c2 2 y+ 2 z+ k2= t2 c2 sIf y= yand y= y,thiswillrequirethat c= cswhichisimpossibleforreal isotropicmaterialssinceithasalreadybeenpointedoutthat cb 2 cs.This calamitycanbecuredby(a)somesuperpo sitionofsolutions,(b)eliminatingone sidebyextendingittoinnity(theinniteplatescenarioassolvedbyRayleigh),or (c)specialmodalsituationsinvolvingonlyshear( =0).Eachoftheseexceptions willbeencounteredinthesequel.

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CHAPTER3 THEORETICALASPECTSOFRECENTNUMERICALMETHODS 3.1StationaryLagrangian Holland[21],Demarest[23],andVisscheretal.[25]eachconstructatheoretical justicationfortheirnumericalapproxim ationmethodsbyidentifyingsolutions withthestationarypointsofaLagrangian.Moreprecisely,thestartingpointis statedtobeHamiltonsprinciple,butthestationarypointsofthetime-independent Lagrangianwereassertedtobeequivalent.Whetherornotthisisavalidsubstitution,however,dependsonwhetherthefunctionspacewithinwhichthestationary functionsarefoundisrestrictedtofunc tionswithaseparableharmonictime dependence.(Demarestmakessomeattemptatvaryingamoregeneralclassof trialfunctions,butfailstoproperlyfactorthevariationinhisequation(4).) Thisisnotaninconsequentialsuspensionofformality.Stationarityofthe Lagrangianwithinafunctionspaceofelasticnormalmodesfollowsimmediately fromstationarityoftheHamiltonianpreci selybecausethekineticandpotential functionshavethesameharmonictimede pendency.Integratinganysuchfunction overtimeproducesafactorof1 / 2sothatstatingvariationstobezeroisequivalent tostatingthatvariationofthetime-independentpartiszero. f2L ft2= t2L t L ( r,t ) dt = 1 2 L ( r ) However,ifthefunctionspaceisnotinsomewayrestrictedtothosefunctions whosespatialpartsareatleastconsistentwiththeirhavingharmonictimedependencies,thenitiselementarythatstationarityoftheirspatialpartsisnota 33

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34 prioriequivalenttothesatisfactionofH amiltonsprinciple.Thedesiredharmonic dependencywillbemanifestbysatisfacti onofawaveequation.Afterreviewing, below,theactualLagrangianvariationalresult,thepointtobeemphasizedwill bethatrestrictingthesetoftrialfunctionstothosewhichsatisfywaveequations isindeedthemostimportantprerequisitetovalidityoftheresult. TheLagrangianvariationalanalysiswithineachoftheaforementionednumericalapproximationattemptsiswellrepr esentedbythatcontainedinthepaperby Visscheretal.Thereasoningiselegantandclear,andsoitishelpfultoreproduce itwhilemakingrelevantobservations. ThederivationbyVisscheretal.takesplacewithinacloseddomainofarbitrary shape.Itthereforeis,inessence,aresonanceproblem.Thesamereasoningcan beextendedtopropagatingmodesbyconsideringsuchamodevirtuallybounded withinarectangularwaveguidebysurfa cesperpendiculartothesidesandseparatedbyaphasedierenceof2 .Ifanysurfaceintegrationisperformed,reversal oftheoutwardnormalsofthesevirtualsurfaceswouldcanceltheircontributions totheoverallsurfaceintegralinsuringt hesameresultasforaphysicallybounded region. Visscheretal.pre-incorporateharmonictimedependencebywritingthekinetic energydensityas 1 2 2uiui(3.1) Theyrepresentthepotentialenergydensityas 1 2 cijkui,juk,(3.2) whichcanbeshowntobeequivalenttoequation(2.3)byexpandingthede“nitionsofthestraintensorsandexploitingsymmetriesoftheelastictensor.The

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35 Lagrangian,then,isjusttheintegraloverthedomainofthedierenceofthese densities. L = 1 2V2uiuiŠ cijkui,juk,dV (3.3) Iwillassume,asdotheauthors,thatthereaderisabletocompletetheexercise bydoingthealgebraandapplyingthedivergencetheoremto“ndtheconditions forthe“rstvariationofthisLagrangiantobezero.Iwillpointoutthattheir particularchoiceofhowtorepresentthepotentialenergydensityappearsnotto havebeenaccidentalsinceitguidesthealgebraandmakestheprocessrathermore transparent.Theresult,asisexpectedfromanyLagrangianminimization,isthe equationofmotion.However,theparticularformofthesurfacetermwhichmust bezeroisimportantandsoIwillwritethefullstatementoftheresulttoinclude it. L =0 V( 2uiŠ cijkuk,j) uidV +S( cijkuk,) uinjdSj=0(3.4) where n istheoutwardnormalsurfacevectorand njdSjisthemagnitudeofthe outwardareavector.Theparenthesizedpartsoftheintegrandsaretheelastic waveequationandarepresentationofstress.Theseparenthesizedtermsshould beseparatelyzerowhenthe“rstvariationoftheLagrangianiszero. Thesametensoralgebrausedtoshowthatequation(2.3)isequivalentto Visscherschoiceofrepresentationforthepotentialenergydensitycanbereversed toshowthattheintegrandofthesurfaceterminequation(3.4)canbereplaced with cijkuk

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36 whichisjusttherighthandsideofequation(2.1)bywhichstressisde“ned.Surface normalstress,byde“nition,isjust ijnjandsothesurfaceterminequation(3.4)iszeroifthesurfacenormalstressiszero. Withrespecttothewaveequationaspectofthiscondition,theassumptionof harmonictimedependenceguaranteedthisinadvanceascanbeseenfromthe factthatinsection2.3Iderivethesamewaveequationfromthatassumptionand theapplicationofNewtonsLaw.Lagrangianformalismissimplyasubstitutefor Newton,butleadstothesameresult. Whatisnewistheappearanceofthestress-freeboundaryconditionasa prerequisite.Thereisnothinginsection2.3thatwilldirectlypermitmetoinfer thatmodalfunctionsformulatedtosatisfytheelasticwaveequationcorrespond toastationaryLagrangian. WhatVisscheretal.concludefromthisoutcomeisthat...thedisplacement functions u ,whicharesolutionstotheelasticwaveequationwithfreeboundary conditionson S ,arejustthosepointsinfunctionspaceatwhich L isstationary.Ž Iwanttoemphasizeyetagainthatthisresultispartlyself-ful“lledbystarting withanassumptionofharmonictimedependenceandthatthesatisfactionofthe waveequationisactuallyanimplicitsta rtingpointratherthanaconditiononthe outcome. Inviewoftheforegoing,Idrawaslightlydierentconclusionthanthatexpressed byVisschersteamnearlyadecadeago.WhatIseeintheresultisthefollowing implication:Somesuperpositionoffunc tionswhichareguaranteedtosatisfy theelasticwaveequationwillalsosati sfyfreeboundaryconditionswhentheir combinationmakestheLagrangianstationary.

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37 Havingrephrasedtheconclusionitbeco meseasiertoseehowlimitationscan arisefromthebasischoicesemployedinactualnumericalcalcualtion.Visscheret al.,likeDemarestbeforethem,chooseabasisforcomputationalconveniencewhich doesnotinfactconsistoffunctionswhichsatisfyanywaveequation.Resonances whichinvolveacombinationofcomponentswhicharehyperbolictogetherwith somehavinglowperiodicitycanbeaccuratelydetermined.Itbecomesmore diculttocomputeresonancesforshapes,suchasthosehavinghighaspectratios, whichrequiremorecomponentshavingextendedperiodicities.Itisharderto assemblesuperpositionsofwavesfromabasisdevoidofwavesolutionsthanitis fromabasismadeupofwavesolutions. Inanycasethepapercontinueswithalogicalinversion.Itisclaimedthat ndingastationarypointinsomerobustfunctionspaceisparamounttonding afunctionthatisagoodapproximationtoanormalmode.If,ofcourse,thebasis isnotmadeupofwavesolutions,thestat ionarypointmustoftensimultaneously produceawavesolution,butfailureto constrainthebasismeansthatmoreis expectedoftheprocessthanneedbe.Itcanonlybesaidthatconcernforplacing extraburdensontheprocessislostinthe factthatresonanceproblemsusually convergenicelyandgiveresultswhichappeartobeaccurateforsituationsof practicali nterest[ 26] Thatconvergenceofsomekinddoesoccurinthecaseofresonanceproblemsis welldocumentedintheliteraturecited.Indeed,theprocesshasnowbeeninused todetermineelasticconstantsforadecade.Ihavepersonallyobservedtheprocess anddiscusseditwithDr.Migliori.Fromhisunpublishedremarks,however,itis clearthatthenumericalprocesswillsometimesfailtoconvergewhenattemptingto matchtheactualeigenspectrumofobjectswithhighaspectratiosagainstestimates generatedbythenumericalprocess.Thisdicultyappearsnottohavebeen

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38 thesubjectofpublishedmethodicalanaly sis,butIhavepersonallyexperimented withrectilinearcoppersampleshavingroughlya1x1x3aspectandwasunable to“ndelasticconstantsusingwhichth enumericalalgorithmcouldproducea resonancespectrumthatmatchedwhatwa sdirectlymeasurable.Thisauthorcan therforecon“rmthatwhateveritssuccess,thereareclearindicationsthatthe numericalalgorithmpremisedontheLagrangianformalismjustreviewedisnot reliableoutsideofsomeconvenientsetofs ituationsdespiteremarkableaccuracy whencon“nedtothatsetofsituations. Mypurposeisnottoburythenumericaltechniques,noreventheirtheoretical underpinnings.Rather,Iwishtopraisethemsincetheyareextremelyhelpfuland usefuldespitethelimitationstheyoper ateunder.Itisimportanttorepeatthat whateverthepreciselimitsofthenumericalalgorithmsturnouttobe,thefactis thattheyaredemonstrablyaccuratewithinthedomaintheyaretypicallydeployed in.Moreover,thesomewhatcircularlogicoftheLagrangianformulationcarries withinittheinspirationforarenewedinves tigationofwhetheranalyticalsolutions arepossibleatleastfortherectilinearg eometry.Speci“cally,theLagrangian techniquehighlightsthatthemostpromisingbasisoughttobeonemadeupof functionswhichsatisfyaconsistentwave equation.ByconsistentŽwaveequation, Isimplymeananywhosewavevectormagnit ude,andthusfrequency,matchesthat ofthenormalmodeinapropagatingproblem. ItisinterestingthatmyobservationsonthelimitationsofLagrangianminimizationactuallyvindicatesHollandsini tialapproachinthathepurposelychose productsofsinesandcosinesasbasis functions.IfHollandhadfoundaway toimplementhisnumericalapproximationbyseparatinghisbasistoexpandthe longitudinalandshearcontributionssepa rately,hisalgorithmwouldhavebeenan appropriatenumericalanalogoftheanalyticapproachwhichIdevelopherein.

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39 3.2NumericalApproximation Equation(3.3)canbewrittenasadiere nceofdistinctfunctionalsintheform L = T [ u ] Š V [ u ](3.5) Iamnowusing u inthesenseoftime-independentdisplacementfunctionswhich inhabitavectorspaceequippedwiththeusualinner-product(i.e.,vectordot product). T and V arelinearfunctionalsoperatingonthespaceof u sandIinvoke atheoremofFunctionalAnalysis(seeRef.[44] § 30),usuallytakenforgranted,that permitsmetoassumetheexistenceofoperatorssuchthatIcanrewrite(3.5)in theform L = u | T | u Š u | V | u (3.6) wheretheelementsofamatrixrepresentationoftheoperatorsarede“ned,as usual,bytheresultsoftheiroperationonabasisforthe u s.Thismeticulous formalityallowsmetoemphasizethatthis isasituationinwhichthatbasisisnot actuallyknownandthatthesituationisthusfarabstract. Actualnumericalcomputationisaccomplishedusinga“niteandde“nitebasis. FollowingVisscheretal.,Iwillexpandthedisplacementsandestimatetheoperatorsusingbasisfunctionsoftheform xpyqzr.Thisisjustacomputationally convenientrealizationofthemoregene raluseofproductsoffunctionsofthe coordinatessuchthateachelementisaseparablefunction.Thesearenot,in general,orthogonal,buttheymaybechosentobelinearlyindependent…asisthe caseherewheresimplepowersoftheco ordinatesareused.Itwillfollowthata Rayleigh-Ritzlikeminimizationoftheir combinationisreducibletoageneralized eigenvalueproblem.Onedistinctandusef ulfeatureofthisparticularbasiswillbe thatitselementshavede“niteparity.

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40 Equation(3.2)whichwillformthebasisforcomputingmatrixelement sof thepotentialenergyoperator,involvesamixingofdierentcomponentsofthe displacementvectorfunctions.Itherefo reexpandthecomponentsofthedisplacementvectorfunctionsinbasesoftheirownratherthanexpandthedisplacement vectorsinabasisofvectorfunctions.Corre spondingly,therearemultiple(though perhapsnotdistinct)basissets { ( i ) k} ( i = x,y,z )andtheseresultinanexpansion oftheform uxuyuz = kc( x ) k( x ) kkc( y ) k( y ) kkc( z ) k( z ) k Intermsofthesenitebases,approximationstothekineticandpotentialoperators havethefollowingrepresentations: T( i )( j )=Volb ( i ) ( j ) ijdV (3.7) V( i )( k )=Volfj( i ) cijkf( k ) dV (sumover j & )(3.8) ThesematrixelementdenitionsareinthesameformasusedbyVisscher. Theyareelementsofapproximate,notexact,operators. Thereisnoassumptionthatbasisfunctionsindividuallysatisfythebulkwave equationsoftheelasticmedia,letalon etheboundaryconditionsofthegeometry. Thereisnorelianceuponorthogonalitypropertiesofthebasisfunctionsinany geometrywhatsoever. OnewaytoexpressthattheLagrangianisstationary,istoassertthatthederivativesofequation(3.6)withrespecttocomponentsof u areallzero.Whenthe u s arediscrete,thiscertainlyisequivalentto thefollowing(functionaldierentiation

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41 willleadtothesameresult): t2T | u V | u =0(3.9) Thisisjustastatementofageneralizedeigenvalueproblemforwhichcomputer algorithmsarewidelyavailable.Thenumericalalgorithmgeneratesabasisto somelimitonthepowersofthecoordinate s,calculatesthematrixelementsofthe operatorsbyintegratingoverthesamplepursuanttoequations(3.7)and(3.8), andtheninvokesstandardsoftwaretocalculatetheeigenvaluesand,ifdesiredthe eigenvectors. Thiseigenvalueproblemformofaminimizationproblemisexpectedtodisplay aconvergenceofeigenvaluestowardsomelimitastheorderoftheoperators increases.Itiscommonexperiencethatthisconvergenceisalwaysofadecreasing naturesothattheestimatesarealwaysupperboundsofthetruevalues.However, whenthebasisfunctionsusedinthenumericalproceduresaremerelyproductsof powersofthecoordinatesandwhenthephysicalrealitybeingmodeleddoesnot involvetheconvergingeigenfunctions,ortheirderivatives,converginguniformally tozeroattheboundaries,theexpectatio nthateigenvaluesareupperboundsmay notbeprovable.AtypicalproofthataRayleigh-Ritzprocedurewillconsistently produceupperboundsdependsuponthetrialfunctionsthemselvesvanishingat theboundaries[45].Inaddition,althought henumericalbasisconsistsofelements thatarelinearlyindependent,theyarenotingeneralorthogonalasisassumed,for example,inamoreelegantapproachtoasimilartheorem[46]proposedbyPeierls. Ofcourse,itmayhappenthatsomeofthemodesarewellrepresentedby functionsthatvanishattheboundaries.InCh.6Iwillderiveallofthe k =0modes ofarectangularwaveguideandasubset ofthesewillbeuncoupledanddened explicitlybytheboundaryzerosofsinesandcosines.Powersofthecoordinates

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42 certainlyconvergerapidlytoisolatedsin eandcosinefunctionsafeatureendemic tonumericalcomputationsoftheirvalues.Ifthesolutionsconsistofsuperpositions ofsinesandcosines,however,andiftheboundaryvaluesarenotwell-represented, therearenosuchregularitiestoassisttherateofconvergenceandnovanishing ofthewavefunctionsortheirderivativestoconvergetoward.Whileitisbeyond thescopeofthisresearchtopursuetheseconcernsfurther,Ihaveattemptedto pointoutthatthechoiceofanon-orthogonalbasiscoupledwiththefactthat theboundaryconditionsdonotgenerallyin volvevanishingofthebasiselements, createsinherentdicultiesfortheeca cyofprevailingnumericalmethods.This strengthensthemotivationfordevelopingananalyticalsolution. 3.3PartitioningtheProblemintoParityGroups Holland[21]carefullyorganizeshisbasis,madeupofproductsofsinesand cosines,intoparitygroupsrelatedtoac oordinatesystemcenteredinhisparallelepipedwithaxesnormalt othefaces.Demarest[23],andVisscheretal.[25] adheretothesamesystem.Thesameparit ypatternclassicationsystemplaysa pivotalroleinmyownanalyticalderivationswhichfollow. Theimportanceofthisclassicationschemederivesfromtheinterplayof arectilineargeometrywiththesymmetrypr opertiesoftheelastictensor(see section2.3)withinthepotentialenerg yoperatordenedinequation(3.8).The kineticenergyoperatorcanbediagonalizedindependently,butcarefulexamination ofequation(3.8)showsthatthepotentialenergyoperatorwillblockdiagonalize wheretheblocksaredenedbyparitypa tternsofanybasiswhoseelementshave deniteparity.Amongtheconsequencesofthisfactisthattheeigenvalueproblem statedinequation(3.9)canbedividedintoadistincteigenvalueproblemforeach

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43 block.Fromananalyticalstandpoint,however,therealvalueisthatthesymmetry classicationsareanorganizingprinciplefortheentireanalyticalapproach. Althoughtheparityclassicationschemewasrstidentiedinthecontextof closedgeometries,thepotentialenergyo peratordoesnotchangeforminthecase ofarectangularwaveguideandsotheclassicationschemeremainsequallyvalid andapplicable.Theonlydistinctionistha ttheparityalongthelongdirectionis oftenarbitraryaswillbesoindicated.Whenturningawaveguideintoaresonator, theresonatormodessymmetricalongtheformerlongaxiswillsimplydivideinto evenandoddgroups.Forthesakeofcomprehensiveness,Ithereforeexhibitthe fullparitypatternusingthenomenclatureofHolland. LetEandOdenoteevenandoddparityforfunctionsrelativetoCartesian coordinatesalignednormaltothesurfa cesandalongthecentralaxisofthewire. Anunspecied,butconsistent,parityasmaybeassociatedwiththelongaxisofa waveguideisdenotedwithaP.Paritycomp lementationofanotherwiseunspecied P(as,forexample,resultingfromdierentiation)isdenotedby P.Directional dependenciesareimpliedbyjuxtapositioninaproductintheorder x,y,z Symmetrypatternsofdisplacementfunctionsarethenlimitedtoonlythe familiesshowninTable3.1shownascolumnvectorsofthe x,y,z components. Thelasttwoofthese(exural)areadegeneratepairinthecaseofsquare rectangularcrosssections,butaspointe doutbyNishiguchi[1],theyaredistinct forthegeneralrectangularcase. Eachoftheseparitypatternscanbegene ratedbyasingleproductrepresenting theparitypatternofthescalarpotentialwhosegradientproducesthelongitudinal partofthedisplacementfunctions.Theshearpartmusthaveamatchingparity patternandsothepatternofthescalarpotentialxesthatofthemodegenerally. InthecaseofDilatation,forexample,observethatthegradientofaproduct

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44 Table3.1:ModeFamiliesinRectangularGeometryDe“nedbyDisplacement ParityPatterns PEE POE PEO Dilatationgroup:D, P=E F1, P=O POO PEO POE Torsiongroup:T, P=E S1, P=O PEO POO PEE Flexionofz-xplane:S2, P=E F3, P=O POE PEE POO Flexionofy-xplane:S3, P=E F2, P=O withparitypatternPEEautomaticallypr oducesvectorswiththeparitypattern oftheDilatationgroup.Similarly,theg radientofPOOwillgeneratetheTorsion group…andsoon.

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CHAPTER4 MATHEMATICALSTRATEGY 4.1NotationandFunctionExtensionIssues Insection3.3Ireviewedhow,forrectangulargeometries,thepotentialenergy operatorblockdiagonalizedtopartitiontheproblemintoindependentsolution familiesbasedonparitypatternsofthebasis.Forthistobethecase,itis onlynecessarythatthebasisfunctionsindividuallydisplayade“niteparityin theirseparableparts.Fromthis,anysuperpositionofbasisfunctionsfromthe samefamilywillalsoexhibitthesamede“niteparitypatterneventhoughthe superpositionbecomesnonseparable. Myapproachinvolvessolvingtheboundaryvalueprobleminawaythattakes advantageofthefactthatsolutionsarepartitionedbyparityfamily.However,this doesnotimplythatitisnecessarytodoadistinctderivationforeachsuchfamily. Rather,asmuchaspossible,eachderiva tionwillencompassallparityfamilies insuchawaythatonealgebraicresult canbeconvertedintoarealizationfor eachdistinctparityfamilybyastraightforwardsubstitution.Itthusisadistinct resultthatthesolutionsforeachfamilyareshowntobemanifestationsofasingle theoreticalresult. Inordertobeabletotranscenddistinctionsbetweenparityfamiliesforeach derivation,itwillbenecessarytoutiliz easpecializednotation.Thenotation willnotviolateotherconventionsofmathematicalnotation.Itwillsharpenrather thanobscureimportantrelationships.Itwillconsiderablyshortentheexpressionof individualrelationships,dramatically shortenderivations,andprecluderedundant 45

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46 derivations.Resultsexpressedinthisnotationwillunifythemanifestationsamong parityfamilies. Initssimplestcharacterization,theb asisfunctionsde“nedbelowwillallbe productsoffunctionsthatarecoincidentwithsinesandcosinesinsideofthe waveguide.Itshouldbeimmediatelypointedoutthatextendingthesefunctions toin“nityisatempting,butinadvisableoption.Itistruethatvanishingof thedensitybeyondtheboundarywouldservetokeepthephysicaldescription realisticevenifthedisplacementperseweretobesoextended.However,there aredistractingadversemathematicalcons equencesofindulgingsuchanextension thatshouldbeavoided.OneoftheseisthattheHelmholtzTheorem,whichis cruciallyreliedupontoseparatethedisplacement“eldintopartsgeneratedbya scalarpotentialandavectorpotential,becomesproblematicwhenthe“eldsdo notvanish.Eithertheyshouldvanishtotallyattheboundary,orifextended, somekindofconvergencefactorshouldbei nserted.This,however,leadstoother complications.Ultimately,allsuchcom plicationswillbeneatlyavoidedbya judiciouschoiceoftransform,butitre mainsthecasethatIwillneedthefreedom toassumethatthedisplacement“eldshaveabehaviorbeyondthesurfaceswhich willnotcontraveneassumptionsoftheHelmholtzTheoremandatthesametime Iwishtoavoidhavingtospecifywhatthatbehaviorisspeci“cally. Henceforth,thefunctionsmultipliedt ogethertoformdisplacementbasisfunctionswillthemselvesbede“nedascosineorsinefunctionsonlybetweenthe surfaces.Withrespecttothestipulatedcoordinatesystem,theseneedtohave de“niteparity,andsotheywillneverhaveconstantosetstotheirphaseatleast inthetransversedirections.ItwillproveanassettointuitionifIchoosetosimply

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47 callsuchfunctions E and O inrespectoftheirbeingeitherevenorodd. Eq = cos q | q | h unde“ned | q | >h (4.1) Oq = sin q | q | h unde“ned | q | >h (4.2) where h istheboundarylimitand q is y z ,orsomecoordinatevalue(suchas h ). Tominimizenotation,subscriptsandsuperscriptscanbeusedtoindicatewave numberanddirectionaldependencies.Distinguishing x y and z dependencieswill sometimesbeinferredbypositionifthereisnoriskofambiguity. Forexample, E ( xx ) E ( yy ) E ( zz )= Ex xEy yEz z= ExEyEz= EEE Toimplementgeneralityinthederivations,Iwillneedtodenotefunctionsof de“nite,butunspeci“ed,paritybyusingaf unctionvariable.Ingeneral,theletter P ,withappropriatesubscriptingtodistinguishvariables,willbeusedforthis purpose.Forexample, PyPzcouldtakeonthespeci“cfunctionvaluesof OO EE OE ,or EO Often,themathematicalstructureofrelationshipswilldependupontherelativeparityofjuxtaposedfunctions.Iw illaccommodatethisbydenotingparity complementationofafunctionvariablebyplacingabaroverit.Forexample, P P couldbe EO or OE Dierentiationof E or O withrespecttocoordinatewillbethedominant operation.Becauseofthesignchangeintroducedbydierentiationofa E within sampleboundaries,itisnotalwayscorrecttoassume,for Piasanabbreviation of P ( ixi),that Pi,i= i Pi.Toneverthelesspermitdierentiationstobeunambiguouslyspeci“edatthehighestlevelofabstraction,paritycomplementation

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48 resultingfromdierentiationperse willbedenotedbyplacingatildeovera functionvariable.Speci“cally,E = Š O whileO = E ,anditisthusalwayscorrect towrite Pi,i= i Pi.Uponeventualsubstitutionof E or O for Pi,theappropriate signchangescanbemade.However,thesymmetriesoftheproblemultimately resultinthecancelationofsigndistinctionsandtherearealsocasesofsuccessive dierentiationwhichinvoketheparity-invariantruleP = Š P Thederivationswilleventuallyreveal thatthesigndistinctionsinchoatein notationslikeP areeliminatedinthe“nalresultswhichcaninvariablybestated strictlyintermsofparityvariablesandt heirsimplecompliments.Solutionsspeci“c toparityfamilieswillberealizedsimplybychoosing E vs O assignmentsforat mosttwofunctionvariables(viz. Pyand Pzinthecaseofawaveguidewith Pxbeing anadditionalvariableonlyifthewaveguideiscappedtobecomearesonator).In orderforthistoberesolvedinthederivations,thefollowingadditionalnotational devicewillbeneeded: x y Pi x,Pi= E y,Pi= O (4.3) Somesimpleexamplesthatillustratehowthiscanbeappliedare:Py= 1 Š 1 PyPy Px= Š 1 1 Px Px(4.4) Therefore,dierentiationofany P or P canbeexpressedwithoutrecoursetothe tildenotationbyusing Pi,i= i Š 1 1 Pi Pi Pi,i= i 1 Š 1 PiPi(4.5)

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49 4.2DeningtheBasisandSuperpositions Mykeyphysicalstrategyistoassumethatsomesuperpositionoftractable fundamentalbasisfunctionswillassemble atractablenon-separablefunctionthat managestosatisfytheboundaryconditions.Whilethisisclearlyanobvious approachtothepredicament,Ihavebeenunabletondexamplesintheliterature thatrevealanyattempttoactuallyapplyittotheanalyticalsolutionofthis problem.Thoughtheassumptionthatsomekindofsuperpositionisneededcan beinferredfromvariousdiscussions,afailuretoevenrealizethepossibilityis sometimesclearlyevident(see,e.g.,Ref.[3] IV).Icanonlyspeculatethatthe unavailabilityofbasisfunctionswhichth emselvessatisfytheboundaryconditions hasbeenviewedassuchaseriousdeparturefromtypicalitythatithasbeenmore temptingtoconcludeunsolvabilitythantopursuesuperpositioninspiteofit. Amorecourageousviewhasbeenexhibitedbyresearcherslookingfortheoreticalunderpinningsofnumericalapproxim ationattempts.Ithereforegivecredit forreinvigoratingananalyticpursuitof superpositionstothosewhodeveloped numericalapproximationmethods.Thesehavebeenreviewedinthepriorchapter inpartbecausetheycreateaframeworkinwhichsuperpositionsandnotionsof howtheirelementsshouldbestructuredtakeconcreteshape.Infairness,itshould thusbesuggestedthattheresultsachievedhereinaretheresultofadaptingthe progressmadeinnumericalmethodstoarevisitofthepresumablyintractable analyticproblem. Myownapproachtoformulatingthebasisfunctionsisstraightforward.Iaccept thatthelongitudinalandshearcontributio nsshouldbeexpandedinseparatebases. IdepartfromrecentnumericalapproachesinthatIrequireallbasisfunctions toatleastsatisfythebulkwaveequatio nwithrespecttolongitudinalorshear expansions.I nfairness,itshouldbepointedoutthatHolland[21],byusing

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50 productsofsineandcosinefunctionsas trialfunctions,approachedtheproblem similarly,buthedidnotsegregatethetrialfunctionsintolongitudinalandshear contributions.Neitherdidhissuccessors,DemarestandtheVisscherteam,eachof whomabandonedanyeortstoconstraintri alfunctionstothosewhichsatis“eda waveequation.Numericalapproachestodatehavesoughtanoverallconvergence ofdisplacementarrangementsguidedbyavariationalprincipleandsothetrend hasbeentoimposeincreasingarbitrarinessonthestructureofthetrialfunctions. Analytically,thisisfruitless…orworse. Thereasonisthatthetrialbasesusedby HollandandhissuccessorsisnotinananalyticallycompleteŽrepresentation. Thethreekeyissuessurroundingtheconstructionofsuperpositionsare(1) whichfundamentalbasisfunctionstouse,(2)howtorepresentthesuperpositionsof these,and(3)howtotransformtheequationswrittenintermsofthesuperpositions sothattheycanbesolvedasa“nitesetorsolvedbysomerecursiveprocess operatingonanin“niteset.IhavealreadyindicatedthatIwillexpandboth longitudinalandshearcontri butionsinbasisfunctionswhichindividuallyrepresent wavesolutions.Whatremainsistospeci fymoreconcretelyboththerepresentationofthesebasiselementsandtherepresentationoftheirsuperpositions.Ina subsequentsectionofthischapter,Iw illtakeupthetransformationissue…also proposingasimpleapproachwhichhasnotappearedheretoforeintheliterature. Inderivationswhichfollow,fundamentalbasiselementswillalwaystakethe formofaseparableproductintheform PxPyPzwherethe P s(seepreceding sectiononnotation)standforparticula revenoroddfunctionswhich,withinthe boundariesofthesample,arecoincidentwithcosineandsinefunctions,respectively.Theeightparitypatternscorrespondingtotheeightmodefamiliesde“ned intable3.1correspondtotheeightpossiblevaluesof PxPyPz.Note,however, thataslongaseachcomponentofasuperpositionhasacommonparitypattern,

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51 anysuperpositionofthemwillexhibitthesameparitypatterneventhoughit isitselfnotaseparablefunction.Also,sinceIamconcernedwithpropagating modescharacterizedbytranslationalinvariance, Pxwillnaturallycanceloutamong therelations.(Forarectangularreson ator,modefamiliescouldbegeneratedby superimposingpropagatingmodesofvarying x -dependencies.) Sincethemodefamilyisde“nedbyaparitypatternoftheoveralldisplacement, theparitypatternoftheshearcomponentsisconstrainedbythenecessitythat theparitypatternof H beidenticaltotheparitypatternof .Itiseasily checkedthatfor PxPyPzthisconstraintwillbesatis“edsolongasthecomponentsof H aremadeupof fundamentalelementswhichhaveparitypatternsintermsoftheseasfollows: Hx Px Py PzHy PxPy PzHz Px PyPzFundamentalbasiselements(eitherlo ngitudinalorshear)arelinearlyindependent,intheintervalsde“nedbythe medium,fortheobviousreasonthat sinesandcosineswithdistinctwavenumbersalonganycoordinateformalinearly independentset.Anylinearcombinationofthebasiselementsmustsatisfythe bulkwaveequationandthiswillbeachievedsolongastheelementsindividually satisfyaŽwaveequationforthesamemagnitudewavenumber…whichthenfactors. Thisisaccomplishedbyrequiringthewavenumbersoftermsineachbasiselement tosatisfythebulkdispersionrelations: k2+ 2 y+ 2 z= 2 c2 forcomponentsof (4.6)

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52 k2+ 2 y+ 2 z= 2 c2 sforcomponentsofeach Hi(4.7) Thisconstraintreducesthedegreesoffreedombyone.Icontemplateamodewith a“xed k ,andchooseonetransversedirectionasrepresentingadegreeoffreedom whileconstrainingthesecondacco rdingtoequations(4.6,4.7). Thesefunctionscanbetreatedasorthogonalwithrespecttoeachcoordinate, butthiswillonlybeexploitedobliquely inthesensethatthispropertyisdeeply buriedinthenatureofthetransformthatwillbeappliedtotheirsuperpositions. Asuperpositioncanbemodeledasdiscreteorcontinuous.Onphysicalgrounds, however,thediscretedistributionisthecorrectoneinthiscase.Thephysical realitybeingmodeledconsistsofelast icwavesinboundedmedia.Althoughthe boundaryconditionswillbeviewedthrought helensofmathematicalabstraction, therealityisthatelasticwavesrefract attheboundariesandanysuperposition modelsthesummationprocessthatsuperimposesalloftheirre”ections.Since therearea“nitesetofsurfaces,eachrefractionisadiscreteevent.Thesuperpositionisthusadiscretesummation. Accordingly,thescalarpotentialwillhavetheform = Px( kx )idiPy( iy ) Pz( iz )(4.8) wherethe superscriptdenotesalongitudinalconjugationŽde“nedby i= 2/c2 Š k2Š 2 ipositiveroot(4.9) Fortheshearsuperpositions,theexpansionstaketheform Hx= Px( kx )jaj Py( jy ) Pz( + jz )

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53 Hy= Px( kx )jbjPy( jy ) Pz( + jz ) Hz= Px( kx )jcj Py( jy ) Pz( + jz )(4.10) wherethe+superscriptdenotesashearconjugationŽde“nedby + j= 2/c2 sŠ k2Š 2 jpositiveroot(4.11) Notethat,intheforegoing,both isand iscanrangeoverrealandimaginary valueswithinthesamesuperposition. Thereisafurther,important,observation.Theforegoingsumscontainweighted termsthataresolelyproductsofsinesa ndcosines…atleastwithintheirboundary domains.Sinceeachsuchtermhasde“niteparitywithrespecttothesignofthe wavenumberinitsarguments,theeectofthesignofthewavenumberisalways factorable,inthesenseof Pi( Š kxi)= 1 Š 1 PiPi( kxi) Aseriesoftermswhichdieronlyinthecombinationofthesignsofthewave numbersintheargumentswillalwaysfactorintoasingletermwithpositive wavenumberargumentsmultiplyingasumanddierenceofcoecients.The combinationofcoecientscanalwaysbeabsorbedintoasinglecoecient. Theresultisthatineachoftheforegoingsums,Ialwayschoosearepresentation inwhichonlypositive(albeitrealorimaginary)wavenumbersaresummedover, butforwhichsomeofthecoecientsmaybenegative.

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54 4.3DimensionlessRepresentations Insection2.1thegroundworkwaslaidforreportingtheresultsofderivations inadimensionlessform.Thespeci“cscalefactorsandnotationforthisareas follows: First,recallthatoftherectangularhalf-widths,the hzwillbearbitrarily notatedasthesmallestonewhenever hy = hz.Theappearanceofanunsubscripted h willreferto hz.Thedimensionlessaspectratio a willbe hy/hz.Ofthetwo isotropicvelocities, cswillusedtorescaleresultsandtheratio c/csis,asnoted, denoted R Frequencywillberescaledbytherule = owhere o= cs h (4.12) Thisisacommonrescalinginthehistor icalliteratureoftheproblemand,in addition,itiscommontoreportinunitsof / 2. Consistentwiththisscheme,wavenumberswillbeputintocompatibledimensionlessunitsbymultiplyingthemby h .Thereisnoabsoluteruleonthechoiceof symbolsforlongitudinalversusshearwavenumbersindimensionlessunits,butthe generalattemptwillbetoshowdimensionlessformsbyconvertingLatinletters fromlowertouppercaseandchoosingdistinctGreeksymbolstoconvertexisting Greeksymbolstodimensionlessform.Forexample, K = kh = h and = h arecommonchoices.Thebulkwavedispers ionrelationsindimensionlessterms wouldthentakeformsasfollows: 2 y+ 2 z+ K2=22 y+ 2 z+ K2= 2 R2(4.13)

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55 Insomederivations,afreewave-numbervariable, ,willbeused.Initsdimensionlessform,itwillbedenoted. Theeasewithwhichdimensionlessforms ofresultscanbewrittenfrominspectionofdimensionaloneswillbecomeapparentasexamplesappear. 4.4DerivationofRayleigh-LambEquation IntheIntroduction,itwaspointedoutthatin1889LordRayleighwasable toderiveatranscendentalfrequencyequationwhichdenesthepropagating modesofanisotropicinniteplate.Thisequationisnowuniversallyreferredtoas theRayleigh-LambEquationandthedispersionpatternsitgeneratesareoften calledLambWaves.Derivations,andin troductionsdesignedtopromotereadercompletedderivations,appearofteninth eliterature,butmostoftheseexpositions areconsiderablymorecumbersomethanthederivationabouttobedemonstrated (seeRayleigh[8],Mikl owitz[10],andCh.10ofAuld[38]).Nevertheless,the formofthisequationisoffundamentalimportanceintheresultstofollowand itwillbehelpfultodemonstratehowitcanbere-derivedsuccinctly.Besides producinganeededresult,thisexercisewillprovideaclarifyingexampleofthe devisedspecializednotationatthesametimeitintroducesthebasicpatternfor novelderivationswhichfollow. Thephysicalscenarioconsistsofanisotro picelasticmaterialsandwichedbetween inniteplanesat z = h .Thesurfacesarestressfree.Aplane-wavesystem propagatesalongthe x direction.Atany x position,therearenovariationsalong the y directions. Referringtoboundaryconditions( 2.22), s z p x p y : At z = hzforall y andall x :

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56 Š k2 +2 ,zz= Š 2( Hy,xŠ Hx,y),z,yz=1 2[( Hx,yyŠ Hx,zz)+( Hz,zŠ Hy,y),x] ,xz= Š1 2[( Hy,xxŠ Hy,zz)+( Hz,zŠ Hx,x),y](4.14) However,since y 0,theforegoingwillsimplifydramatically.Inaddition,the simplestvectorpotentialthatwillgenerateashearwavewithno y displacement, isjust: H = 0 Hy0 andsoIcanset Hx= Hz=0.Theboundaryconditionsabovenowcollapseto Š k2 +2 ,zz= Š 2 Hy,xz,zx= Š1 2( Hy,xxŠ Hy,zz)(4.15) AssumingthatIwillnotneedsuperpositions,individualbasiselementsthatproducepotentialsabletosatisfythewaveequationare(accordingtomyalready reasonedbasischaracterizationabove)simply = DPx( kx ) Py( y ) Pz( z ) Hy= A Px( kx ) Py( y ) Pz( z ) With D and A asunknownconstants.Butif y 0,itmustbethat = =0, and Py(0)= E (0)=1andsothepotentialscanbefurthersimpli“edto = DPx( kx ) Pz( z ) Hy= A Px( kx ) Pz( +z )(4.16)

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57 Substitutingequations(4.16)into(4.15) andperformingthedierentiationsat z = h producesthefollowingsimultaneousequations: Š D ( k2 +2( )2) Px( kx ) Pz( h )= Š 2 Ak 1 Š 1 PxPx( kx ) + 1 Š 1 PzPz( +h ) Dk Š 1 1 Px Px( kx ) Š 1 1 Pz Pz( h )= A 1 2 ( k2Š ( +)2) Px( kx ) Pz( +h )(4.17) Atthispointitistrivialtodivideoneequationintotheothertoeliminatethe unknowncoecientsaswellasallofthe x -dependencies.Theresultis ( k2 +2( )2) Pz( h ) k Pz( h ) = 4 k+Pz( +h ) ( k2Š ( +)2) Pz( +h ) (4.18) Itisgratifyingtonoticethatthetwocon”ictingsigncontingenciesfor Pxsimply eachresolvetoanassuredminussign…andthe Pzsigncontingenciesneutralizeeach other.Thisisapatternthatwillrepeatit selfinthemoreinvolvedderivations. Theresultisalreadyintheformofafrequencyequation,Žbutbesidessome rearrangement,thereisa“nalanalytica lsteptobeperformed.Sincethewave systemsaremadeplaneŽbyvirtueof = =0,theconjugationsare: ( )2= 2/c2 Š k2( +)2= 2/c2 sŠ k2(4.19) Recallthat = R2Š 2with R = c/cs.Applyingthesetotheparentheticalonthe leftofequation(4.18),itcanberestatedmoreusefully. ( k2 +2( )2)=( R2Š 2) 2 R2c2 s+2( 2 R2c2 sŠ k2)= Š2 k2Š 2 c2 s= Š ( k2Š ( +)2)

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58 Thedetailsofsuchstepsareindulgedhereb ecausetheyareprototypicalofrearrangementsthatrecurinsubsequentderivati ons,butwhichwillnotbehereafterpresentedindetail.Withthisparticularre-ex pressionsubstituted,andfollowingsome rearrangementofterms,equation(4.18)canbewrittenintheform Pz( h ) Pz( +h ) Pz( h ) Pz( +h ) = Š 4 k2+ ( k2Š ( +)2)2(4.20) Equation(4.20)is,infact,theRayleig h-Lambfrequencyequationinarepresentationwhichencapsulatesbothitsalternativeforms.Theso-calledsymmetricformfollowsfromsetting Pz= E =cosinwhichcasethelefthandside becomestan( h ) / tan( +h ).If Pz= O =sin,thelefthandsidebecomes tan( +h ) / tan( h )andtheequationissaidtobeinitsantisymmetricform. Accordingly,theRayleigh-Lambeq uationisoftenwrittenintheform tan( h ) tan( +h ) = Š4 k2+ ( k2Š ( +)2)2 1(4.21) Inthisrepresentation,thesymmetricandantisymmetricformscorrespondtothe exponentontherightbeingpositiveornegativerespectively.Thenamesgiven theseformsobviouslymatchtheparityof Pzinmyspecializednotation,butthat isnotwhytheywereso-named.Ifthedisplacementpatternscorrespondingtothese equationsaremapped,itisreadilyseenthatthesymmetricformcorrespondsto dilatationsŽinwhichtheplatesurfaces areeitherextendedorindentedtogether ateach x position.Intheantisymmetriccase,thesidesoftheplatearealternately extendingorindenting…givingrisetoarippleeect.Indeed,theantisymmetricŽ solutionsare”exuralinnature.

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59 Finally,theRayleigh-Lambequationindimensionlessform(seesection4.3) wouldbe Pz( n ) Pz( ) Pz( n ) Pz( ) = 4 K2n ( K2 2)2(4.22) 1 i 2 i 3 i 4 i 5 i 0 1 2 3 4 5 6 K hk x 0. 1. 2. 3. 4. 5. 6. h c s inunitsof 2 R 2 3 P z E Figure4.1:Rayleigh-LambDilatationalModes.Inthehistoricaldevelopmentof thesolution,thesearereferredtoasthesymmetricmodes. Therootsofthisequationcanbemappedasdispersioncurvesusingcontour plotting.Therewillbeasetofbranchesfor Pz= E andanotherfor Pz= O .A rearrangementoftheequationsisnecessarytoprecludefataldivergencesinthe numericalcomputationinvolvedintheplotting.Intermsofthedimensionless representation,numericalplottingisbasedonndingtherootsof cos( n )sin( ) sin( b ) bcos( ) Pz[( K2 2]2+4 n PzK2 Pz( n ) Pz( )=0(4.23)

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60 Figure4.1showsbranchesofthedilatationalorsymmetricmodesresultingfrom substituting Pz= E =cosintodimensionlessRayleigh-Lambequation.Figure 4.2showsthebranchesoftheexuralorantisymmetricmodesresultingfrom substituting Pz= O =sinintotheequation. 1 i 2 i 3 i 4 i 5 i 6 i 0 1 2 3 4 5 6 K hk x 0. 1. 2. 3. 4. 5. 6. h c s inunitsof 2 R 2 3 P z O Figure4.2:Rayleigh-LambFlexuralModes.Inthehistoricaldevelopmentofthe solution,thesearereferredtoastheantisymmetricmodes. TheRayleigh-Lambequationdenescoupledmodesoftheinniteplate,but therearealsoasetofuncoupledmodesthatcanpropagateinthisgeometry.Recall thatsurfacecouplingisaninteractionbetweenlongitudinalandthatpolarization ofshearwaveshavingdisplacementcomponentsperpendiculartothesurface.Ican set =0intheboundaryequations(4.14)an dcontemplateplanewavesthatonly havedisplacementparalleltothe z = h surfaces.Thiscanberealizedsimplyby

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61 setting Hx= Hy=0.Theboundaryconditionsthenreducesimplyto 0= Hz,zy0= Hz,zxat z = h (4.24) Shearwaveswithpolarizationresultingin displacementssolelyparalleltoencounteredsurfacesareoftencalledSHwaves(forshear,withdisplacementshorizontal tothesurfacesŽ)incontradistinctiontoSVwaves(forshear,withdispacements verticaltothesurfacesŽ)thatarecoupl edtolongitudinalwavesatsurfaces. 0 1 2 3 4 K hk x 0. 1. 2. 3. 4. h c s inunitsof 2 Even&OddSHModes IndependentofR Figure4.3:In“nitePlateSHModes.Shearwaveswithdisplacementsparallelto thesurfaces,andwhichvanishthere,formanuncoupledpropagatingsysteminan in“niteplate.Thelowestevenandoddsubbandsareshowntogether. Now, Hz,bymychosenbasisrepresentation,musthavetheform Hz= B Px( kx ) Py( y ) Pz( +z )

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62 Thus,itiseasytoseethatthenontrivialsolutionsforthisSHsystemfollowsimply bysettingthe z -derivativeof Pz( +z )at z = h tozero. Pz( +h )=0 t sin( 2 K2) cos( 2 K2) Pz=0(4.25) IwillmakecomparisontotheseSHsolutionsinthesequel.Subbandsofthis solutionssetareshowninFigure4.3.Theyareanalagoustotorsionalmodes ofawaveguidewhosedominantdisplacementpatternarealsocharacterizedby displacementsparalleltothesurfaces. 4.5HowtoTransformSuperpositions There-derivationoftheRayleigh-Lamb equationwasanexerciseinorganizing theproblemintoanalgebraicformamenabletothesimpleeliminationofunknown constants.Thethreeindependentbounda ryvalueequationsconstituteatmost threeconstraints.Absentsuperpositions,thescalingconstantsforthepotentials willconstituteonedegreeoffreedomforthescalarpotentialandasmanyadditional degreesoffreedomastherearedistinctcomponentsofthevectorpotentialto resolve.However,ateachsurfaceonlyone directionalcoordinatewillbexedand sothereispossiblyoneadditionaldegreeoffreedomtoberesolvedwithrespect totheother.TheRayleigh-Lambscenarioreducesenoughdegreesoffreedom tobalancetheconstraints.Specically ,restrictingthescenariotoplanewaves eliminatesthedirectionaldegreeoffreedomateachsurfaceandreducesthenumber ofconstantsfromamaximumoffourtoamanageabletwo.Withthenontrivial constraintsreducedtothesamenumber,asolutionfollows. Intherectangularwaveguidecase,thedirectionaldegreeoffreedomateachsurfacepersistsbythefactthatplanewavesnolongersuce.Barringsomefortuity,

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63 theconstraintequationswillgotosixthreeforeachsurfacewithnoeliminations. Aquestionarises:howmanyindependentvectorpotentialcomponentsmustthere beandthushowmanydegreesoffreedomduetothem?Twodirectionaldegrees offreedomplusthescalarplusatleastonevectorpotentialcomponentmakes theminimumdegreesoffreedomtobefour.IfIcoupletheadjacentsides,the independentconstraintsmaybereducedandifIincludeadditionalvectorpotential componentsIcanincreasethedegreesoffreedom.ButasIhavedemonstrated (section2.5)abstractly,nonseparabilityc ompelstheintroductionofsuperpositions andanewproblemarisesofhowtoresolvetheirexpansioncoecients.Thegoal ofdevisingatransformationistod ealwiththislattercomplication. Oncetransformsareappliedtoasuperpositionsubstitutedintoarectangular system,itwillbeseenthatthereareelementaryRayleigh-Lambrelationships betweencomponentsofthecoupledpotentials.Thisresultmightbeanticipated qualitativelybynoticingthatmanydis persionsubbandsrevealedbynumerical approximationsoftherectangularcasebearuncannyresemblancetoRayleighLambdispersionsofaplate.(Thiswas noticedsomewhatbyNishiguchi[1] 3). GiventhatIhaveexplicitlychosenarepresentationofthepotentialsassums ofproductsofsinesandcosines(overonlypositivewavenumbers),theboundary conditionsarereadilyvisualizedaseq ualitiesamongsumsofexponentials.A Fouriertransformisthennaturallysu ggested.Thedomainofthetransformed termsis,however,nite,andsoitisnecessarytoeitherperiodicallyextendthe functionsbeyondtheboundaryortomodifythedenitionofthefunctionssothat theyvanishorconvergetowardszerobeyondtheboundary.Asimpleperiodic extensionwouldhavethedesirableconseq uenceofproducingdeltafunctionsunder aFouriertransformcorrespondingtoeachtermbutonlyifthewavenumbers wererealandthecoordinatewasbeingmappedtorealtransformvariablesor

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64 imaginaryandbeingmappedtoimaginarytransformvariables.However,itisto beexpectedthatagivenpotentialisrepresentedbyasumcontainingbothrealand imaginarywavenumberswiththeresultthatastandardFouriertransformwould diverge.TheinescapabilityofthisexpectationcomesfromexaminingRayleighLambsolutionsforwhichthefundamentalmo desdistinctlyinvolvetransversewave numberswhicharepureimaginary.Inaddition,asanticipatedinsection4.1,a periodicextensionwould,atleastabstractly,rendertheassumptionssupporting applicationoftheHelmholtzTheoreminvalid. Preservationoftheabilitytotransformin todeltafunctions,despitedivergence oftheFouriertransformandotherissues,canbesecuredbyasimpleexpedient. InsteadofthegeneralFouriertransform,Iuseasimpli“edŽtransformwhichis validonlyonexponentials(thoughwithoutregardtowhetherthewavenumbersare realorimaginary)andwhichavoidsthedivergenceproblembyvirtueofthedetails ofitsde“neddomain.Moreover,sincethedomainoffunctionstobetransformed isexplicitlyconstrainedbythephysicalb oundariesofthesample,thereisnothing illogicalorrestrictiveinde“ningtheapplicabledomainforthetransformtoinclude onlyexponentialsde“nedonthecoordinateintervalsthatmeasurethesampleand thusthereneednotbeanyconcernoverperiodicextension.Thetransformtobe usedthenhasasimplede“nitiondeterminedbytheelementmapping aeiq 2 a ( Š )(4.26) Š hq q hq( q = x,y,z ) a C Thefactorof2isaconveniencetodisposeofthefactor1 2intheexponential representationofthesineandcosine.

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65 Itisalmostself-evidentthatthetransformis1-1betweenthesetofexponentials andsetofdeltafunctionssorestricted.Since a canbezero,anadditiveidentity existsoneachsideandwehaveanisomo rphismbetweentwogroups.Becausethe rangeof and isde“nedtoencompassbothrealandpureimaginaryvalues,the transformoperateswithoutdicultyonanycombinationofexponentialswithreal orimaginarywavenumbers. Itistroublesometowritedownanintegralformofthistransformwhich smoothlyadaptstowhethertheargumentisrealversusimaginaryandwhich limitsitselftothecoordinateboundaries .Ofcourse,theunderlyingmechanismis atrivialFouriertransform.Fortunately ,sincethedomainoffunctionsisstrictly limitedandtheelementmappingfromth atdomaintothetransformdomainis clearandunambiguous,thetransformca nbeperformedwithoutdiculty.Itis interesting,therefore,thattheinversetransformcanbetriviallywrittendownin anintegralformthatisnottroublesomeand whichappliesadaptivelytotransforms involvingimaginarydeltaargumentsaswellasrealones.Oneexamplewouldbe f ( y )= TŠ 1 y { f ( ) } = 1 2 =+ = Šeiyd + =+ i = Š i eiydOfcourse,thisisonlyvalidfor f ( )thatareproducedbythesimpli“edŽtransforminthe“rstplace. Theleftorrighthandsideofanyboundaryvalueconstraintwillinvolveoneor moresumsinthefollowinggeneralform(whichomitscommonfactorsof Px( kx ) whicharealsosubjecttodierentiation):jajf ( j, j) Pq1( jq1) Pq2(  jq2)(4.27) Here,Ihavegeneralizedthevariouscases:

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66 q1,q2standfordistinctcoordinates y or z ; P1,P2aredistinctfunctionvariables,orderivativesofthem: Pyor Pz; willbean or forrepresentationsofthescalarorcomponentofthevector potentialrespectively; isanabbreviationfortheconjugatewavenumberbasedupontheapplicable velocity: = 2/c2 Š k2Š 2+= 2/c2 sŠ k2Š 2; f ( j, j)willbeaprefactorresultingfromoneormorederivativestaken.These willalwaysbeasingleproductorsumsofproductsof jand/or  j.To characterizetheeectoftransformso ntheboundaryexpressions,itwillbe sucienttocontemplate f ( j, j)asbeingasinglesuchproductsinceasum ofsuchtermscanbedistributedtoproducesumsofsummations. Bysummarizingbelowhowthesimpli“ edŽtransformaectsboundaryvalue termsgenericallyde“nedbyequation(4.27),itwillbepossibletoimmediately writedownthetransformsoftheactualbo undaryconditionswithoutelaboration. First,with y chosentomaketheexampleconcrete,considerthegeneraleect ofthetransformonafunctionvariable. Ty{ Py( y ) } = 1 Š i Py ( Š )+ 1 Š 1 Py ( + ) (4.28) Icannowwritedownthetransformofequation(4.27)withrespectto q1 Notethat standsforwhichevertransformdi mensionvariableismatchedto q1. Myconventionhenceforthwillbethat y and z .Intherenderingof transforms, q1and q2couldbeeither y or z ,thoughalwaysdistinctinagivencase.

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67 WhilethereaderispresumedcapableofwritingdownFouriertransformsof sinesandcosinesonhis/herown,theusualresultsaresomewhatsimpli“edand adjustedinthiscasebyvirtueofthesti pulationthatsuperpositionswillalways bechosentoinvolveonlypositive(albeitpossiblyimaginary)valuesofwhatever wavenumbervariable designates.Theresult,therefore,isthatonlyoneofthe twodeltatermssurviveineachcaseandthespeci“cformoftheresultdepends uponthesignofthetransformvariableinawaythatcanbeneatlysummarized. So,forconvenienceIlisttheresultsindetail: Tq1 jajf ( j, j) Pq1( jq1) Pq2(  jq2)=jajf ( ,) 1 Š i Pq 1 ( Š j) Pq2( q2)for > 0(4.29)jajf (0 0) 1 Š i Pq 1 2 0 Pq 1 ( j) Pq2(0q2)for =0(4.30)jajf ( | | ,) 1 Š i Pq 1 1 Š 1 Pq 1 ( | |Š j) Pq2( q2)for < 0(4.31) Similarly,Icannowwritedownthetransformofequation(4.27)withrespectto q2.Thereareagainthesamethreecasesdependinguponthesignor Tq2 jajf ( j, j) Pq1( jq1) Pq2(  jq2)=jajf ( , ) 1 Š i Pq 2 ( Š  j) Pq1( q1)for > 0(4.32)jajf (0, 0) 1 Š i Pq 2 2 0 Pq 2 (0 Š  j) Pq1(0q1)for =0(4.33)

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68jajf ( , | | ) 1 Š i Pq 2 1 Š 1 Pq 2 ( | |Š  j) Pq1( q1)for < 0(4.34) Havingbeenmeticulousinmotivating,t henjustifying,andnowdemonstrating theeectsofthissimpli“edŽtransform.Itcanbedrasticallysimpli“edinpractice withthefollowingobservations: Thepriorstipulationthatexpansionswillonlybeoverpositive,thoughpossibly includingimaginary,wavenumbershasallowedtheresultsoftransformsinthe contextoftheproblemtobemoreeasilysummarizedintermsofsingledelta functions.Oneadditionalfactcannowliter allytrivializetheuseofthistransform inpractice.Namely,thefactthatanygivenderivationtakesplaceinthecontextof aspeci“cparitypatternguaranteesthatwithinanyderivationtheparitypattern oneachsideofaboundaryequationwill beidentical.Thisparityagreement guaranteesthatwheneithercoordinateistransformed,the P functiontransformed oneachsidewillbethesame.Thatbeingthecase,alloftheprefactorswhich dependontheparityof P willcancelbetweenthesidesinallcaseswherethat P functioniscommontoalladditiveterms. Ingeneral,thisconditionnearlyalways ful“lled.Moreover,withthosedistinct ionsgone,anexaminationofequations (4.29-4.34)willrevealthatreplacing with | | throughoutissucienttocoverall cases.Therfore,asapracticalmatter,theonlyrulethatwillbeneededisjust Replaceevery Pq( qq )tobetransformedwith ( | |Š q). Itwillnotmatterwhether qisawavenumberoritsconjugate. Itwillnotmatterwhether qisrealorimaginary(itisguaranteedto bepositive). Thetrivialityofthisruleisadirectconsequenceofhavingimposedameticulous seriesofspeci“cchoices.Itisneitherfortuitousnorcouldithavebeenreadily anticipatedthatitwouldreducetothis.

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69 Onceaboundaryequationthathashadas uperpositionsubstitutedintoit becomestransformedinthisway,itbeco mesanequalityintermsofthetransform variable.Giventhebehaviorofdeltafunctions,eachvalueofthetransformvariable will,oneachsideofthesummation,selectouteitheraspeci“ctermofthesum, orbeidenticallyzero.Thiscollapsestheequalityoffunctionsofsumsintoa constraintbetweencomponentsofsumsfr omeachside.Correlatingtransforms over y withthoseover z isaremainingissue,buthowthismustbedonewillbe developedinthederivationswhichfollow.

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CHAPTER5 DERIVINGNORMALMODESOFPROPAGATION 5.1GeneralConsiderations Thefullboundaryconditionsrevealedgen ericallybyequations(2.22)inferthat allthreecomponentsoftheshearvectorpotentialaremixedtogetherinsatisfying thestress-freesurfaceboundaryconstraint. If,however,Irestrictattentiontojust thosemodeswhichpropagate,itisnotimmediatelyclearwhetherallthreevector componentsareneeded.SomeinspirationcanbedrawnfromtheRayleigh-Lamb derivationofsection4.4.Thatderivationinvolvesaninniteplateboundedin the z directionsandonlyrequiresthe Hycomponentofthevectorpotential.The intimationisthatboundingalsointhe y directionsmightsimplyinvoketheneed forthe Hzcomponent,butthereisnoapriorireasontoexpecttoneedan Hxcomponentaswell. Theforegoingmotivatesanattempttondtheessentialrelationshipbetween Hxandtheothervectorcomponentswhichp articipateinsatisfyingtheboundary conditionsalongthesurfaceofawaveguideinwhichnormalmodespropagatein the x direction.Unlesssomeconstraintcanbefoundthateliminatescomponentsor establishessomedependencyamongthem ,therewillbemoredegreesoffreedomin thepropagatingproblemthenconstraints availabletoresolvethem.Ithusproceed toinvestigatethisrelationshipamongcomponentsinawaythatisindependentof theboundaryconditionspersesoastoco ndentlynarrowtheapproachesusedin solvingtheboundaryproblems. 70

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71 5.2AcousticPoyntingVectorofaNormalMode Propagatingmodescarryenergy.Inanalogywithelectrodynamics,therewill beavectorthatindicatesboththedirectionandmagnitudeoftheenergy”ux.This vector,theacousticPoyntingvector,describesaphysicalrealitywhoseinvariances undermanipulationofthecoordinatesyst emconstrainitsmathematicalform.The componentsof H participateintheconstructionofamodeviaantisymmetries arisingfromthefactthatsheardisplacementis H .Moreover,satisfyingthe boundaryconditionsostensiblyinvolvesmixingupthecomponentsof H .This suggestsIlookforconstraintsontherelationshipbetweencomponentsof H that mightberequiredtomaintaininvariancesofthePoyntingvectorwhilepreserving theantisymmetriesbuiltintotheshearcontributionstothemode.Somefamiliarity withtheacousticPoyntingVectorrevealsittobeacomplicatedobjectwhen expressedintermsofstrainandthisprovokesacuriosityoverwhetheritmay harborsuchconstraints.Itisdiculttoarticulatefurtherwhatis,intheend,an intuitionthatthismightbeso.Theintu itionwillultimatelybejusti“edbythe result. ArepresentationoftheacousticPoyntingVectoritselfcanbereadilyderived. ThePoyntingVectorwillbedenotedhereinas J since P shavebeenextensively usedforanotherpurpose.Ititsde“nedbythepropertythatintegratingitovera surface S producestheenergy”uxthroughthatsurface. E,t=SJinids (5.1) where n isthesurfacenormalvector.Componentsoftheforcedensityatapoint onthesurfaceare Fj= Š ijni(5.2)

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72 Thesearetheinternalforceswhicharearesponsetostrain.Thoughinconsequentialinwhatfollows,theminussignwhichappearsabovepropagatestooneinthe expressionfor J whereitislikelytoseemcounterintuitive. Forcedensitytimesdisplacementisjustworkdensityandthusthemeasure ofenergytransportperunitvolume.Takingtheforcedensityasconstantover innitesimaldisplacements,thetimerateofchangeof Fjujisjust Fjuj,t.Theresult isthatpowerdensityissimplythescalarproductofforcedensityandvelocitya resultfamiliarfromelementarymechani cs.Applyingthistoequations(5.1)and (5.2),theexpressionforuxandPoyntingvectorcomponentsitinfersare E,t=S( ijnj) ui,tdS t Ji= ijuj,t(5.3) Ishallbeconcernedonlywithinvaria nceofdirectionalityandsymmetrypropertiesand,sinceonlynormalmodesarerelevant,Iwilldispensewiththetime derivativeandignoretheminussign. Forani sotropicmaterial,dividingequation(2.17)bymaterialconstant s csand b expressestheproportionalityofstresstostrainindependentofthematerial. ijr nijrurr+2 uij(5.4) where n = R2 2.Fromthediscussionsfollowingequation(2.18),Icanreplace theinvariantsumwith 2 = k2 ijissymmetricinitsindices.Substituting thisresultintotherepresentationfor J inequations(5.3),Icanexpresstheproportionalityofthetime-independentpartof Jitoafunctionoftheundierentiated scalarpotential,totalstrain,anddisplacement. Jir nk2 ui 2 ujuij(5.5)

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73 uijseparatesintoshearandlongitudina lpartsandthelongitudinalpartis immediatelyexpressibleintermsofthescalarpotentialusing u( ) ij= ,ij(5.6) whereasapplyingde“nition(2.2)to u( s ) ijandreplacingthedisplacementsthereof bycurlsoftheshearvectorpotentialrequiresthemoreinvolvedsubstitution u( s ) ij=1 2( iH,j+ jH,i)(5.7) Correspondingly,componentsoftherema iningdisplacementtermcanbeexpanded intermsofpotentialsby uj= jabHb,a+ ,j(5.8) Withtheforegoingsubstitutions,the ujuijontherightsideofequation(5.5) expandsto ujuij=1 2( jabiHb,aH,j+ jabjHb,aH,i) +1 2( iabHb,aj+ jabHb,ai) ,j+ jabHb,a,ij+ ,j,ij(5.9) Afteratediousamountoftensoralgebra,the“rsttermontherightofequation (5.9)reducesto 2 H[ a,i ]H[ b,a ] ,b+2 H[ a,b ]Ha,bi(5.10) Theremainingtermsdonotsimplifyinusefulwaysatthislevelofexpression. Collectingtheforegoingresults,equation (5.5)cannowbefullyexpressedinterms

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74 ofthescalarpotentialandcomponentsofthevectorpotentialby Ji ( k2 l )( iabHb,a+ ,i) Š 4( H[ a,i ]H[ b,a ] ,b+ H[ a,b ]Ha,bi) Š ( iabHb,aj+ jabHb,ai) ,jŠ 2 jabHb,a,ijŠ 2 ,j,ij(5.11) Theactualphysicaldirectionofenerg ypropagationshouldbeinvariantunder exchangeofthetransversecoordinatesandso,usingequation(5.11),Ishowthat inorderforthistobetrue,the y and z componentsof H cannotbeallowedto mixwiththe x component. Speci“cally,sincethePoyntingvectoriscomposed,inpart,ofcurlsof H ,any exchangeof y and z willinteractwiththehandednessofthecoordinatesystemto requireanappropriateantisymmetry.Tobeconsistentwiththis,thewavenumber inthe x directionmustalsobetakentochangesignwithany y z interchange andsoallderivativeswithrespectto x willalsoberequiredtochangesign.Inview ofthetranslationalinvarianceofthemodes,thisissimplyare”ectionoftheneed tochange k ,thecommonwavenumberinthe x direction,to Š k toaccommodate inversionofthe x direction. ThefullexpansionofthePoyntingvectorrevealsacomplexmixingof x versus y,z componentsof H ,butIshallshowthatfailuretoseparatesolutionstoavoid thismixingresultsinfailuresinthe y z symmetriesofsomeofthetermsand thereforeimpliesthatsolutionsmust bebuiltdistinctlyfromcasesinwhich Hx=0 versusthoseinwhich,alternatively Hy= Hz=0.Itshouldalsobenotedthatthis willturnouttobeconsistentwithananalysisofthe k =0casetofollow.Inthe k =0caseallderivativeswithrespectto x vanishandtheboundaryconditions

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75 naturallytakeaformwhichre”ectsadecouplingofthe x versus y,z components of H The“rstadditivetermontherighthandsideofequation(5.11)contributesa termtothe x componentofthePoyntingvectorthatisproportionateto xabHb,a+ ,x=2 H[ z,y ]+ ,x(5.12) whichisfullyantisymmetricunder y z onceweincorporatetherulethat ,x Š ,x.Moreover,thisantisymmetryispreservedevenifeitheroneoftheterms vanishes. Thesecondtermontherighthandsideofequation(5.11)contributesaterm tothe x componentofthePoyntingvectorequalto Š 4H[ x,y ]( H[ x,y ] ,x+ H[ y,z ] ,z) + H[ x,z ]( H[ x,z ] ,x+ H[ z,y ] ,y)+2 H[ y,z ]H[ y,z ] ,x(5.13) andbeforesignchangesduetodierentiationby x ,thisexpressionistotally symmetricunder y z .Expansionoftheantisymmetricpartsentailsproduction ofthepair ... Š Hy,xHx,yx... Š Hz,xHx,zx... (5.14) andthusdoesnotuniformallyassembleoddnumbersof x dierentiationswith symmetric yz termsandsotheneededantisymmetryisnotfullyrealizedunless sometermsvanish.If Hxissettozero,thenitiseasilycheckedthatantisymmetry willberealized,towit: ŠHy,xHy,xxŠ Hy,xH[ y,z ] ,z+ Hz,xHz,xxŠ Hz,xH[ z,y ] ,y+8 H[ y,z ]H[ y,z ] ,x(5.15)

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76 Similarly,if Hxdoesnotvanish,but Hyand Hzvanishsimultaneously,then, again,theresultingexpansionwillbecomeantisymmetricunder y z oncethe x -derivativesareconsidered,towit: Š [ Hx,yHx,yx+ Hx,zHx,zx](5.16) Thethirdtermontherightofequation(5.11)contributesthefollowingterm tothe x componentofthePoyntingvector: Š 2H[ z,y ] ,x,x+ H[ z,y ] ,x,y+ H[ z,y ] ,z,z+ H[ z,y ] ,x,x+( H[ x,z ] ,x,y+ H[ x,y ] ,x,z)(5.17) Under y z thisistotallyantisymmetricwhenthederivativesof x areconsidered.Moreover,if Hxvanishesor,alternatively Hyand Hzvanishtogether,the antisymmetryoftheresultispreserved. Finally,thelasttwotermsontherightofequation(5.11)contributethe followingtermstothe x componentofthePoyntingvector: Š 4H[ z,y ],xx+( H[ x,z ],xyŠ H[ x,y ],xz)Š 2[ ,x,xx+( ,y,xy+ ,z,xz)](5.18) Itiseasilycheckedthatthedesiredantisymmetryispreservedandthat,again,the vanishingeitherof Hxaloneor Hyand Hztogetherdoesnotchangethisresult. Theconclusionisthat,whenassemblingapropagatingnormalmode,theshear contributionmustbemadeoutofcomponentsforwhichallthe Hxsarezero,or forwhichallthe Hyand Hzpartsarezero.Inconsideringhowtorepresentthe shearsuperpositionsofapropagatingnormalmode,thereisnocaseinwhicha

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77 superpositionfor Hxwillbemixedwithonesfor Hyand Hz.Thisremovesatleast onedegreeoffreedomfromtheproblem. 5.3PropagatingModesInvolving Hy, HzShear 5.3.1DerivingtheFrequencyEquations TheeasewithwhichtheRayleigh-Lambso lutionisderivedinspiresaderivationthatfollowsthesamepattern.Armedwiththeconclusionthat Hxcannot evenbeaccommodatedinanormalmodesolutionthatalsoincludes Hyand Hzcomponents,Iproceedtoderivethespectrumofpropagatingmodeswith Hx=0. Accordingly,fromequations( 2.22),theboundaryconditionsat z = hzwith s z p x p y ,and Hx=0become nk2 +2 ,zz=2 Hy,xz,zx= 1 2[( Hy,xx Hy,zz)+ Hz,zy] ,zy=+1 2( Hz,zx Hy,yx)(5.19) Asareminder,underthebasisrulesdevisedforthisproblem(seesection4.2),the representationsofpotentialswillhavethefollowingforms: = Px( kx )idiPy( iy ) Pz( iz )with i= 2/c2 k2 2 iH = Px( kx ) 0jajPy( jy ) Pz( + jz )jbj Py( jy ) Pz( + jz ) with + j= 2/c2 s k2 2 j

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78 Substitutingintothe“rstboundaryco nditionofequations(5.19)Iobtainidi( k2 +2( i)2) Py( iy ) Pz( ihz)= 2 k 1 Š 1 Pxj+ jPy( jy ) 1 Š 1 PzPz( + jhz)(5.20) ThesimpleŽtransformdevisedins ection4.5isthenappliedsothat Py( iy ) ( | |Š i)and Py( jy ) ( | |Š j).Bychoosingavalue oofthetransform variable suchthat o{ i}{ j} ,thesumsonbothsidescollapseleavingthe followingequality: do( k2 +2( o)2) Pz( ohz)=2 k 1 Š 1 Px 1 Š 1 Pzao+ oPz( + ohz)(5.21) Thisprovidesoneconstraintonpossiblecombinationsof( ,o)atagivenvalue of k .Hereandinsubsequentsteps, o| o| Repeatingthisprocessforthesecondboundaryconditioninequations(5.19), thetransformedversionoft hesecondconstraintbecomes dok Š 1 1 Px o Š 1 1 Pz Pz( ohz)= 1 2 Pz( + ohz) ao( k2Š ( + o)2) Š boo( + o) 1 Š 1 Py Š 1 1 Pz (5.22)

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79 Substitutingintothethirdboundaryconditioninequations(5.19),thetransformedresultis doo o Š 1 1 Py Š 1 1 Pz Pz( ohz)= 1 2 k 1 Š 1 Px bo+ o Š 1 1 PzŠ aoo Š 1 1 Py Pz( + ohz)(5.23) Equations(5.22)and(5.23)canbereconciledintooneconstraintby“ndinga relationshipbetween aoand bothatrendersthemequivalent.Dividingthetwo equationswilleliminatethetranscendentaltermsandsomecommonfactors,leaving Š k2 o= ao Š 1 1 Py( k2Š ( + o)2)+ bo Š 1 1 Pzo+ o bo Š 1 1 Pz+ oŠ ao Š 1 1 Pyo(5.24) Solvingthisfortherequiredrelationshipbetweencoecientsyields bo Š 1 1 Pz= ao Š 1 1 Pyo+ o k2+ 2 o(5.25) Substitutionofequation(5.25)intoeither equation(5.22)or(5.23)toeliminate bowillproducethefollowingresult: do Š 1 1 Px o Pz( ohz)= 1 2 aok( k2+ 2 o) Š ( + o)2 k2+ 2 o Pz( + ohz)(5.26)

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80 Havingreducedthesystemtotwoequation swithtwounknowncoecients,those coecientscanbeeliminatedbydividingequation(5.26)intoequation(5.21). First,however,inspiredbyasimilarstepintheRayleigh-Lambderivation,the parentheticalontheleftsideofequation(5.21)canbemanipulatedandfoundto haveaconvenientequivalentrepresentationwhichcoincideswithasubexpression inequation(5.26). ( nk2 +2( o)2)= (( k2+ 2 o) ( + o)2) Substitutingthisandthenperformingthe divisionfollowedbytheusualrearrangementproducesthefollowingfrequencyequation: Pz( ohz) Pz( ohz) Pz( + ohz) Pz( + ohz) = 4 o+ o( k2+ 2 o) [( k2+ 2 o) ( + o)2]2(5.27) Comparingwithequation(4.20)thisisseentobeaRayleigh-Lambequationwith thequantity k2+ 2 oplayingtheroleofthemagnitudeofthepropagationvector. Thisequationisparameterizedby oandisvalid,atagiven k ,onlyforcertain combinationsof oand t .Consider,nevertheless,whathappenswhen k o(or, eectively o 0fornite k ). o= 2/c2 k2 2 o k= 2 c 2 k2= (inequation(4.21)) + o= 2/c2 s k2 2 o k+= 2 c 2 s k2= (inequation(4.21)) (5.28) Therefore,atlarge k ,itisexpectedthatequation(5.27)becomesmorelikethe classicRayleigh-Lambequation(4.21) forplanewavesinaninniteplate.Atthis intermediatepoint,theconstraintofboundaryconditionsatthe y = hysurface

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81 hasnotyetbeenimposed,buttheimplicationneverthelessisthatforlarge k thesysteml ookssubstantiallylikeaRayl eigh-Lambsystem.Thisimpliesthat as k 0theeectfromthesidestendtodecoupleandthepropagationl ooks increasinglylikesimpledecoupledRayleigh-Lambplane-wavepropagation. Itnowbecomesimportanttofocusonasimpleobservation.Theforegoingstep producedanequationintermsofthetransformvariable whosepermissiblevalues (denoted o)comefromthesetofvaluesin { i} and { j} (whatmathematicians wouldcallthesupportfor and Hy, Hz).Therecouldbemoreelementsof thosesupportsetsthanpossiblevaluesof oandtheseadditionalvaluesmaybe uniquelyconnectedtothetransformvariable appliedtoboundaryconditionsat theadjacentsurface.However,connectingthe y = hyand z = hzsurfaceswill bepossibleonlytotheextentthatsomeofthevalueswhichthetwotransform variablestakeonareindeedsharedselectionsfrom { i} and { j} Fromequations( 2.22),theboundaryconditionsat y = hywith s y p z p x ,and Hx=0willbe nk2 +2 ,yy=2 Hz,xy,yx=1 2[( Hz,xx Hz,yy)+ Hy,yz] ,zy= 1 2( Hy,yx Hz,zx)(5.29) Thetransformedrstboundarycondition fromequations(5.29),beforeapplicationofthedeltafunctions,willbe idi( nk2 +2 2 i) Py( iahz) ( i )= 2 k 1 1 Pxjj 1 1 PyPy( jahz) ( + j )(5.30)

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82 Tolinktheadjacentsurfaceconditions,Irelyuponthefactthateachrootof equation(5.27)necessarilycorrespondstotheexistenceofspecicelementsofthe supportsets { i} and { j} .Infact,if ois,withsomevalueof t ,arootofequation (5.27),itissolelybecause o{ i} and o{ j} suchthat o= o= o. Now,theleftsuminequation(5.30)awillnecessarilyencounter o= o. Suppose,then,thatIcontemplatethevalueof correspondingto o= ofor whichthedeltafunctionontheleftofequation(5.30)isnonzero.Obviously,itwill be o.If,forthatvalueof ,therighthandsideofequations(5.30)isnottrivially zeroforthesamevalueof ,theremustexistsome 1suchthat + 1= = o. Inthealternative,Icouldrstcontemplateavalueof forwhichtheargument ofthedeltafunctionontherightofequation(5.30)iszero.Obviously,itwillbe + o.If,forthatvalueof ,thelefthandsideofequations(5.30)isnottrivially zeroforthesamevalueof ,theremustexistsome 1suchthat 1= = + o. TheupshotofthisreasoningisthatIcanconnectthetransformsofthetwo setsofboundaryconditionsbycontemplatingsimultaneous( t,o)rootsofboth ofthem.Towritethetransformedboundaryconditionsfortheadjacentsidein termsof o,Irequireeitherthat osothat,byoperationthedeltafunction,thewavenumbervariablestakeonvalues: i o= odi dowhichwillbeeliminated i oj 1i.e.presumedtoexist ai a1unknown,buttobeeliminated

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83 + j oj=( o)+(5.31) orIrequirethat + osothat,byoperationthedeltafunction,thewavenumbervariablesalternatively takeonvalues: j o= obj bowhichwillbeeliminated j + oi 1i.e.presumedtoexist ai a1unknown,buttobeeliminated i + oi=( + o)(5.32) Iwillnametherstofthesealternatives(i.e.,equations(5.31))L-Conjugation sinceitispremisedonequatingthelongitudinalconjugationofthe z -surface solutionwiththe y -surfacesolution.Thesecondalternative(i.e.,equations(5.32)) S-Conjugationsinceitispremisedonequatingtheshearconjugationofthe z -surfacesolutionwiththe y -surfacesolution. BecauseIhavestipulatedthatallmembersofthesupportsetsarepositive andbecauseonlypositivesquarerootsareused,alloftheprecedingmappingsare guaranteedtobeunambiguous.Thereaderisinvitedtoreviewthedenitionsof conjugationdenotedwith*and+superscriptswhichwereintroducedinconnectionwithequations(4.8)and(4.10).Fromthosedenitions,itcanbenotedthat

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84 theseconjugationshavetheproperty ( )= ( +)+= Adetailedexpansionofthenalrelationinequations(5.31)is ( o)+=! " # t2 c2 s ( o)2= ( + o)2+ 2 o ( o)2(5.33) andofthenalrelationinequations(5.32)is ( + o)=! " # t2 ( R2c2 s) ( + o)2= ( o)2+ 2 o ( + o)2(5.34) Theseexpansionsillustratethegeneral rulethatL-ConjugationandS-Conjugationsolutionsarerelatedbystraightforwardsubstitutionsofvariables.Itwill thusbesucienttocompletedetailsoftheo ngoingderivationfortheL-Conjugate caseandthenstatetheanalogousresultsfortheS-Conjugatecase.Applying L-Conjugationtoequation(5.30),andaftercollapsingthesums,theresultis do( nk2 +2 2 o) Py( oahz)=2 k 1 1 Px 1 1 Pyb1( o)+Py[( o)+ahz](5.35) Similarly,thesecondandthirdboundaryconditions(5.29)underL-Conjugation caneventuallybeputintothefollowingforms: dok 1 1 Pxo 1 1 Py Py( oahz)=1 2 Py[( o)+ahz] b1[ k2 (( o)+)2]+ a1 o( o)+ 1 1 Py 1 1 Pz (5.36)

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85 doo o Š 1 1 Py Š 1 1 Pz Py( oahz)= Š 1 2 k 1 Š 1 Px a1( o)+ Š 1 1 PyŠ b1 o Š 1 1 Pz Py[( o)+ahz](5.37) Dividingequations(5.36)and(5.37)toputone a1intermsof b1,Iobtain a1 Š 1 1 Py= b1 Š 1 1 Pz( o)+ o k2+(( o)+)2(5.38) Itmaybenotedthatequation(5.38)isnotidenticalto(5.25).Thishighlightsthe factthat a1and b1areexpectedtobedistinctfrom aoand bo.Ifequation(5.38) issubstitutedintoeitherequation(5.36)orequation(5.36)toeliminate a1,the resultisidentical,towit: do Š 1 1 Pxo Py( oahz)= Š 1 2 b1k( k2+( o)2) Š (( o)+)2 k2+( o)2 Py[( o)+ahz](5.39) Inwhatisbynowaritual,theparentheti calontheleftofequation(5.35)can bemanipulatedtoshowitsrelationtothenumeratorinthefractionontheright handsideof(5.39). ( k2 +2 2 o)= Š [( k2+( o)2) Š (( o)+)2](5.40) Havingreducedtheunknowncoecientstoonly doand b1,equation(5.39)canbe dividedintoequation(5.35)toeliminatet hemandproducethefrequencyequation

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86 associatedwiththe y = hyside. Py( oahz) Py( oahz) Py[( o)+ahz] Py[( o)+ahz] = 4 o( o)+( k2+( o)2) [( k2+( o)2) (( o)+)2]2(5.41) Onceagain,thisisaRayleigh-Lambequation.Her e,thequantity( k2+( o)2) playstheroleofthesquareofthemagnitudeofpropagation. Equation(5.41)ismanifestedinthec lassicRayleigh-Lambstructurewhich showstheconsistencyoftheresultwithprecedingderivations.H owever,inthis case,thatconsistentm anifestatio nmasksaninterestingfeature.Namely,all k dependencyinequation(5.41)cancelsinternally.Toseethis,expandtheterms whichostensiblyappeartoshowa k dependencyandobservethat k iseliminated. ( o)+=! " # t2 c2 st2 c2 k2 2 o=! " # t2 c2 s$1 1 R2%+ 2 o( k2+( o)2)= k2+t2 c2 k2 2 o= t2 R2c2 s 2 oEquations(5.27)and(5.41)dene( t,o)rootsystemsforindependentequations.Thecoincidencesofthesesimultaneousconstraintsdenevaluesof t that constitutetheeigenspectrumwithrespecttotheL-Conjugationcase.Thereisa setofcommonrootsforeachdistinctvalueof k .Plottingthe k versus t dispersions requiresmethodicalplottingatdierentvaluesof k ,but,inprinciple,thespectrum (forL-Conjugation)hasbeenanalyticallyandpreciselyspecied. Theentirederivationisalsoinvariantunder y z exchangethoughthismight notbeimmediatelyapparent.Thekeytor ealizingthatitmustbesoistorealize thatrelabelingthedirectionsmustalso beappliedtotherepresentationsofthe potentialsassuperpositionsandthatwit hdirectionsrelabeledtheaspectratiois

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87 denedby hz= ahy.Thereadercaneasilyverifythattheresultsofderivation willbeequivalenttotheforegoing. Repeatingthesamesequenceofstepsfromequation(5.30)tothepresent point,usingS -Conjugationdenedbyequ ations(5.32),theS-Conjugateanalog toequation(5.41)is Py(( + o)ahz) Py(( + o)ahz) Py[ oahz] Py[ oahz] = 4 o( + o)( k2+( + o)2) [( k2+( + o)2) 2 o]2(5.42) 5.3.2Interpretation Hadanyonebeeninsightfulenoughtoanticipatethatcoupledmodesofa rectangularwaveguidecouldbecharacterizedbyacoincidenceofRayleigh-Lamb solutions,provingthatitwassowouldh averemainedaselusiveashistoryshows themainproblemtohavebeen.Moreover,thereareaspectsoftheresultwhich, haditbeensomehowforseenaspossible,wouldhavearguedagainstbelievingit. Themai nimpedimentwouldhavebeenthattherei sani ntrinsicinterferencebuilt intotheresultwhichprecludesboundarysatisfactionatbothsurfaceswithout therestofthesuperposition.Althoughtheelegant-lookingresultinvolvesa coincidencewiththefullsuperposition itisstillnotthefullsuperposition.What isintriguingisthatIdonotneedtohaveafulldescriptionofthesuperpositionin ordertondtheeigenspectrum. Thederivationtellsusthatanysuperpositionofshearandlongitudinalcomponentsthatsatisfyboundaryconditionsata djacentsidesmustincludeaparticular combinationofcomponentsthat,inapartialsense,mimicaRayleigh-Lambwave system.TheL-Conjugationa ndS-Conjugationcasesaremerelytwodierentways ofrealizingthis.Figures5.1and5.2illus tratetheessenceofthesealternatives. Theyimplytworecipesforbuildingthesuperpositions.

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88 Figure5.1:IllustratingtheL-ConjugationCaseforModalSolutions.Shownarethe relationshipsamonglongitudinalandsh earwavevectorcomponentsofthede“ning physicalwaveswhichmustbeamongthosemakingupthetotalsuperposition neededforasolution.Thecommon x directionalcomponent( k )isnormaltothe page. TherecipeimpliedbyL-Conjugationbeginswithalongitudinalwaveata desired k value.Conceptually,onecouldimaginestartingoutwithsome and some closetoamodalsolution.Adda shearwavewiththesame“xed k Thepolarizationofthisinitialshearcontributionissuchthatsheardisplacement isnotparalleltothesides.Nowadjust untiltheRayleigh-Lambequationis satis“edwithrespecttothe z = hzsides.Therewillbearangeof sforwhich Rayleigh-Lambcanbesatis“edattheseparallelsides.Foreachpossible thebulk dispersionrelationswill“x and +wavevectorcomponentspointingalongthe z directions.Nowaddasecond(co njugateŽ)shearwaveatthesame k andclose

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89 Figure5.2:IllustratingtheS-ConjugationCaseforModalSolutions.Shownarethe relationshipsamonglongitudinalandsh earwavevectorcomponentsofthede“ning physicalwaveswhichmustbeamongthosemakingupthetotalsuperposition neededforasolution.Thecommon x directionalcomponent( k )isnormaltothe page. tothe impliedbytheprocesssofar.Thepolarizationofthisshearwaveshould alsoresultinadisplacementnotparalleltothesides.Theinitial + conjissetequal to .Justasthe“rstshearwavehadawavevectorcomponentcommonwiththe foundationlongitudinalwavealongthe y direction,thisconjugateshearwavehas awavevectorcomponentcommonwiththe foundationlongitudinalwavealongthe z direction.Bulkdispersionwill“xthevalueof conj.Nowadjust overtherange ofvaluesthatcontinuetosatisfyRayleigh-Lambatthe z = hzsurfacesuntilone isfoundforwhichRayleigh-Lambisalsosatis“edforthefoundationlongitudinal waveandthejust-addedconjugateshearwaveatthe y = hysurfaces.When

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90 suchan t isfound,itdenesaneigenfrequencyofapropagatingsystemattheset valueof k .Therewillbeasmanysuch t sastherearesubbands. TherecipeimpliedbyS-Conjugationisprocedurallythesameastherecipe forL-Conjugationexceptthatthefoundationwaveisashearwaveinsteadofa longitudinaloneandtwolongitudinalone sareaddedinsteadoftwoshearwaves. Therecipeseachfocusoncombiningwavesintotaldisregardoftheirreections atadjacentsurfaces.SatisfactionofRayleigh-Lambincorporatesreectionsonlyat parallelsurfaces.Thereectionsfromadjacentsurfacesbeingignoredconstitute therestofthesuperpositions.Whatthederivationshowsisthattondthe eigenspectrum,onecanignorethesereections.Tobuildacompletedescriptionof thewavefunction,however,thesereectionsmustbeincluded.Incorporatingthe ignoredreectionsisnottrivial.Therear etwoprinciplecomplications.Therst isthatshearandlongitudinalwavesscatte rintoeachotheruponreectionandthe scatteringamplituderatiosarenontrivi altranscendentalrelationsevenwhenthe surfaceenvironmentisfreeofotherinteractionswhichtheyarenot.Secondly,from plottingtherootsoftheequation(5.27)pairedalternatelywithequation(5.41)or equation(5.42)thereappeartobec aseswhere,foragi ve neigenfrequency,there aremultiplerootcoincidencesanditisnotclearwhethereachisanindependent foundationfromwhichthereectionscanbetakenasemanating,orwhetherthe multiplerootcoincidencesare,insomesense,resonancesofeachotherandbuilding reectionsfromonlyoneofthemissucient. Sincetheamplitudeofreectionintoe ithershearorlongitudinalisbounded byunity,successivereectionsmustprogressivelydissipateinamplitudeandthe implicationisthateachfoundationsetconst itutesarst-ordercharacterizationof theentirewavefunction.

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91 Inspiteofthepracticaldicultiesinvolvedinbuildingacompletedescription ofthewavefunctionthatfullydescribesthesuperposition,theimplicationofthe derivationremainsastrongone.Namely ,nomatterhowcomplicatedthedetails ofthewavefunctionare,thedispersioni spreciselydenedbythebehaviorofonly oneoverlappingsetofcomponentswhichmustexistasadominantpartofthefull superposition. 5.3.3ModeDispersions Themostimportantphysicalfeatur ewhichcannowbeexhibitedarethe modebandsandparticularlythedisper sions.AsthetermsL-Conjugationand S-Conjugationwillnowappearmorefrequ entlyandtogether,thecontractions L-ConjandS-Conjwillbegintobeusedroutinely. Equation(5.27)pairedalternatelywithequation(5.41)orequation(5.42) denessimultaneoustranscendentalrelationshipswhichdeterminethepropagatingnormalmodesofanelasticisotropicrectangularwaveguide.Extractingthe actualsubbanddispersionsisaccomplishedbysubstitutingsuccessive K values intoadimensionlessformofequation(5.27),contourplottingitsr ootsystem,and superimposingthatr ootsystemontopofther ootsystemplottedforequation (5.41)or(5.42).Itisofsomehelpthatthelatterequationsare K -independent andneedonlybeplottedoncei ndimensionlessform. ThesubbandsresultingfromL -Conjugation(viz.equations(5.27)and(5.41) combined)haveastrongsimilaritytothestandardRayleigh-Lambbands.This canbeseenbymethodicallyplottingthelowestsubbandsoverarangeof K values andsuperimposingthemonthelowestsubbandsofthestandardRayleigh-Lamb solution.Thisi sshownfordilatationalmo desinFig.5.3wherethedispersionfor therstthreesubbandsofthesepropagat ingwaveguidemodesareshownagainst

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92 0 1 2 3 4 K hk x 0. 1. 2. 3. 4. h c s in units of 2 R 2 3 P y E P z E Figure5.3:PlottedK-DispersionforPropagatingDilatationalModes-LConjugateCase.Heavylinescorrespondtowaveguidemodes(solidlinesare squarecrosssection,dashedlinesare1x2crosssection)andthinreferencelines areRayleigh-Lambinniteplatesymmetricsolutions. thebackgroundoftherstthreeRayleigh-Lambbranches.Waveguidemodesfor thesquarecrosssection( solidlines)aresh owntogetherwithmodescorresponding toa1:2cross-sectionalaspectratio(dash edlines).(L-ConjandS-Conjdilatational modesincombinationarecomparedwithnumericalmodecomputationsinFig. 6.4.) Thedilatationalsubbandsresultingfro mS-Conjugationarenoti ntrinsically similart oinniteplatemodesofthesameparitypattern.Figure5.4showsthe dispersionofthelowerS-Conjsubbands forthedilatationalfamilyagainstthe backgroundoftherstthreeRayleigh-Lambbranches.Waveguidemodesforthe squarecrosssection(solidlines)areshowntogetherwithmodescorrespondingto a1:2cross-sectionalaspectratio(dashedlines).

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93 0 1 2 3 4 K hk x 0. 1. 2. 3. 4. h c s in units of 2 R 2 3 P y E P z E Figure5.4:PlottedK-DispersionforPropagatingDilatationalModes-SConjugateCase.Heavylinescorrespondtowaveguidemodes(solidlinesare squarecrosssection,dashedlinesare1x2crosssection)andthinreferencelines areRayleigh-Lambinniteplatesymmetricsolutions.ForcombinedL-Conjand S-Conjdilatationalmodescomparedton umericalresults,pleaseseeFig.6.4. Figure5.5showsthedispersionforL-Conjugatedexuralsolutionsagainsta backgroundofRayl eigh-Lambexuralmodes.Thedashedlinescorrespondtoa1:2 cross-sectionalasp ectratiowheretheexingmotionofthefundamentalmodesis ofthesameplane( y x )inboththeRayleigh-Lambinniteplateandrectangular waveguidescenarios. Althoughnotorsionofaninniteplateispossible,thedisplacementpatternof low-ordertorsionmodes(vi z.dominantlyparalleltothesurfaces)i sanalogousto theuncoupledSHmodes(viz.alsoparalle ltothesurfaces,butvanishingthere) oftheinniteplatederivedattheendofsection4.4.Figure5.6showstorsional modesofasquarewaveguideagainstabackgroundofinniteplateSHmodes.

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94 0 1 2 3 4 K hk x 0. 1. 2. 3. 4. h c s inunitsof 2 R 2 3 P y E P z O Figure5.5:PlottedK-DispersionforPr opagatingFlexuralModes-L-Conjugate Case.Heavylinescorrespondtowaveguidemodes(solidlinesaresquarecross section,dashedlinesare1x2crosssection)andthinreferencelinesareRayleighLambinniteplateantisymmetricsolutions.Flexingisofthe y x plane. NotethatthisgurecombinesinonedisplaybothL-ConjugateandS-Conjugate solutionsandshowsthemagainstabackgroundofcombinedevenandoddSH platemodes.Thisguredoesnotattempttoshowtorsionmodesotherthanfora squarecrosssection. Itissurprisingthattherearemorethanonefundamentaltorsionmodei.e., morethanonetorsionmodethatgoestozeroas k 0.Thetorsionaldispersion forcircularcrosssectionsareanalyti callyknownandhaveasinglefundamental mode(seeRefs.[38,5,10]).Whiletherectangularcrosssectionbreakstheazimuthal symmetryofthecircularcrosssection,itwasnotanticipatedthatconsequentnew modeswouldbefundamental(cf.,assumptionsinRef.[47]).

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95 0 1 2 3 4 K hk x 0. 1. 2. 3. 4. h c s in units of 2 R 2 3 P y O P z O Figure5.6:PlottedK-DispersionforPr opagatingTorsionalModes-L-Conjugate andS-ConjugateCases.Heavylinescorrespondtosquarewaveguidemodes(solid linesareL-ConjanddottedlinesareS-Conj)andthinreferencelinesareuncoupled SHmodesofaninniteplate. Twoofthetorsionalmodebranchesarequadraticnear k =0andtheir correspondinghighdensityofstatesatl owfrequenciesmakethemparticularly interesting.Itisworthwhiletoextract somegreaterstructuralunderstandingof thesemodebranchesandsimultaneouslydemonstratethatthepresenttheorydoes allowmorethanjusttheeigenspectrumtobeexhibited. Asexplainedbelow,mod edispersionsareobtainedbymeticulouslyplotting rootfamiliesofthederivedtranscendent alfrequencyequationsandtrackinghow thefrequenciescorrespondingtotheirintersectionschangeas k isvaried.Thisis accomplishedusingrootplotslikethosedi splayedliberallythroughout,butFigs. 5.7and5.8showanactualprogressionofsuchintersectionsfortheL-Conjand S-Conjcasesoftorsionalmodesforverysmall k .Iwillnowshowthattheseroot

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96 1. i 0 1 0.1 0.2 0.3 K 0.3 R 2 3 a 1 P y O P z O 1. i 0 1 0.1 0.2 0.3 K 0.5 R 2 3 a 1 P y O P z O Figure5.7:MappingFundamentalTorsionalModes-L-Conjugate.Intersections ofheavysolidandheavydashedlinesde netwofundamentalL-Conjtorsional modes.Intersectionstotherightofdotte dreferencelinecorrespondtoimaginary inthefoundationlongitudinalwave.In tersectionstotherightofthethin connectedreferencelinecorrespondtoreal()+.CorrelatewithFigure5.1 plotsareabletoprovideadditionalusefulinformationaboutthestructureofthe foundationwavesthatcharacterizethemo des(seeFigs.5.1,5.2andaccompanying text). Therootplotsshowthezerocontourindi mensionless-spacecorresponding to f ( ,t )=0,where f isoneofthefrequencyequations.Inthesamespace,Ican generatecontoursforotherconditions. Forexample,thecontourscorresponding to +=0, =0,( +)=0,and( )+=0canalsobeplotted.Thesequantities areeachasquarerootofsomesumofpurerealtermsoneofwhichis 2.Now,as changes,thezerosofthesetermsalwaysliebetweenapurerealandpureimaginary value.Plottingthezerocontoursofsquareofthesetermsalwayspartitionsthe spacebetweenregionsforwhichthetermisimaginaryorreal.Fortermscontaining + 2,thepurerealregionwillbeclockwisefromthezerocontourandforterms containing 2,thepurerealregionwillbecounterclockwisefromthezerocontour. Hence,foragivenrootintersection,Icanalwaysestablishthenatureof from thehorizontalaxisandbysuperimposingazerocontourofthesquareofanyof

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97 1. i 0 1 0.1 0.2 0.3 K 0.3 R 2 3 a 1 P y O P z O 1. i 0 1 0.1 0.2 0.3 K 0.5 R 2 3 a 1 P y O P z O Figure5.8:MappingFundamentalTorsio nalMode-S-Conjugate.Intersections ofheavysolidandheavydashedlinesde nesthefundamentalS-Conjtorsional mode.Intersectionstotherightofdottedreferencelinecorrespondtoimaginary +inthefoundationshearwave.Intersect ionstotherightofthethinconnected referencelinecorrespondtoreal(+).CorrelatewithFigure5.2 theforegoingterms,Icandeterminethe natureofthosetermswithrespecttothe modedenedbythati ntersectionaswell. Applyingthistechniquetothetorsionalmodesastheychangewith k (asshown inFigs.5.7and5.8)Icann owdistinguishthesemodesintermsofthenatureofthe wavecomponentsoftheirL-ConjandS-Conjcharacteristiccomponents.Interms ofthenomenclatureofFigs.5.1,5.2,thedistinguishingcharacteristicsamongthe fundamentaltorsionalmodes canbesummarizedasfollows: L-ConjLinearBranch isreal; and += +areimaginary; conj=( )+isreal; + conj= isimaginary. L-ConjQuadraticBranch isimaginary; and += +areimaginary; conj=( )+isimaginary; + conj= isimaginary. S-ConjQuadraticBranch isimaginary; +and = areimaginary; conj=( +)isimaginary; + conj= +isimaginary.

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98 Obviously,thethingthatdistinguishesthebrancheswhicharequadraticclose tozero k isthefactthatalloftheirtransversewavecomponentsareimaginary. InthissectionIhavenotshownextens iveexamplesofthecontourplotsof rootfamiliesforallofthedierentpari tyfamilies.Ihavechoseninsteadtoshow modedispersionsderivedfromthemsincethesearetheresultsofphysicalinterest. However,arobustsamplingofsuchcontourplotsforthe k =0caseareshownin section6.4.2andtheinterestedreaderisrecommendedtoexaminetheminthat context. 5.3.4CommentsontheProcessofMappingModeDispersions Extractingthe K vsbranchstructureispresentlyalabor-intensiveprocess. Ifitwerenotfortherapidevolutionofaordablecomputing,eachcontourplot itselfwouldbetemporallyexpensivetoproduce,butinthecourseofthisresearch, desktopcomputingpowerhasincreasedby afactorofapproximatelyfortyandso theplotsthemselvescanberenderedec ientlybyacompetentuseofcommercial software.Consequently,inspectingtheov erlaidplotsandvisuallyidentifyingthe intersectionshasbecometherate-limitingstep. Theprecisionwithwhicheigenfrequenciesaredetermineddependsuponthe plottingprecision.Thebestcontourplotswhichappearinthisdissertationwere createdbyevaluatingthecontourspaceat400x400points.Atthatpointdensity, eachcontourplottakesbetweentwoto“veminutestocompute(viz.16,000 equationcomputations)usinga600MH zPentiumcomputer.Theoverlaysmust berenderedathighresolutionandfeaturesmustbezoomablesothatanoperator canvisuallyidentifytheintersectionpointwithadequateprecision.Ifthreedigits ofprecisionaredesired,thisprocesspresentlyconsumesbetweenoneandtwohours perdispersionbranchper20points.

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99 Inprinciple,theprocessofextractingtheeigenspectrumand K -dispersionscan beautomatedoratleastrenderedmoreco mputerassisted.Thecontourplotsare alreadystoredasPostscriptleswhichen capsulateaproceduraldescriptionofthe intendedimageandsoitisconceivablethatspecializedsoftwarecouldbedeveloped toacceptmultipleinputsofthistypeandthenfollowacustomalgorithmtoproject theintersectionpointsandthenoutputthem.Unlikenumericalapproximation methods,thereisnotheoreticallimitontheaccuracyofthisprocessandthereis noestimatepersewhichdegradesinhigherregionsofthespectrum. 5.4Disposingofthe Hx=0 Possibility If Hyand Hzaretakentobezeroandsolutionsaresoughtfor k> 0using Hxalone,thennosystematicsolutionse tswillbefound.Ratherthandetailthe attemptsatderivationwhichultimatelyf ailtoproducesolutions,Iwillidentify theunderlyingdiculties. Fromequations( 2.22),theboundaryconditionsat z = hzwith s z p x p y Hy= Hz=0and Hx=0willbe nk2 +2 ,zz=2 Hx,yz,zx=1 2Hx,xy,zy=1 2( Hx,yy Hx,zz)(5.43) Fortheboundaryconditionsat y = hywith s y p z p x ,equations (2.22)takethefollowingform: nk2 +2 ,yy= 2 Hx,yz,yx= 1 2Hx,xz

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100 ,zy=1 2( Hx,yyŠ Hx,zz)(5.44) Thesuperpositionstobesubstitutedareagain ,de“nedasbefore,and Hx, whichhastheform Hx= Px( kx )jaj Py( jy ) Pz( + jz ) Anearlysignofdicultycanbeseenimme diatelybynoticingthatintheprevious derivationbasedon Hy, Hz,therewerethesamenumberofboundaryconditions, buttwomoredegreesoffreedomforthemtoresolve.Inthepreviousderivation,the transformsof Hyand Hzexhibitedoneeachunknownexpansioncoecientateach surfaceandthetwobottomconditionswereappliedtoresolvetheirrelationship andthuseliminateoneofthem.Here, Hxhasnocompanionandatthetransform leveltherewillbeonlyoneunknowncoecientpersurface,butstilltwoapplicable independentboundaryconditions.Thisimbalancewillresultinaconstrainton therelationshipsamongvariablesinstea doftheeliminationofunknownconstants. Ultimately,thisleadstothefollowingconstraints: L-Conjcase: +=0 ( )+=0(5.45) S-Conjcase: +=0 =0(5.46)

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101 Whenthesearesubstitutedbackintothe boundaryconditions,afrustratingseries ofcontradictionsarisejustifyingtheconc lusionthattherearenonon-trivialsolution sets. Iftheparticularcaseof k =0isconsidered,oneofitseectswillbetoeliminate entirelyoneoftheboundaryconditionsandthusrestorethebalancebetween boundaryconstraintsandunknowncoeci entsinthetransformedequations.In thenextchapter,thiswillbeexploredanditwillbeshownthatwhile Hx =0does notgenerateanypropagatingmodeswithinthewaveguide,itdoesgenerate k =0 transverseresonanceswhichturnouttobethe k 0limitofthepropagating solutionswhen Hyand Hzexist.

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CHAPTER6 K =0MODESOFARECTANGULARWIRE 6.1k =0 BoundaryConditions As k 0,propagatingmodesapproachcutolimitsandtherearealsonon propagatingmodesresonatingtransverselyinthewaveguide.Anostensiblythree dimensionalproblemisreducedtotwodimensionsforwhichtheboundaryconditionsandrepresentationofthepotentialssimplifysigni“cantly.Speci“cally, as k 0,thestipulatedcommon x -dependencyresultsinthevanishingofall derivativeswithrespectto x Therefore, ,x 0,andthecurlofthevectorpotential H andstraincomponentsderivedfromitscomponentssimplifyforthesamereason, yeildingthefollowing: u( s )= HxHyHz Hz,yŠ Hy,zHx,zŠ Hx,y (6.1) u( s ) zzŠ Hx,yzu( s ) yy Hx,yzu( s ) xz1 2( Hz,yzŠ Hy,zz) u( s ) xy1 2( Hz,yyŠ Hy,yz) u( s ) yz1 2( Hx,zzŠ Hx,yy)(6.2) 102

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103 Applyingequation(6.2)and ,x=0toequation(2.22)at z = hzand y = hysurfacesproducestheboundaryconditionsat k =0: at z = h forall y [ h,h ] nk2 +2 ,zz=2 Hx,yz,yz= 1 2( Hx,zz Hx,yy) ,xz=0=1 2( Hz,yz Hy,zz)(6.3) andat y = h forall z [ h,h ] nk2 +2 ,yy= 2 Hx,yz,yz= 1 2( Hx,zz Hx,yy) ,xy=0=1 2( Hz,yy Hy,yz)(6.4) 6.2Uncoupled(Separable)k =0 Modes Thefactthatthe k =0boundaryequations(6.3)and(6.4)includeone conditionindependentof ,suggeststhatIlookforgeneralsolutionstothecase =0.Thesewillbesolutionsforwhichthereisnocouplingoftheshearand longitudinalpartsandthesuperpositio nsrepresentingcomponentsoftheshear potentialcancollapsetosingleterms.Sinceallderivativeswithrespectto x arealreadyexpressedintheformof theboundaryconditions,thecommon x dependencycanbesuppressedentirely.Inrespectoftherepresentationalscheme speciedinsection4.2,the H componentswillhavethefollowingform: Hx= A Py( ay ) Pz( + az ) Hy= BPy( by ) Pz( + bz )

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104 Hz= C Py( cy ) Pz( + cz )(6.5) Substitutingintoboundaryconditions(6.3)and(6.4)at =0producesthe followingsurface-pairedconditions(recallmyconventionthat hy= ahz): 0= a+ aPy( ay ) Pz( + ahz) 0= a+ aPy( aahz) Pz( + az ) A =0(6.6) 0=(( + a)2Š 2 a) Py( ay ) Pz( + ahz) 0=(( + a)2Š 2 a) Py( aahz) Pz( + az ) A =0(6.7) Cc+ c 1 Š 1 PyPy( cy ) Š 1 1 Pz Pz( + chz)= Š B ( + b)2Py( by ) Pz( + bhz) C2 c Py( cahz) Pz( + cz )= Š Bb+ b Š 1 1 Py Py( bahz) 1 Š 1 PzPz( + bz )(6.8) Therearetwoindependentcasesde“nedbythealternativesof A =0versus B,C =0. Becausesinesandcosineshavenocommonzeros,theonlynontrivialsolution forthe A =0casescomesfrom a= + a= with Py( hz)= Pz( ahz)=0.For thesamereason,thiseliminatessolutionswhen Py = Pz…whichmeanssimplythat therearenosolutionsofthistypeforthe”exuralfamilyofmodes. Inthe B,C =0cases,equations(6.8)harborsomesubtleties.First,consider thespecialcaseof b= c.Itcanbereadilyestablishe dthatalgebraicallythis satis“esbothequationssimultaneously.However,inthecutolimit,theonly contributiontodisplacementof Hyand Hzistode“ne u( s ) x=2 H[ z,y ].Aftercarefully consideringthecasesof Py= Pzand Py= Pz,itcanbeshownthat B/C ratio

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105 thatresultsfromequatingthesigmasalsoforces u( s ) xtovanisheverywhere.So, b= cturnsouttobeanullsolution. Thenexamineequations(6.8)andnoticethatifeither B or C arezero,that thesurviving willparameterizesolutionsdenedby Pz( +hz)= Py( ahz)=0 regardlessofwhichconstantiszero. If B and C arenonzerosimultaneously,Icandividetheequationstoeliminate thecoecients.Thenitappearspossibletondrootsdenedby Pz( + ahz)= Pz( + bhz)=0or,independently,by Py( aahz)= Py( bahz)=0.This,however,isillusorysinceinthelimits z hzand y hytheratiosofthese willapproachnonzeroconstants.Apparently,solutionswhenbothconstantsare nonzeroaregeneratedpurelybythesimultaneityof Pz( + ahz)= Py( aahz)=0 and Pz( + bhz)= Py( bahz)=0whichisjustasuperpositionofindependent solutions. Usingdimensionlessunits(seesection4 .3),theworkedoutuncoupledsolutions canbesummarized. First,the A =0caseisrestrictedtothedilatationalandtorsionalmodefamilies andyieldsthefollowingresults: dilatationalfamily( Py= Pz= E ) =n (2 p 1)2+ (2 q 1)2 a2$r 2%p,q =1 2 3 ... (6.9) Subjecttothefollowingconstraint: q ( integer )= 1+(2 p 1) a 2 ( a =1 t p = q )

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106 torsionalfamily( Py= Pz= O ) =2n p2+ q2 a2$r 2%p,q =0 1 2 ... (6.10) Subjecttothefollowingconstraint: q ( integer )= ap Thisgroupofmodesisfragileinthesensetheycanonlyexistforspecialexact valuesof a startingwith a =1.Theseareanenumerationofso-calledLam emodes whichcorrespondtoshearwavespolarizedinaplanenormaltothesurfacesatan incidenceof45o( = +).Atthatpreciseincidenceangle,itiswellknown(see e.g.,VolIICh.9Ref.[38])thatthescatteringamplitudeforverticallypolarized (SV)shearwavestoreectaslongitudinalwavesiszeroconsistentwiththe =0 conditionstipulated. Secondly,the B,C =0casesforeachmodefamilyyieldsthefollowing: dilatationalfamily( Py= Pz= E ) =2n p2+ q2 a2$r 2%p,q =0 1 2 ... (6.11) torsionalfamily( Py= Pz= O ) =n (2 p 1)2+ (2 q 1)2 a2$r 2%p,q =1 2 3 ... (6.12) exuralfamily( Py= Pz) =n (2 p 1)2+ q2 4 a2$r 2%p =1 2 3 ...,q =0 1 2 ... (6.13) =n (2 p 1)2 a2+ q2 4$r 2%p =1 2 3 ...,q =0 1 2 ... (6.14)

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107 Thedegeneracyoftheexuralmodesforasquarecrosssectionisbrokenwhen a =1andtheasymmetryoftheaectedequationaugmentstheresulting distinction.Fortheotherfamilies,thefactthat p ,and q havethesame enumerationobviatesrestatingtheequationwith a juxtaposed. Whilethe A =0caseinvolvesverticallypolarized(SV)shearsurfaceincidence, the C,B =0caseinvolvesonly u( s ) x=0shearwavespolarizedparalleltothe surface(SHwaves).SHwavesreectentirelyasthemselvesatallangles(provided theygotozeroatthesurface)andthusdonotinvolvecouplingtolongitudinal waves.(again,seeCh.9Ref.[38])Thesearethetwodimensionalextensionsof theSHsolutionsforaninniteplatedescribedattheendofsection4.4. 6.3UncoupledModesnottheLimitofPropagatingModes Theforegoinguncoupledsolutionsat k =0arenotcontinuouslyconnectedto anysetofpropagatingmodes.Thisconclusionisinconictwiththeresultsof numericalmethodsdescribedinsection3.2. Onewaytorealizethatuncoupled k =0modesdonotconnectcontinuouslyto propagatingmodesbeginswithaphysicalargument.Theseuncoupledsolutions areuncoupledeitherbecausenoneoftheirsheardisplacementshavecomponents perpendiculartothesurfaces,orbecausetheirverticalcomponentsariseatthe oneexceptiontocouplingwhichoccurswhenthewavevectorangleofincidenceis precisely45o.IfIperturbeithersituationbyaddingawavevectorcomponentinthe x directionhoweversmall Iwillnecessarilyinducesomedisplacementcomponent verticaltothesurfacesthatwillresultinapartialrefractionoflongitudinalwaves. Inotherwords,perturbationbyadding k> 0requires =0.If cannot beidenticallyzero,thentheboundaryco nstraintsnolongersimplifyandthe derivationofpropagatingmodes,alreadypresented,denesthesolutions.The

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108 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 TransformVariable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. unitsof 2 K 0 R 2 3 a 1 P y E P z E Figure6.1: k =0PropagatingModesComparedto k =0UncoupledModes… L-ConjDilationalModesinSquareCross Section.Intersectionsofcurvesde“ne propagatingmodes.Horizontallinesateigenvaluesofuncoupled k =0(SHand Lam e)modesdonotgenerallycorrespondto k 0limitsofpropagatingmodes. k 0limitsofthesolutionsetforpropagatingmodes,asshowninthedispersion plotswithinsection5.3.3,doesnotco rrespondtothesolutionsforuncoupled modes. Thatthe k 0limitofpropagatingmodesarenotcoincidentwiththe uncoupledsolutionscanbeseendirectlybysuperimposingtheeigenspectrumof uncoupledmodesontothe k =0rootcoincidencesofanyparityfamily.Figures6.1 and6.2showL-ConjandS-Conjrootintersectionsforthedilatationalsystemof asquarewaveguidewithhorizontallinesatvaluesoftheuncoupledSHandLam e uncoupledeigenvalues.Asthe“guresshow,therearenosystematicagreements betweentheuncoupledeigenvaluesandth erootsystemintersections.Thesame disparityoccursforallofthetorsionaland”exuralmodes,though“guresshowing theseareomitted.

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109 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 Transform Variable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. units of 2 K 0 R 2 3 a 1 P y E P z E Figure6.2: k =0PropagatingModesComparedto k =0UncoupledModes S-ConjDilationalModesinSquareCross Section.Intersectionsofcurvesdene propagatingmodes.Horizontallinesateigenvaluesofuncoupled k =0(SHand Lam e)modesdonotgenerallycorrespondto k 0limitsofpropagatingmodes. Numericalsolutionsofwaveguidemodeswhichapplythenumericalprocedurediscussedinsection3.2doshowacontinuousconnectionbetweenuncoupled solutionsandpropagatingones.Thiscanbeseen,forexample,bycomparingthe smallk partofthenumericallygenerateds ubbandstotheuncoupledeigenvalues. Figure6.3showsthatsomeofthenumericallyproducedmodefamiliesconverge touncoupledvaluesas k 0.Nishiguchisapplication[1]ofthenumerical procedureusingabasisformedfrompow ersofthecoordinatesshowsthesame result. Theinabilityofthenumericalprocesstodistinguishnonpropagatingmodes frompropagatingoneswasactuallyanimpedimenttodevelopingananalytic methodsincenumerouseortstocontri veananalyticsolutionthatagreedwith

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110 0 1 2 3 K hk x 0. 1. 2. 3. 4. h c s inunitsof 2 R 2 3 P y E P z E Figure6.3:NumericallyApproximatedModesComparedto k =0Uncoupled Modes…DilationalModesinSquareCro ssSection.Dilatationalsubbandscomputedusingthepowersofcoordinatesasabasisfornumericalmethoddeveloped byVisscheretal.andutilizedbyNishiguchi.Horizontallinesateigenvaluesof uncoupled k =0(SHandLam e)modeseachcorrespond(incorrectly)to k 0 limitofapropagatingmodes. numericalresultswereunfruitful.Thisdisparitycannowbeunderstoodanda signi“cantlimitationofthenume ricalprocess“nallyappreciated. At k =0,thenumericalmethodconverges…atleastatthelowerpartofthe spectrum…tomodesconsistentwiththatvalueof k .Thesewillinclude,among others,theuncoupled k =0modes.Infact,Ihavefoundthatthemodesproduced bythenumericalprocesswhichappeartocoincidewithuncoupled k =0modes, agreewithuncoupledmodesdetermineda nalyticallytoatleastsixsigni“cant digits.Theseuncoupledmodesinvolvezerosofsimplesinesandcosinesand, aslongastheperiodicityislow(i.e., h/< 2)powersofthecoordinateswill convergestronglytosuchsolutions.Thenumericalprocessisbasedon“nding theeigenvaluesofa“nite-ordermatrix.Inthiscase k isjustaparameterof thecalculation.As k> 0,theeigenvalueswillbeperturbed,butfor k 1 onlyslightly.Butthe“niteorderofthematrixdoesnotchangewith k and

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111 0 1 2 3 K hk x 0. 1. 2. 3. 4. h c s inunitsof 2 R 2 3 P y E P z E Figure6.4:AnalyticallyDerivedPropagatingModescomparedtoNumerically ApproximatedModes…DilationalModesi nSquareCrossSection.Starsonaxis denoteuncoupled k =0modes.Thicklinesareanalyticmodes(solidareL-Conj, brokenS-Conj)andthinlinesarefromnumericalmethodasusedbyNishiguchi. therelaxationisnotabsolutelyconsis tentwithstress-freeboundaryconditions, butonlyapproachesitasbestthe“nitebasiswillallow.The“xed“niteorder precludesentirelynewmodesfromsuddenl yappearingandthenumericalprocess makesitappearasiftheeigenfunctionsaresmallperturbationsfromtheir k =0 forms.Whatisactuallyhappening,however,isthatcombinationsofthe“nite basiselementsapproximatingthepropagatingfunctionsisbeginingtobemixed intotheeigenfunctionswhichdonotpropagategivinganillusionofcontinuity. Eventually,as k continuestogrow,thesubbandswhichappearedtostartout fromtheuncoupled k =0modes,takeonmoreofthecharacteristicsofthetrue propagatingmodes.ThisisconsistentwithFigure6.4whichshowsL-Conjand

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112 S-ConjDilatationmodesofasquarewireagainstabackgroundofnumerically approximatedsubbands. 6.4Coupled(nonseparable)k =0 Modes 6.4.1Derivation…AllParityFamilies Iftherightsidesofboundaryequations(6.3)and(6.4)aresettozero,itis quicklyseenthatabsentasuperpositionofsomekind,noparitypatternof willbeabletosatisfytheequations.Lookingatthederivativestobetaken,the impossibilityofcommonzerosofsinesandco sinesofthesameargumentisdecisive. Furtherprogresscanonlybeachievedbyfacingthenonseparabilityissueand substitutingsuperpositionsofbasicfunctions.However,thiscannowbeaccomplishedusingasomewhatabbreviatedver sionoftheprocessusedtoderivepropagatingmodes. Substitutingrepresentationsfromequat ions(4.8)thru(4.10)intotheboundary conditions(6.3)at z = hz,Iobtain: Š Pxidi( k2 +2( i)2) Py( iy ) Pz( ihz)= 2 Pxjajj+ j 1 Š 1 PyPy( jy ) 1 Š 1 PzPz( + jhz)(6.15) Pxidii i Š 1 1 Py Py( iy ) Š 1 1 Pz Pz( ihz)= Š1 2Pzjaj( Š ( + j)2+ 2 j) Py( jy ) Pz( + jhz)(6.16)

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113 Applyingthesimpletransformfromsection4.5anddenotingwith oanyofthe possiblevaluesofthetransformvariablewhichsimultaneouslymatchelementsof both { i} and { j} ,thesumscollapseandtherelationshipsbecome:(Reminder: underthetransform, i| | etc.,buttoavoidclutter,Iwillomittheabsolutesign notationssothat of| o| .Recallalso,thatallsquarerootsareassumedpositive. Both oandanysquarerootsshowncould,however,bepositiveimaginary.) do( nk2 +2( o)2) Pz( ohz)=2 aoo+ o 1 1 Py 1 1 PzPz( + ohz)(6.17) doo o 1 1 Py 1 1 Pz Pz( ohz)= 1 2ao(( + o)2 2 o) Pz( + o)(6.18) InspiredagainbysimilaritieswiththeRayleigh-Lambderivation(seesection4.4), Icanmanipulatetheparentheticalont heleftofequation(6.17)toshowthat ( nk2 +2( o)2)= (( + o)2 2 o) Ithendivide(6.17)by(6.18),applytheforegoingmanipulation,andrearrange termstoproduce Pz( ohz) Pz( ohz) Pz( + ohz) Pz( + ohz) = 4 2 o o+ o ( 2 o ( + o)2)2(6.19) Inthisequation, t isdenedimplicitlyby oand + oandsor ootsofthisequation denethevalidcombinationsof t and o. Comparingequation(6.19)toequation(4.20)revealsittobeaRayleighLambsolutionwith oplayingtheroleofthewavenumberinthepropagation direction.Inthe k 0limitthismakesevenmorephysicalsensethanitdidwhen

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114 encounteredinderivingthepropagatingmodes.BeforeIimposeadjacentsides, thephysicalsituationispreciselytheRay leigh-Lambscenario.Ifwaveactivity survivesthecutolimit,itmustbelimit edtothetransversedirection…whichis y atthe z = hzboundary.Thetransformofthe y -dependencyhasmappedtoa transformvariablethatis,ineect,awavenumberinthatdirection.Ifweareina cutomode,thenthereshouldbenonon-uniformdisplacementinthe x direction… whichispreciselythesameconditionasrequiringplanewaves.Wealreadyknow thattheRayleigh-Lambequationde“nesa solutioninthissetofcircumstances andsotheresultisconsistent. Tocompletethederivation,Itakethesamerepresentationsof and Hxandinsertthemintotheboundaryequations(6.4)atthe y = ahzsurface.In ordertoseeclearlythelogicusedtolinkthetransformedboundaryconditions betweenadjacentsurfaces,Iwillstartbyw ritingdownthesituationfollowingthe transformationoftheconditions,butbeforethedeltafunctionshavebeenusedto transformwavenumbervariablesorthesumshavebeencollapsed.idi( k2 +2 2 i) Py( iahz) ( iŠ )= 2jajj+ j 1 Š 1 PyPy( jahz) 1 Š 1 Pz ( + jŠ )(6.20)idii i Š 1 1 Py Py( iahz) Š 1 1 Pz ( iŠ )=1 2jaj(( + j)2Š 2 j) Py( jahz) ( + jŠ )(6.21)

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115 Inthecourseofder vingthepropagatingmodes,Ihavedevelopedthelogicfor linkingthetransformedboundaryvalueconstraintsatthe z = hzsurfacewith thetransformedboundaryconstraintsattheadjacent y = hy( hy= ahz)surface. (Seesection4.4)Thesamelogicappliesher eand,aswasdonei nthepropagating modecase,IshowdetailsonlyfortheL-Conjcaseandsimplystatetheanalagous resultfortheS-Conjcase. Applyingthemappings(5.31)toequations(6.20)and(6.21)andcollapsingthe sums,Iobtainthefollowing: do( nk2 +2 2 o) Py( oahz)=2 a1 o( )+ 1 1 PyPy(( o)+ahz) 1 1 Pz(6.22) doo o 1 1 Py Py( oahz) 1 1 Pz= 1 2 a1[( o)2 (( o)+)2] Py(( o)+ahz)(6.23) Ashasbeenthepatternthusfar,theparentheticalontheleftofequation(6.22) canbeshowntobethenegativeofthebracketeddierenceontherighthandside ofequation(6.23).Withthisapplied,divi dingthetwoequationsandrearranging termsproducesthefollowing: Py( oahz) Py( oahz) Py(( o)+ahz) Py(( o)+ahz) = 4( o)2o( o)+ [( o)2 (( o)+)2]2(6.24) Comparingequation(6.24)toequation(4.20)revealsthatitalsoisaRayleighLambsolution,thistimewith oplayingtheroleofthewavenumberinthe propagationdirection.Thistoo,inisolation,makesphysicalsense.IfIignore theadjacentsides,thephysicalsituatio ncanbeinterpretedasaRayleigh-Lamb scenario.Thetransformofthe z -dependencyhasmappedtoatransformvariable thatis,ineect,awavenumberinthatdirection.Ifweareinacutomodeand

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116 pretendtheadjacentsidesdonotexist,t henthereshouldbenonon-uniformdisplacementinthe x directionwhichisthesamecondi tionasrequiringplanewaves. Again,theRayleigh-Lambequationdene sasolutioninthissetofcircumstances andsotheresultisconsistent. 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 Transform Variable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. units of 2 K 0 R 2 3 a 1 P y E P z E Figure6.5:PlottedRootFamiliesforL-Co njDilatationalCutoModesSquare CrossSection.Solidcurvesrepresentrootsofequation(6.19)anddashedcurves representr ootsofequation(6.24). Applyingmappings(5.32)toequations(6.20)and(6.21)leadstothefollowing S-Conjsolutionanalagoustoequation(6.24): Py(( + o)ahz) Py(( + o)ahz) Py( oahz) Py( oahz) = 4( + o)2o( + o) [( + o)2 2 o]2(6.25)

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117 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 Transform Variable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. units of 2 K 0 R 2 3 a 1 P y E P z E Figure6.6:PlottedRootFamiliesforS-Co njDilatationalCutoModesSquare CrossSection.Solidcurvesrepresentrootsofequation(6.19)anddashedcurves representr ootsofequation(6.25). Sincetheforegoingsolutionswerepremisedu pon Hy= Hz=0with Hx=0,it iscounterintuitivethatthesesolutionswouldbeaspecialcaseofthepropagating modesforwhich Hx=0.Thatis,however,preciselythecase.Itcanbechecked evenbysightthatequation(6.19)ispreciselyequation(5.27)with k =0,that equation(6.24)ispreciselyequation(5.41)with k =0,andnally,thatequation (6.25)ispreciselyequation(5.42)with k =0. 6.4.2ManifestationsByParityFamily InthesamemannerasRayleigh-Lambdispersioncurveswereplotted(see section4.4)therootsystemsofthetwotranscendentalequations(6.19)and(6.24) canbefoundbycontourplotting.Whereth erootsystemsintersect,eigenvaluesof

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118 thecoupled k =0modeswillbefound.Thesesolutionswillbemadespecictothe threeparityfamiliesbymakingappropriates ubstitutionsofsineorcosinefunctions into Pyand Pz.Topreservegeneralityoftheresult,graphicalresultswillbesh own indimensionlessunits(seesection4.3 ).AswasthecaseforplottingRayleighLambsolutions,arearrangementoftheequationsi snecessarytoprecludefatal divergencesinthenumericalcomputatio ninvolvedintheplotting.Forexample, equation(6.19),indimensionlessform,isrearrangedforcomputationintothe followingform: cos()sin(+) +sin() cos(+) Pz[(+)2 2]2+4 + Pz2 Pz() Pz(+)=0(6.26) 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 K hk x 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. h c s in units of 2 K 0 a 1 P y E P z E Figure6.7:RootFamiliesforUncoupled ShearModesSquareCrossSection.This gureillustrateshowtherootfamilieswou ldappearforS-Conjsolutionsifall couplingtolongitudinalwavesweretoberemoved. Figure6.5showstheresultofsubstitutingthedilatationalparitypatterninto theforegoingsolutionsinthecaseofasquarecrosssection.Intersectionsofthe

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119 rootsystemsoccuratvaluesofwhichareeigenvaluesoftheproblem.S-Conj modeshavethesamerootsystemarisingfromthe z = hzboundaryasdo L-Conjsolutions(viz.equation(6.19)),butther ootsystemarisingfromthe y = hyboundaryshiftsfromequation(6.24)toequation(6.25).Figure6.6 showstheintersectingr ootsystemsforS-Conjdilatationalmodesi nasquarecross section.UnliketheL-Conjcases,theseonlyinvolvevaluesofwhicharereal. Theconjugaterootsystem(i.e.,rootsofequation(6.25))alsodisplaysgreater regularitythanL-Conjmodesexceptnearthemaximalvaluesof o. 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 Transform Variable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. units of 2 K 0 R 2 3 a 2 P y E P z E Figure6.8:PlottedRootFamiliesforL-Co njDilatationalCutoModesx2Cross Section.Solidcurvesrepresentrootsofequation(6.19)anddashedcurvesrepresent rootsofequation( 6.24). TheS-Conjmodesotherthanalongtheedgedenedbymaximalvaluesof, appeartoreectweakercouplingofsheartolongitudinal.Toseethis,compare thesmalllargeareaoftherootcontoursinFigure6.6totherootcontours

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120 withinthesamer egionofanuncoupledsyst eminFigure6.7forwhichtheonly boundaryconditionisthatthederivativeofthewavefunction(i.e., EE familyor cos( y )cos( +z )withintheboundaries)inthesurf acenormaldirectioniszeroat thesurfaces. 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 Transform Variable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. units of 2 K 0 R 2 3 a 1 P y O P z O Figure6.9:PlottedRootFamiliesforL-Co njTorsionalCutoModesSquareCross Section.Solidcurvesrepresentrootsofequation(6.19)anddashedcurvesrepresent rootsofequation( 6.24). Figure6.8showstheresultoftransitioningfromasquarecrosssectiontoone withanaspectratioof2:1.A stheaspectratiochanges,therootsystemat hzdoesnotchangesince hyisdenedrelativetothissurface.Astheaspectratio increases,therootsystemassociatedwiththe hysystemhasmorebranchesinthe samespaceandthetotalnumberofmodesincreases. Figure6.9showstheresultofsubstitutingtheTorsionalparitypatternintothe foregoingsolutionsinthecaseofasquarecrosssection.Intersectionsoftheroot

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121 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 Transform Variable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. units of 2 K 0 R 2 3 a 2 P y O P z O Figure6.10:PlottedRootFamiliesforLConjTorsionalCutoModesx2Cross Section.Solidcurvesrepresentrootsofequation(6.19)anddashedcurvesrepresent rootsofequation( 6.24). syst emsoccuratvaluesofwhichareeigenvaluesoftheproblem.Figure6.10 showstheresultoftransitioningfromasquarecrosssectiontoonewithanaspect ratioof2:1. Figure6.11showstheresultofsubstitutingtheFlexuralparitypatterninto theforegoingsolutionsi nthecaseofasquarecrosssection.Intersectionsofthe rootsystemsoccuratvaluesofwhichareeigenvaluesoftheproblem.Ther oot structureofthesquarecaseismappedinFig.6.11. Theexuralmodesaredegenerateinasquaregeometry.Inthelowfrequency fundamentalmodes,onecanimagineexingofthe x z or x y planeswhich aregeometricallyindistinguishablewhe nthecrosssectionissquare.However,as theaspectratioincreasesfromunity,theexureofthosesameplanesisacross

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122 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 Transform Variable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. units of 2 K 0 R 2 3 a 1 P y O P z O Figure6.11:PlottedRootFamiliesforL-Co njFlexuralCutoModesSquareCross Section.Solidcurvesrepresentrootsofequation(6.19)anddashedcurvesrepresent rootsofequation(6.24)Transposingthe Pyand Pzparityassignmentswillnot changethemodesetinthesquaregeometry. dieringwidthsandjuxtaposingtheparityassignmentsof Pyand Pzwillproduce adierentsetofmodes.ThiscanbediscernedbycomparingFigures6.12and 6.13.

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123 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 Transform Variable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. units of 2 K 0 R 2 3 a 2 P y E P z O Figure6.12:PlottedRootFamiliesforLConjFlexuralCutoModesx2Cross Section.Solidcurvesrepresentrootsofequation(6.19)anddashedcurvesrepresent rootsofequation( 6.24).Thesemodefa miliesaregeneratedbysubstitutions Py= E and Pz= O .

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124 2 i 4 i 6 i 8 i 10 i 12 i 0 2 4 6 8 10 12 Transform Variable 0. 0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.5 10. units of 2 K 0 R 2 3 a 2 P y O P z E Figure6.13:PlottedRootFamiliesforLConjFlexuralCutoModesx2Cross Section.Solidcurvesrepresentrootsofequation(6.19)anddashedcurvesrepresent rootsofequation( 6.24).Thesemodefa miliesaregeneratedbysubstitutions Py= O and Pz= E .Themodesdenedbyrooti ntersectionsaredistinctfromthose generatedby Py= E Pz= O .

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CHAPTER7 FRACTALPHASESPACEOFCOUPLEDMODESAT K 0 7.1Motivation:LowTemperatureHeatConductance RegoandKirczenowhaveshown[48]thattheLandauerformalismcanbe appliedtoheattransportintheballisti cregimeandthatinthisregimethermal conductancedependsuponthecarriers ubbandstructurewithoutregardtothe detailsofdispersion.Ineect,atveryl owtemperatures,phononthermalconductancedependsonlyonthecutomodes.Inaddition,althougheachsubband isacontributingchannelofthermalconduc tance,thefundamentalmodesthose whosedispersionsgotozeroatzero k contributelinearlyin T .Atthelowest temperatures,contributionsothertha nthosefromthefundamentalmodesare exponentiallysuppressed.Therefore,as T 0,theoverallconductancedueto phononsinamesoscopicwirereducestoaquantumlimitatwhichconductanceiscomputedasaconstantcoecientmultiplyingthetemperature.In2000, Schwab,H enriksen,WorlockandRoukespublishedtheresultsofanexperiment[49] verifyingthat,atverylowtemperatures, thermalconductanceb yballisticphonons inmesoscopicchannelssaturatesatavalueconsistentwiththisquantumof thermalconductance. Itisgenerallyassumedthatastheback groundtemperatureincreasesbeyond therangeofeectivesaturationassociatedwiththequantumlimit,thatthe temperaturedependencyofthermalconductancebecomescubic.Iwillshow thatforasmuchofthetransitionrange ascoupledphononsremainballisticin 125

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126 arectangularwire,thatthetemperatu redependencyapproachesafractalpower lessthanthree. Sincetheresultisrecent,butstraightforwardinitsderivation,Iwillrederive asmuchoftheRegoandKirczenowresultasIwillneed.Followingtheoutlinein reference[48],theLandauerheatuxin awireduetoballisticphononsbetween reservoirsattemperatures TLand TRisgivenby Q = 1 2 rm 0dk htm( k )[ nR( tm( k )) nL( tm( k ))] m( k ) vgm(7.1) where ni( tm( k ))istheaveragemodeoccupationnumberat TigivenbythePlanck distribution, m( k )isthetransmissionprobability,and vgmistheenergytransport velocity( ftm/fk ).Thesubscript m indexesdispersionbranchesacrossallfamilies ofmodes. ItisassumedthatconnectionstothereservoirareecientenoughthatIcan set m( k )=1.Thenequation(7.1)isconvertedtoanintegraloverfrequency.In convertingtoafrequencyintegral,a1-Ddensityofstateswillplace ftm/fk in thedenominatorwhichcancelsthegroupvelocityremovinganydependencyon thedispersion.Theresultis Q = 1 2 rm m cdt ht [ nR( tm( k )) nL( tm( k ))](7.2) where tmcisthe k =0valueof t inthe m thphononsubband.Notethatforeach ofthefourfundamentalmodes, tmc=0. Sinceactualmeasurementsinvolveinducingextremelysmalldierencesbetween TLand TR,consideringonlythenetenergytransportinthe T 0limit willsuce.Inthislimit,theconductance,denedas Q/ T ,canbefound bydierentiating ni( t )withrespectto T undertheintegralattheaverage

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127 temperature.Theresult(expressedsimilarlyinRef.[47])is K T 0= k2 BT hm xmx2ex ( exŠ 1)2dx (7.3) wherethelowerboundofintegrationforthe m thmodeisgivenby xm= hm c kBT. Inthisform,theconductanceatagiventemperatureinducedbyverysmall temperaturedierencesisdeterminedentirelybythesetofcutofrequenciesof thephononsubbands. Ihaveheretoforeexpressedcutofrequenciesindimensionlessunits(seesection 4.3)whichabsorbthephysicalscaleofthewireandtheshearvelocity.Inthis context,however,itwillbehelpfultoavo idconfusingthesymbolforhalf-width withPlancksconstantandIwillrepeatthede“nitionintermsof d asthefull widthofthesmallestwirewidth. mc= mc oo= 2 cs d (7.4) Iwillalsorescalethetemperaturecomponentofequation(7.3)toabsorbphysical sizeandmaterialproperties(otherthan R = c/cs)byusing t = T ToTo= 2 hcs kBd (7.5) Therelationshipbetween Toand othenhasthemnemonicform kBTo= ho(7.6) Itistobenotedthat Towillbecloseto1Kfor d ontheorderof200Angstroms and csontheorderof103m/sec.Inanticipationoflargervaluesof d ,Iwill computeconductanceoverarangeof0to10unitsof T/To.Thisdoesnotimply

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128 anassertionaboutthelikelihoodthatpho nonswillremainballisticoverthesame rangeforagivensituationandIexpectth attheballisticassumptionbreaksdown primarilyasafunctionofactualtemperature.Thisrangeischosenmerelyto demonstrateconsistencyoftheresultan dpreservepossibleapplicabilitytoawide rangeofphysicallyrealizablesituations. Finally,tofullyisolatetheuniversal T -linearcontributionto K fromthose contributionswhichreectthephonons ubbandstructure,Ifocusoncomputing K/T rescaledtodimensionlessunitsas = Kh k2 BT = Krd cskBt (7.7) sothattheequation(7.3)isnowrearrangedintothecompletelydimensionless form: =m xmx2ex ( ex 1)2dxxm= mc t (7.8) Theonlymaterialpropertieswhichsurviveinequation(7.8),thoughnotexpressed explicitly,aretheassumptionofisotropyandtheratio R oflongitudinaltoshear velocitieswhichwillbechosenforrepresentativecomputationsbelowtobe 3. Therighthandsideofequation(7.8)hasallofitstemperaturedependence inthenonzerocutoboundswithin { xm} .Moreover,thesetof { mc} ,and thus { xm} ,includeszero-valuedelementscorrespondingtothebeginningofeach fundamentaldispersionbranch.Forea chsuchzero,thereisacommon-valued temperature-independentterminthesumontherighthandsideofequation(7.8). Forcoupledmodesinrectangularwires,Ihaveshownthattherearesixsuchterms whichcorrespondtothefundamentaldisp ersionbranchesfordilatational(one), exuralmodes(two),andtorsional(three)modefamilies.Eachsuchzeroin { xm} contributesaquantumofconductancewhosedimensionlessvaluei s r2/ 3.[48,49].

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129 7.2EectiveDimension&DensityofPropagatingModesas k 0 Fromequation(7.8),ballisticphonont hermalconductanceisdeterminedbythe levelspacingdistributionofthefullsetofcutomodes.Wereitnotforcoupling, theselevelscouldbeenumeratedasacollectionofpointsetseachofwhichoccupies atwo-dimensionalphasespacewitheachofthesecharacterizedbytwoindependent quantumnumbers.The k =0uncoupledmodeenumerationsderivedinsection 6.2areexamples. 200 400 600 800 m 1.85 1.90 1.95 2.00 effective p 0,1,2...;q 1,2,3... p 0,1,2...;q 0,1,2... p 2 q 2 m Figure7.1:Eectivecapacitydimensionfortypical2-Dphasespaces.Foramodest numberofeigenvalues(m),theeectivedimensionalityofthephasespacereveals thegraininessofthediscreteeigenvaluesandthegeneraltrendissubjecttosmall systematicdistortionwithminorchang esinthedetailsofmodeenumeration. Shownisamovingaverageof100nondecreasingmodes.Therulefor m was totakethemaximumof15%ofmor15modes. Theactualsetofpointsde“nedbythe k 0limitofcoupledpropagating modesarenotsubjecttoasimpleenumerationandgoodquantumnumbersare notknown.Aspriorchaptersshow,couplingresultsinacomplexbandstructure andanunobviouslevelspacingdistributio n.Sincephasespacedimensionalityand constancyofphasespacedensitynormallyaremanifestasintegralpowerlawsfor

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130 physicalproperties,i tshouldbeexpectedthatthecomplexityofcoupledmodes willdistortpower-lawbehavior.Iwill showthatdimensionalityandphasespace densityi sirregulari nthelowesteigenvalueregionsofcoupled k 0valuesandthat ashighervaluesofthesetsareconsidered,theeectivephasespacedimensionality tendstowardsafractalvaluedistinctlylessthantwo. 200 400 600 800 m .60 .80 1.00 1.20 1.40 1.60 1.80 2.00 effective n p 0,1,2...; q 1,2,3... p 0,1,2...; q 0,1,2... p 2 b q 2 m 4 n Figure7.2:Eectivedensityfor2-DphasespacesofFig7.1. FordenitenessIwillfocusonasquarecrosssection.AttheendofthesectionI willalsoshowtheeectivedimensionanddensityofstatesinfrequencyassociated withanaspectratioof1:2which,despitehavingsomegreatervariability,retains thesamegeneraltrendsexhibitedforthesquarecrosssection. Whenthenumberofactivemodesislargeandthedensityofmodesinphase spaceisaconstant,thereisnodangerinrelatingthenumberofmodes m tothe energyorfrequencymeasureviaapower-lawrelationsuchas m = b where isthenumberofequallyspacedquantumnumbersthatenumeratethemodes.In thesekindsofcases,wecansimplydenethedimensionalityofthesetofmodes asthedimensionalityofthephasespaceusedtoenumeratethem.Thisisroughly

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131 equivalenttowhatisalsocalledthecapa citydimension,or,sometimes,theboxcountingdimensiononaccountofaformaldenitionthatinvolvescountingthe minimumnumberofboxesofadiminishingsizeneededtocontai npoi ntsi na regionspannedbysomemeasure[50].Initsphysicale ect,itisobviouslythe measureofhowfastcapacityincreaseswith. 100 200 300 400 m 1.7 1.8 1.9 2.0 2.1 2.2 2.3 effective aggregate coupled modes as k f 0 square cross section 50 mode moving avg Figure7.3:Eectivecapacitydimensionforpropagatingmodesat k 0. m wascomputedbytakingthemaximumof50%ofmor15modes.Comparewith Fig.7.1 Ifthephasespacedensitychangesorregularityislostintheenumeration scheme,itshouldstillbemeaningfulto considertherateofchangeincapacity withrespectto.Thisexibilityisusua llyirrelevantincasescharacterized byalargenumberofpointsandperiodicboundaryconditionsthatresultina constantphasespacedensityofpoints.However,whenthenumberofexcited modesissmall,Iwilldemonstratethatthis exibilitycaneasilybecomerelevant eveninphasespaceswithconstantdensit y.Toaccommodateallsituationswhere theeectivedimensionalityanddensitymightchange,Iwilldeneeective

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132 dimensionality asafunctionofaccumulatedcapacitybysimplyconnectingthe pointsofprogressivecapacityfromsomeaccumulatedbase m tosomehigher capacitylevel mfsuchthat m = bm mand mf= bmf mfsimultaneously.If Iassume b tobeeectivelyunchangedover m ,andifIimposesomerulefor systematicallycomputing m asafunctionof m ,then at m canbeprecisely denedby = log( mf/m ) log(f/ ) (7.9) Inaddition,onceInd Icandeneaprecise b ( m )ateachcapacitylevelby b ( m )= mf m mf m(7.10) Toillustratethemeaningfulnessofthis exibilityandthefragilityofdimension anddensitywhenthenumberofavailablemodesabovethegroundstateissmall, considerthesimplecasesof = r 4 b p2+ q2 p =0 2 3 ... ; q =1 2 3 ... p,q =0 1 2 ... Thephasespacesofthetwocasesisidenticalexceptfortheremovalofpointsalong the q axisinoneofthem.Figure7.1showsacomputationofthee ectivedimension forbothcasesshownasamovingaverageover100points.Themodestcapacities resultinconsiderablegraininesswhichaveragingdiminishes.Thesystematicuctuationsarearesultofdegeneraciesoverwhichbrieystallsas m continuesto increase.Figure7.2showstheeective b overthesamecapacityrangeforthe twocases.Ihavefactoredthegeometricf actorapplicabletoaquarter-circlefrom b sothatithasunitswhichcomparedirectlytoahighm valueof1.

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133 100 200 300 400 m 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 n aggregate coupled modes as k f 0 square cross section Figure7.4:Eectivephasespacedensityofpointsforpropagatingmodesat k 0. b iscomputedwithrespecttotheeective sshowninFig.7.3.Allgeometric factorsareabsorbedinto b .ComparewithFig.7.2 Theforegoingcasesshouldnowbecomparedqualitativelyandquantitatively tosimilarcomputationswithrespecttothe k 0coupledmodesforasquarewire. Toexaminetheeectivedimensionanddensityofthecoupledmodesinthe k 0 limit,rootintersectionswereplotteduptoanof50 r/ 2forallmodefamiliesand ther esultsaggregatedintoasinglesetwhichwasthensortedbyeigenvalue.Since zeroeigenvaluesaregivenspecialtreatme ntintheheatconductancecalculations, onlynon-zeroeigenvalueswereincluded.Figure7.3showsamovingaverage progressionoftheeectivecapacitydimension(perequation(7.9))oftheresulting set. Incontrastwiththeeective for2-DsetsshowninFig.7.1,theeective forcoupledmodesinthe k 0limitshowsadecreasingoscillationandthelack ofsystematicdegeneraciesresultsinasmoothercurvealbeitwithlargeroverall

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134 0 5 10 15 20 10 20 30 40 50 60 70 r m r aggregate coupled modes as k f 0 square cross section Figure7.5:Eectivedensityofstatesinfrequencyforpropagatingmodesat k 0. variation.Insteadoftrendinguptoadimen sionalityof2,thesetofcoupledmodes appearstobetrendingdowntowardadimensionalitydistinctlylessthan2. Astheeectivedimensionalityunde rgoesoscillationforincreasing m sotoo doesthedensity.AsFig.7.4shows,thereis,ingeneral,acompensationfor increaseddimensionalityintheformofd ecreasingphasespacepointdensity.Asa resultofthiscompensationthedensityofstatesinfrequencyadominantfactorin thecalculationofmanyphysicalpropertiesissparedthelargevariationsevident fore ectivedimensionan dphasespac edensityatsmall m .A sFig.7.5shows,the densityofstatesinfrequencyisroughlylin earalbeitwithaslightoverallcurvature. Inthecaseofaperfect2-Dphasespacewithconstantphasespacedensity,the densityofstatesinfrequencywouldbeperfectlylinear. Surprisingly,increasingtheaspectratiofrom1:1to1:2isnotaccompaniedby anobviousdissipationofthedimensionalreductioneect.Theoveralldensityof pointsrouglydoubles,andthevariabilityofeective increases,butasFig.7.6 shows,forthesamegeneralcapacitylevelthetrendistowardsadimensionality

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135 100 200 300 m 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 effective coupled modes as k f 0 1 x 2 cross section 50 mode moving avg Figure7.6:Eectivephasespacedimensionalityforpropagatingmodesat k 0 1x2c rosssection. comparabletothatforasquarecrosssect ion.Similarly,asshowninFig.7.7,the densityofstatesinfrequencystilllaveragestowardsalinearform. 7.3LowTemperatureHeatConductance Applyingresultsofthepriorsection,Iproceednowtocomputethelowtemperatureconductanceduetocoupledpho nonsinasquarewire.Equation7.8 involvesaninnitesumoverafunctionofthediscretesetofmodesat k 0and inthepriorsectionthee ectivecapacitydimensionandphasespacedensityofthe low-lyingmembersofthatsethasbeenexamined.Iassumethatthenumberof pointswhichbringsto20 r/ 2(approximately770points)issucienttowarrant theassumptionthat and b willnotdeviatesignicantlyfromtheirvaluesat thehighestcapacitiesshowni nFigs. 7.3and7.4.Tofacilitateaconsistent methodofcomputation,Ithereforeextra polatethepointsetbeyondthecapacity

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136 0 5 10 10 20 30 40 50 60 70 80 m coupledmodesask 0 1x2crosssection Figure7.7:Eectivedensityofstatesforpropagatingmodesat k 0…1x2cross section. correspondingto=20 / 2byaddingpointsgeneratedby m=( m )1 /using =1 593and =1 793…thedensityandeectivedimensionsofthesetat m=20 / 2.Extrapolatingtom=50 / 2bringsthetotalcapacityofthesetto approximately3970points. Itwillbehelpfultodenotethedimensionlessintegralinequation(7.8)with thefollowingnotation: F ( y )= yf ( x ) dxf ( x )= x2ex ( exŠ 1)2(7.11) ,ofcourse,isasumof F ( y )overallpossiblediscretevaluesof y containedin { xm} .Sincetheextrapolationprocesssetsanupperboundon { m} of50 / 2,at eachvalueof t therewillbeamaximumavailable xmax=(50 / 2) /t .Then,asa

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137 0 1 2 3 4 5 6 7 8 9 10 T T o 0 250 500 750 1000 1250 1500 1750 2000 K Tinunitsofk b 2 h Figure7.8:ThermalConductanceduetopropagatingmodesat k 0. precaution,Iwillcontemplatecomputingupto t =10andconsideranalytically estimatingthetailofthissumwhenIreachthisvalueforeach0 5 f ( x ) x2eŠ x(7.12) sothatfor y> 5: F ( y ) (2+ y (2+ y )) eŠ y(7.13) Assumingthe xmareinnondecreasingorder,Ic anconverttheupperrangeofthe sumtoanintegral.Then,for xm xlim> 5:m = limF ( xm) t xlimy Š 1(2+ y (2+ y )) eŠ ydy (7.14)

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138 .02 .05 .10 .2 .5 1.0 T T o 20 30 50 70 K T in units of k b 2 h Figure7.9:LowTemperatureRegionofFi g.7.8(Log-Log).Thesaturationtoa constantvaluenearthequantumlimitisclearandconspicuous.Thedottedline showtheconductancerecomputedwiththe rsttwononzeroeigenvaluesremoved. Thesevaluescomefromthediscoveredquadratictorsionalmodes.Removingthem doublesthelengthofthequantumplateau. For t atitslargestconsideredvalueof10, xlim=(50 r/ 2) /t isapproximately 7 85guaranteeingthatapproximations(7.12,7.13)areappropriateatalllower temperatures.Evaluatingequation(7.14)for xlim=(50 r/ 2) /t overthewhole range0
PAGE 146

139 1 2 3 4 5 6 T T o 1.2 1.3 1.4 1.5 1.6 1.7 1.8 effective Figure7.10:EectivePower-LawExponentforThermalConductance.Inthe absenceofscattering,theexponentfort emperaturedependencywillbefractal andapproachthedimensionalityofthecoupledmodesinthe k 0limit. where x1= lowestnonzerom t xlim = 50 r/ 2 t Theresultingthermalconductance,ch aracterizedby,isshowninFig.7.8. Thesummationwithinequation(7.8)the rmallysmearsdiscretecontributionsto heatconductanceandprecludesmanifesta tionofanystepwiseprogression. Equation(7.14)incidentallyshowsthattheeectivedimensionofthesetof eigenvaluestranslatesdirectlyintothep ower-lawtemperaturedependencyofthe thermalconductanceonceclearofthelow temperatureregion.Theprogressofthe powerlawfortemperaturedependencecanbedirectlyexhibitedusingthesame techniqueusedtomeasureeectivedimensionalityviaequation(7.9).Theresult isshowninFig.7.10.Itis,ofcourse,basedonanassumptionthatthepropagating phononsremainballisticovertheentire range.Ifthewireisnotuniforminits crosssectionorifimpurityscatteringis relevant,assumptionsoftheLandauer modelwillweakenandtheeectsoffractaldimensionalityarelikelytobelost.

PAGE 147

CHAPTER8 CONCLUSION Themaingoalofthisstudyhasbeentodemonstratethatitispossibleto analyticallyderivethemesoscopicphononeigenspectrumofarectangularisotropic elasticwaveguide.Thesuccessofthisenterpriseshouldimpactarangeofapplicationsandinterests,butperhapsthestrongestcontemporaryrelevancecomesfrom broadinterestinverysmallscalestructuresinnanocircuitry.Quantumwires arecommonlyrectangularandsmalleno ughtoexhibitdimensionalconnement ofelectrons.Atthatphysicalscale,however,theyarealsosmallenoughthat theirlowestphononfrequenciesarehighenoughtoconstituteasignicantsource ofinteractionwiththeelectrons.Ingeneral,atmesoscopicscales,phononsare consideredtobethemostimportantsourceofelectricalresistivity. Toputtheimportanceofscaleinperspective,considerequation(4.12).This simpledenitionofdimensionlessrescalin gusedinthestudyalsoshowsthatactual frequencyscaleswiththeinverseofthesm allestcrosssectionalhalfwidth.Typical dimensionlessfrequenciesshowninthisstudyarelessthan10 r/ 2.Assuming,for thesakeofdiscussion,ashearvelocityontheorderof103m/s(appropriatefor metals),correspondingactualfreque ncieswillbeontheorderofMHzforhalf widthsofacentimeter,tensofGHzfor halfwidthsofamicrometer,andonthe orderof1013sec 1forhalfwidthsofananometer.Since kB/ h isonlyapproximately 1 3x1011,itisnothardtoseethatmesoscopicpho nonsinsmall-to-nanoscaleobjects havethepotentialforelectronicinteractionatquitereasonabletemperatures. Asidefromtheapplicabilitytothermalan delectronicpropertiesonsmallscales, thisstudyhighlightsgenerallythatcoupl edphononsliveinafractal-dimensional 140

PAGE 148

141 phasespace.Asoneentersthemesoscopic regime,surface-coupledphononseclipse bulkphononsinimportanceandoneexp ectationshouldbethatanyproperty connectedtothedimensionalityofthephononphasespacewillbeaected.I havedemonstratedthiseectbycalculat ingafractaltemperaturedependency duetomesoscopicphononsinlow-temper aturerectangularenvironments.In general,power-lawbehaviorstiedtophono nphasespacedimensionwillexperience areductionintheapplicableexponenttosomenon-integralnumberatleastupto thepointatwhichinteractionserodethedirectaectsofthesurface. Thisstudyalsodemonstrateswhatmaywe llbeagenerallyhelpfultechnique fordealingwithnonseparableproblems. Itisthenonseparabilityoftheboundary conditionswhichjusti“edthewidespreadbeliefthattheproblemhadnoanalytic solution.Itissurprising,therefore,to seetheproblemyieldtoanextremelysimple transformtogetherwithacarefullychosen,thoughultimatelyquitesimple,setof representations.Theabilitytocharacter izethespectrumwithoutfullysolvingthe superpositionsissurprising…butnotatallesotericinitsdemonstration.Inthe end,itwascarefulattentiontoveryele mentaryaspectsoftheproblemwhichled tothisresultandthatmaywellprovehelpfulinanothernonseparabilitysituation. Finally,thisstudylaysthegroundworkfor furtherresearch.Itisreasonableto expectthattheresultsofthisstudywillbeastartingpointforseveralendeavors. Theseincludebuildingathoroughmathemat icaldescriptionofthesuperpositions, andextendingtheresultstononisotropicmaterials.

PAGE 149

REFERENCES [1]N.Nishiguchi,Y.Ando,andM.N. Wybourne,AcousticPhononModesof RectangularQuantumWires,Jour nalofPhysics,CondensedMatter 9 ,5751 (1997). [2]B.Auld,in AcousticFieldsandWavesinSolids ,1sted.(Wiley,NewYork, 1973),Vol.II,Chap.10,inwhichunavailabilityofsolutionsforfree-standing rectangularwaveguidesisremarkeduponspeci“cally.Thesameobservation doesnotappearinthe2ded.becausetheentiresubsectioncontainingitwas removed…ostensiblytomakeroomforanewtopic. [3]R.W.Morse,TheVelocityofCompressionalWavesinRodsofRectangular CrossSection,TheJournaloftheAcousticalSocietyofAmerica 22 ,219 (1950). [4]S.Yu,K.W.Kim,M.A.Stroscio,G.J .Iafrate,andA.Ballato,Electron… Acoustic-PhononScatteringRatesinR ectangularQuantumWires,Physical ReviewB 50 ,1733(1994). [5]A.I.Beltzer, AcousticsofSolids (Springer-Verlag,Berlin,1988). [6]L.Pochammer,BeitragzurTheoriederBiegungdesKreiscylinders,Journal F¨ urReineUndAngewandteMathematic(Crelle) 81 ,33(1875). [7]C.Chree,LongitudinalVibrationso faCircularBar,QuarterlyJournalof Pure&AppliedMathematics XXI ,287(1886). [8]L.Rayleigh,OntheFreeVibrationsofanIn“nitePlate,&c.,Proc.London Math.Soc.Series1 XX ,225(1889). [9]H.Lamb,OnWavesinElasticPlate,Proc.RoyalSocietySeriesA 93 ,114 (1917). [10]J.Micklowitz, ElasticWavesandWaveguides (NorthHolland,NewYork, 1978). [11]R.D.Mindlin, TheCollectedPapersofRaymondD.Mindlin (SpringerVerlag,NewYork,1989),reprintedfrom StructuralMechanics ,Pergamon Press(1960). [12]A.N.Holden,LongitudinalModesofElasticWavesinIsotropicCylindersand Slabs,BellSystemTechnicalJournal 30 ,956(1951). 142

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143 [13]G.E.Hudson,DispersionofElast icWavesinSolidCircularCylinders, PhysicalReview 63 ,46(1943). [14]D.Bancroft,TheVelocityofLongitudi nalWavesinCylindricalBars,Physical Review 59 ,588(1941). [15]S.Timoshenko, VibrationProblemsinEngineering ,3ded.(D.VanNostrand, NewYork,1955). [16]M.A.Medick,One-DimensionalTheoriesofWavePropagationandVibrations inElasticBarsofRectangularCrossS ection,JournalofAppliedMechanics Sept66 ,489(1966). [17]K.Washizu,OntheVariationalPrinciplesofElasticityandPlasticity,MIT AeroelasticStructureResearchLaboratoryTechnicalReport15.18,1955. [18]H.C.Hu,OnSomeVariationalVibrationalPrinciplesintheTheoryof ElasticityandtheTheoryofPlasticity,ActaPhysicaSinica 10 ,259(1954). [19]L.Rayleigh,in TheTheoryofSound (Macmillan,London,1877),Vol.i. [20]E.T.Whittaker,in ATreatiseontheAnalyticalDynamicsofParticleand RigidBodies ,4thed.(UniversityPress,Cambridge,1937). [21]R.Holland,ResonantPropertiesofPiezoelectricCeramicRectangularParallelepipeds,TheJournaloftheAcousticalSocietyofAmerica 43 ,988(1968). [22]R.HollandandE.P.EerNisse, DesignofResonantPiezoelectricDevices (MIT Press,Cambridge,Mass.,1969). [23]H.H.Demarest,Jr.,Cube-Resona nceMethodtoDeterminetheElastic ConstantsofSolids,TheJournaloftheAcousticalSocietyofAmerica 49 768(1971). [24]G.Arfken, MathematicalMethodsforPhysicists ,3ded.(AcademicPress,San Diego,1985). [25]W.M.Visscher,A.Migliori,andT.M.Bell,OntheNormalModesofFree VibrationsofInhomogeneousandAniso tropicElasticObjects,Journalofthe AcousticalSocietyofAmerica 90 ,2154(1991). [26]A.Migliori, ResonantUltrasoundSpectroscopy (Wiley,NewYork,1997). [27]V.B.Campos,S.DasSarma,an dM.A.Stroscio,Phonon-Con“nement EectonElectronEnergyLossinOneDi mensionalQuantumWires,Physical ReviewB 46 ,3849(1992).

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144 [28]R.Mickevi cius,V.V.Mitin,K.W.Kim,M.A.Stroscio,andG.J.Iafrate, ElectronIntersubbandScatteringby Con“nedandLocalizedPhononsinReal QuantumWires,JournalofPhysics;CondensedMatter 4 ,4959(1992). [29]M.A.StroscioandM.Dutta, PhononsinNanostructures (CambridgeUniversityPress,CambridgeUK,2001). [30]S.DasSarmaandV.B.Campos,Low-temperatureThermalRelaxationof ElectronsinOne-dimensionalNanometer-sizeStructures,PhysicalReviewB 47 ,3728(1993). [31]S.DasSarmaandV.B.Campos,LO-phononEmissionsbyHotElectrons inOne-DimensionalSemiconductorQ uantumWires,PhysicalReviewB 49 1867(1994). [32]J.SeylerandM.N.Wybourne,AcousticWaveguideModesObservedin ElectricallyHeatedMetalWires,PhysicalReviewLetters 69 ,1427(1992). [33]J.R.SennaandS.DasSarma,GiantMany-BodyEnhancementofLow TemperatureThermal-Electron-Acoust ic-PhononCouplinginSemiconductor QuantumWires,PhysicalReviewLetters 70 ,2593(1993). [34]R.Mickevi cius,V.Mitin,andV.Kochelap,RadiationofAcousticPhonons fromQuantumWires,JournalofAppliedPhysics 77 ,5095(1995). [35]G.D.Sanders,C.J.Stanton,andY.C.Chang,TheoryofTransportinSilicon QuantumWires,PhysicalReviewB 48 ,11067(1993). [36]R.W.Morse,TheDispersionofCompressionalWavesinIsotropicRodsof RectangularCrossSection,Ph.D.thesis,BrownUniversity,Providence,R.I., 1949. [37]B.Auld, AcousticFieldsandWavesinSolids ,1sted.(Wiley,NewYork, 1973),Vol.I&II. [38]B.Auld, AcousticFieldsandWavesinSolids ,2nded.(KriegerPublishing Company,Melbourne,Florida,1990),Vol.I&II. [39]K.W.Kim,S.Yu,M.U.Erdo gan,M.A.Stroscio,andG.Iafrate,Acoustic PhononsinRectangularQuantumWires:ApproximateCompressionalModes andtheCorrespondingDeformationPotentialInteractions,Proceedingsof SPIE 2142 ,77(1994). [40]C.HerringandE.Vogt,TransportandDeformation-PotentialTheoryfor Many-ValleySemiconductorswithAnis otropicScattering,PhysicalReview 101 ,944(1956).

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145 [41]G.D.Mahan,TemperatureDepende nceoftheBandGapinCdTe,Journal ofPhys.Chem.Solids 26 ,751(1964). [42]L.D.LandauandE.M.Lifshitz, TheoryofElasticity ,Vol.7of Courseof TheoreticalPhysics (Wiley,NewYork,1970). [43]E.H.Love, ATreatiseontheMathematicalTheoryofElasticity ,4thed. (Dover,NewYork,1944),reprintof1927Cambridgepublication Mathematical TheoryofElasticity [44]F.RieszandB.Sz.-Nagy, FunctionalAnalysis (Dover,NewYork,1990),“rst publishedbyFrederickUngarPublishingCompany,NewYork,1955. [45]H.Sagan, BoundaryandEigenvalueProblemsinMathematicalPhysics (Dover,NewYork,1989reprintof1961Wiley,NewYorkpublication),see TheoremVII.8,section3.3. [46]R.Peierls, SurprisesinTheoreticalPhysics (PrincetonUniversityPress,New Jersey,1979),section3.5. [47]D.E.Angelescu,M.C.Cross,andM.L.Roukes,HeatTransportinMesoscopicSystems,SuperlatticesandMicrostructures 23 ,673(1998). [48]L.RegoandG.Kirczenow,QuantizedThermalConductanceofDielectric QuantumWires,PhysicalReviewLetters 81 ,232(1998). [49]K.Schwab,E.Henriksen,J.Worlock,andM.Roukes,Measurementofthe QuantumofThermalConductance,Nature 404 ,974(2000). [50]B.B.Mandelbrot, TheFractalGeometryofNature (Freeman,SanFrancisco, 1982),reviseded.of1977Title.

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BIOGRAPHICALSKETCH StevenE.Patamiawasbornin1947inNewOrleans,Louisiana,butgrewup asamilitarydependent.ConsequentlyhehaslivedinEngland,thefarEast,and variousstatesideŽlocations.HecompletedhighschoolinLasVegas,Nevada whilehisfatherwasstationedatNellisAFBoutsideofLasVegas. Stevenhashadstrongscienti“candtec hnicalinterestssincechildhood.Among themanifestationsofthisinterest,hewasanactiveamateurrocketeerŽinthe 1960sandhadtworockets“redattheArmytestrangeatFortSill,Oklahoma. Upongraduationfromhighschool,StevenwasthelocalrecipientoftheBauschand LombScienceAwardgiventorecognizestudentswiththemostscienti“cpromise. Stevenbeganundergraduatestudieswhile stillinNevada,buttransferredto UCLAatthesametimehisfatherwasassignedtodutyinVietnam.Eventually, herejoinedhisfamilyinTampa,Florida,wherehisfathereventuallyretiredfrom theAirForce.Hecompletedhisundergra duatedegreeinmathematicsfromthe UniversityofSouthFloridain1972.Completionofthatdegreewasinterrupted bymilitaryserviceduringwhichhewasa ssignedtoaDefenseIntelligenceAgency facilityinHawaii. Priortobeingdraftedintomilitaryservicein1970,andwhileworkingonhis BA,Stevenwasadirectorofsystemsprogrammingatwhatwasthenknownas theComputerResearchCenterattheUniversityofSouthFlorida.Heremained professionallyinvolvedincommercialsoftwaredevelopmentandsupportovera periodofapproximately30years. 146

PAGE 154

147 Followingmilitaryservice,Stevenpursued ”ighttrainingandultimatelybecame anactivegeneralaviationinstrument”ightinstructor. StevenattendedtheUniversityofFloridaLawSchoolandreceivedaJurisDoctordegreein1977.HepracticedcommerciallawintheMiamiarea,buteventually concentratedonvariousconsultingvent ures.Outofasenseoffrustrationatnot havingful“lledhisearlyscienti“cambitions,Steveneventuallyappliedtoandwas admittedintothePhDprograminphysicsattheUniversityofFlorida.Research onhisPhDtopicwascompletedattheLosAlamosNationalLaboratoryinNew MexicoandStevenhopestobeabletocompletefurtherdevelopmentofthetopic atthatlocation.


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SPECTRUM AND PROPERTIES OF MESOSCOPIC
SURFACE-COUPLED PHONONS
IN RECTANGULAR WIRES












By

STEVEN EUGENE PATAMIA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2001
































Copyright 2001

by

Steven E. Patamia















ACKNOWLEDGMENTS


I would like to thank my advisor, Prof. Pradeep Kumar, for indulging my

interest in pursuing this vestigial problem. I want to acknowledge publically that

I understand the risk to both our careers that accompanied the continued pursuit

of a fundamental and yet elusive problem which, despite its significance, has been

declared without solution.

I want to thank Dr. Albert Migliori and the Los Alamos National Laboratories

for supporting the completion of this dissertation and for confirming in a material

way that there were physicists and scientific enterprises for whom the results truly

mattered.

I want to thank my children, Mandy and Sarah, for their tolerance of my

determination to pursue a PhD late in life.

On the chance that she will orn, i- read this page, I want to express appre-

ciation to Michelle for reasons only she will know.
















TABLE OF CONTENTS


ACKNOWLEDGMENTS ..........

ABSTRACT . . . . . . .

CHAPTERS

1 INTRODUCTION . . . . . .

1.1 Nature of the Problem .........................
1.2 History ................... ..............
1.3 Acoustic Phonons in Quantum Wires .................
1.3.1 Confinem ent . . . . . . .
1.3.2 Mesoscopic Electron-Phonon Interactions ...........
1.3.3 Confined Acoustic Phonons and the Deformation Potential


2 ASSUMPTIONS AND CONVENTIONS .. .........


Physical Model and Coordinate System . .....
Symbolic Consistencies and Adopted Tensor Notation .
Linear Elasticity Theory . ..............
Free Boundary Conditions . .............
Nonseparability of Boundary Solutions . ......


. . 19
. . 21
. . 22
. . 27
. . 30


3 THEORETICAL ASPECTS OF RECENT NUMERICAL METHODS


Stationary Lagrangian . .........
Numerical Approximation . ......
Partitioning the Problem into Parity Groups .


4 MATHEMATICAL STRATEGY .. . ...........


Notation and Function Extension Issues
Defining the Basis and Superpositions .
Dimensionless Representations . .
Derivation of Rayleigh-Lamb Equation .
How to Transform Superpositions ..


. 33
. 39
. 42


. 45
. 49
. 54
. 55
. 62











5 DERIVING NORMAL MODES OF PROPAGATION ........

5.1 General Considerations .........................
5.2 Acoustic Poynting Vector of a Normal Mode .............
5.3 Propagating Modes Involving Hy, H, Shear .. ..........
5.3.1 Deriving the Frequency Equations ..............
5.3.2 Interpretation . . . . . . .
5.3.3 M ode Dispersions .. ....................
5.3.4 Comments on the Process of Mapping Mode Dispersions
5.4 Disposing of the H, / 0 Possibility .. ...............


6 K = 0 MODES OF A RECTANGULAR WIRE .. ......


k = 0 Boundary Conditions . ...........
Uncoupled (Separable) k = 0 Modes . ....
Uncoupled Modes not the Limit of Propagating Modes .
Coupled (nonseparable) k = 0 Modes . .....
6.4.1 Derivation-All Parity Families . .....
6.4.2 Manifestations By Parity Family . ....


. . 102
. . 103
. . 107
. . 112
. . 112
. . 117


7 FRACTAL PHASE SPACE OF COUPLED MODES AT K -- 0 .

7.1 Motivation: Low Temperature Heat Conductance . . .
7.2 Effective Dimension & Density of Propagating Modes as k -- 0 .
7.3 Low Temperature Heat Conductance . . .


8 CONCLUSION .. .....


REFERENCES .. .......

BIOGRAPHICAL SKETCH .. ...


125

. 125
129
. 135










Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SPECTRUM AND PROPERTIES OF MESOSCOPIC
SURFACE-COUPLED PHONONS
IN RECTANGULAR WIRES

By

Steven Eugene Patamia

December 2001


C'!i iii i'I: Pradeep Kumar
Major Department: Physics


This dissertation presents original analytical derivations of the propagating

modes of coupled mesoscopic phonons in an isotropic rectangular wire with stress-

free surfaces. Incidental to the derivations, novel consequences of the derived

cutoff modes are presented as they affect the low-energy heat conductance of

such wires, or indeed .niv property that depends upon the dimensionality of the

phase space within which the modes reside. Owing to nonseparability of the

free-surface boundary conditions, an analytic description of coupled mesoscopic

modes has heretofore been presumed to be underivable. Results presented herein

show that the mode spectrum of coupled mesoscopic phonons is both subtle and

rich, but considerable success in their analytic derivation is achieved. Using

numerical methods developed for resonance problems, at least one contemporary

researcher has purported to exhibit the lowest dispersion branches of propagating

mesoscopic phonon modes in GaAs-which is not isotropic. The accuracy of these

branches has not been measured, but they bear a qualitative consistency with

isotropic modes derived herein. Since before the beginning of the 20th century,

analytical solutions have been known for the infinite thin plate and even the










case of waveguides with circular cross sections. Solutions for these special cases

take the form of transcendental relations among the wavenumber and boundary

parameters, but the underlying wavefunctions are separable in the coordinates.

The analytical results presented herein for the general rectangular case involve

nonseparable solutions whose separable components do not individually satisfy

the boundary conditions. These solutions also take the form of transcendental

relations, but there are sets of transcendental relations for each family of the

cases that partition the problem. Consequently, the eigenspectrum, while defined

by exact forms, must be enumerated by identifying plotted intersections of the

root families of these transcendental relations. The resulting spectrums are more

complex and have less apparent order than the spectrum produced using either

periodic boundary conditions or rigid boundary conditions for uncoupled phonons.















CHAPTER 1
INTRODUCTION


1.1 Nature of the Problem


Despite some progress in exhibiting the low-lying band structure of meso-

scopic phonons by numerical methods, the study of phenomena involving phonon

interactions has been hampered by an inability to analytically derive phonon

modes involving surface coupling within rectangular geometries [1]. In unbounded

bulk media, long-wavelength phonons are adequately represented by independent

families of elastic plane waves distinguished by polarization. However, at nonrigid

surfaces, elastic boundary conditions cannot in general be satisfied except by a

coupled superposition of shear and longitudinal vibrations. In some geometries

particularly those involving edges and corners-this surface constraint cannot even

be satisfied by separable wave functions let alone a derivable superposition of simple

plane waves. Reflecting this difficulty, no general analytical solution has heretofore

been published for the eigenvalues or wave functions of phonons in a mesoscopic

wire of rectangular cross-section [2-4].

As contemplation shifts from bulk material to bounded samples with elongated

geometries, the manifestation of long-wavelength phonons will include torsion and

flexing of the sample. Bulk phonon models based on longitudinal scalar potentials

alone are intrinsically incapable of exhibiting any flexural or torsional behavior

since these require both some mechanism for shear and specific kinds of distortion

of a specified surface geometry. Torsional and flexural behavior can be modeled to

some extent using bending moduli and minimizing the potential energy associated










with the distortion of finite objects (See, e.g., [5] 4.12). Yet, correct torsional

and flexural modes ought to be automatically included within a comprehensive

phonon model which correctly identifies phonons as mixed longitudinal and shear

vibrations which satisfy nonrigid boundary conditions within a specific geometry.

In modern terms, the nature of the problem can be succinctly described in

the following useful way. Within the limits of linear elasticity, phonons in bulk

are distinguished by polarization direction and propagation velocity and they do

not interact. At .,niv non-rigid surface, the reduction of external stress to zero

is a boundary condition that can be satisfied in general only if the two species

of modes become coupled at the surface. In effect, each species scatters into the

other until an energy transfer balance is achieved. A detailed examination of the

boundary conditions clarifies an important feature of this coupling. Namely, the

nondegenerate longitudinal phonons polarized along their propagation direction

(I am ignoring quasi-shear and quasi-longitudinal situations which can occur in

non isotropic materials) only couple to shear phonons which have a polarization

component normal to the surface. The surface interaction that arises as the surfaces

are permitted to distort acts as a perturbation that splits the natural degeneracy of

the shear phonons. Shear phonons polarized parallel to the surface are not affected

and reflect specularly subject to their displacements vanishing at the surface. With

respect to the phonon band structure, this degeneracy breaking is manifest as level

repulsions among phonon dispersion subbands.

It should immediately be noted that degenerate perturbation techniques are

not fruitful in deriving solutions for the coupled phonon problem. The reason for

this is that such an approach depends upon an ability to find eigenfunctions of the

perturbation itself. Pursuing this inevitably encounters the fundamental source of

mathematical difficulty which plagues the problem generally. Namely, solutions










that incorporate the full boundary conditions at .,i.] i:ent surfaces in rectangular

geometries are easily shown to be non-separable. Defining a perturbation that

reflects coupling of.,l.i i:ent sides and then identifying a basis for its eigenfunctions

become intractable.



1.2 History


Early in the history of elasticity theory, vibrational modes which ac.':i|' l,111

full relaxation of applied surface forces (formally, as the projection of the stress

tensor onto a vector normal to the surface vanishes) were successfully derived for

certain geometries. A solution was first obtained for circular, infinite-length bars

by Pochammer [6] in 1875 and independently by C'! ..' [7] in 1886. In 1889 Lord

Rayleigh [8] published the analogous solution for an infinite plate with stress free

surfaces.

In the foregoing cases, the "solutions" obtained took the form of so-called

"frequency equations" which define a transcendental relationship among shear

and longitudinal wave vector components. The roots of these equations, for each

propagation wave number, constitute the eigenspectrum. Reflecting the difficulties

inherent in analytically exposing the roots of such transcendental relations, only

some .,-i i >! totic roots of the Pochammer-C'!I .- frequency equations accompanied

them immediately, and .,-i-, i id ic solutions to Rayleigh's plate solution did not

appear until it was revisited by Lamb [9] in 1917. As Lamb explores .,-i-iI. Iltic

roots and the displacement functions, he makes perhaps the first observation that,

at increasing propagation wave numbers, two of the fundamental modes converge

to form Rayleigh Surface Waves. Over succeeding decades, frequency equations

for plates and rods were extensively studied and numerically derived roots of the










underlying transcendental relations are extensively characterized and well known

in engineering applications [10-14].

It might seem surprising that despite difficulties identified above, that the

frequency equation for circular waveguides was found at all, let alone before,

the published analogous solution for the infinite plate. The reason is that a

cylindrical coordinate system readily accommodates the fact that surface coupling

only involves shear waves whose polarizations result in displacements normal to

the surface. For cylindrical environments, the shear displacement field is naturally

decomposable into a radial component coupled to the longitudinal field and a

component parallel to the surface which is not. By contrast, for surfaces with

sharp corners, only an unobvious superposition of waves forms (not necessarily

plane) will accommodate coupling at .,i.i i:ent faces. Making the geometry finite

whether with flat or rounded ends-further complicates this difficulty, as was noted

even by C'i.,

Once the wave guide takes on a rectangular cross section, the resultant bound-

ary value problem develops pathologies linked to the fact that solutions are gen-

erally nonseparable in the coordinates. The lack of an analytical solution to the

rectangular cross section problem has consequently been widely conceded in the

literature (see e.g., [2-4]) In view of this perceived limitation, approximate solutions

for various subsets of the modes for a bar have been developed-Timoshenko [15]

"beam theory" being an important example in the realm of engineering applica-

tions.

In the absence of direct analytical solutions for the rectangular cross section, it

remained possible to apply a variational approach to generate solutions that coin-

cidentally satisfy the boundary conditions. The possibility of this tactic emerges

in the literature at least as early as 1966 when Medick [16] attempted to develop a








5

1-dimensional wave propagation theory for rectangular bars. Medick reviewed the

important result [17,18], already then a decade old, that solutions which render

the Lagrangian stationary are precisely those which simultaneously satisfy the

bulk wave and natural boundary conditions. Medick, however, did not apply the

variational principle directly to numerical computation.

The basic observation that the eigenfrequencies of a freely vibrating elastic body

are stationary in the space of functions was deduced early on by Lord Rayleigh

who incorporated it into his treatise The Theory of Sound [19] in 1877. This result

is usually referred to as "Rayleigh's Principle" [20]. Note that using Rayleigh's

Principle to support a variational solution can be distinguished from approaches

discussed within that rely explicitly upon stationarity of the time-independent

Lagrangian as a manifestation of Hamilton's principle.

Early direct application of a variational principle to compute the low-order

modes of parallelepipeds was presented by Holland [21] in 1968. Beginning with

stationarity of the time-independent Lagrangian, Holland articulates an essen-

tially Rayleigh-Ritz approach wherein finite combinations of trial functions are

minimized. Holland's particular interest is in piezoelectric materials, but his

computations are not in any way restricted by this. Holland examines the resulting

eigenfunctions in an attempt to describe the displacement pattern of the modes

themselves along with the spectrum. Holland's trial functions for components

of the displacement field are products of sines and cosines of the coordinates.

Later, Demarest (below) believed that such choices could precipitate inaccuracy

in the results of a variational calculation. In recognition of the importance of this

convergence issue, Holland and EerNisse examine this factor critically in a later

monograph [22].










Building on Holland's 1968 result, Demarest [23] in 1971 improved the accuracy

of Holland's mode computations for a free-surface cube-in part by changing the

nature of the trial functions. The trial functions of Demarest's own Rayleigh-

Ritz expansion are products of Legendre Polynomials of the coordinates. Along

with identifying an inadequacy in Holland's choice of trial function, Demarest

makes the observation that while Legendre Polynomials exhibit a computationally

useful orthogonality in the relevant space, in principle they could be replaced by

a power series in the coordinates of which Legendre Polynomials are just a linear

combination.

Neither Holland's 1968 paper nor Demarest's 1971 paper derive the general rela-

tionship between satisfaction of the boundary-value condition and the efficacy of

the variational principle utilized. Since they are able to demonstrate a satisfactory

correspondence between actual measurement results and the numerical predictions,

they appear to consider the theoretical, albeit important, question of boundary

conditions to be moot. Demarest instead focuses on the fact that derivatives of his

trial functions constitute a more "relatively complete" set of basis functions than

Holland's. By "more relatively complete," he means that linear combinations of

selections from this finite basis are able to converge to a broader range of possible

functions. Holland, as Demarest points out, manipulates his trial function choices

to favor convergence of their derivatives to zero at the boundary. Demarest's

choice of Legendre Polynomials enriches the basis so that convergence both to

zero displacements and zero derivatives of displacement are possible. While this

is sufficient to improve the accuracy of numerical results as compared against

measurements, it cannot be decisive. The actual boundary conditions which

will be explored herein cannot be simply partitioned into cases involving zero

displacements versus zero derivatives of displacements. Moreover, in most cases










unobvious superpositions are needed to assemble boundary solutions which are

provably nonseparable and the nature of their boundary behavior is impossible

to discern in advance of actually exhibiting them. Whatever its incremental

improvement, Demarest's choice of basis also thwarts the expectation that the

final solution, and presumably its components, must be constituted of no more

than two parts which separately satisfy a basic wave equation-as is provable from

Elasticity Theory.

Both Holland and Demarest utilize a Rayleigh-Ritz kind of procedure which

approximates the solution to an eigenvalue problem by finding stationary points

among a combination of trial functions. The conventional Rayleigh-Ritz procedure

is implemented using trial functions which individually satisfy applicable boundary

conditions (see any text on mathematical physics, e.g., [24] 17.8). This, however,

is not possible under circumstances where there are no available basic solutions

for the geometry that are known to satisfy those boundary conditions in the first

place. It thus was openly expressed by Visscher, Migliori and Bell [25] in 1991 to

be "fortuitous" that displacement functions which could be found to minimize the

time-independent Lagrangian would automatically satisfy the bulk-wave and free-

boundary conditions simultaneously. As I will point out more particularly within,

this fortuity is less dramatically useful than its discovery infers. The reason is that

it is only as helpful as the set of trial functions is versatile and is literally true only

if the set of trial functions happens to include a combination that is a solution.

Again, the central impediment is that basis functions which at least satisfy the

boundary conditions do not even exist. Visscher et al., therefore, concentrated

upon assembling a basis chosen for its versatility and the convenience with which

numerical computations could incorporate its elements. In fact, the basis elements

they choose, while linearly independent, are not orthogonal, and do not even satisfy










a wave equation let alone boundary conditions. Remarkably, at least in terms of

resonances of closed objects, the basis is nevertheless sufficient to produce useful

results to a sometimes high degree of accuracy [26].

Possibly because engineering and physics communities do not consistently over-

lap, Visscher et al. were probably not aware that Medick had previously noticed

the same fortuity a quarter-century before and had been reviewing results available

a decade before that. Nevertheless, Visscher, Migliori and Bell offer an extremely

elegant derivation of the principle and clarify greatly the theoretical underpinnings

of the algorithm developed by Holland and Demarest. Moreover, equipped with

access to some of the earliest tools to bear the designation "supercomputer,"

Visscher et al. tested directly the efficacy of a particularly useful and elegant

choice of basis functions. Specifically, they implement Demarest's -,ii-:. -I. li that

products of powers of the coordinates should constitute a set of basis functions

that would be sufficiently "complete" to converge to accurate solutions-at least

for the frequency eigenvalues in resonance problems.

Like Holland and Demarest, Visscher et al. were motivated by the need to find

accurate low-order eigenmodes which could be verified by comparison to actual

measured resonances of samples. They showed that an algorithm employing a

basis built from simple products of powers of the coordinates would accomplish

this even for geometric shapes far more challenging then the parallelepipeds of

Demarest and Holland. Subsequently, Migliori has perfected this technique into

an accurate standard means of finding the microscopic elastic constants of materials

having arbitrary i--l I1 or composite structure [26].

The directness of computability and broad applicability of the computational

algorithm refined by Visscher, Migliori, and Bell has recently been adapted and

applied to exhibit the modes of an infinite rectangular wire. Through the simple










expedient of replacing one coordinate factor of the basis functions with a common

periodic function of the long coordinate alone,


i.e., "Pyr v -+ x"r ei q


Nishiguchi, Ando and Wybourne [1] resolved the transverse cross section com-

ponents using the same variational methods as Visscher et al. while retaining

translational invariance along the length of the infinite wire. Instead of obtaining

a finite set of resonant eigensolutions for a resonator, he obtains a finite family of

solutions for each value of q in a waveguide. they then generate the dispersion rela-

tions along the z-axis by plotting the resulting solutions which remain continuous

in the parameter q. Since the wires are infinite length, there are no considerations

of mixing from end-effects. There is also no need for the variational algorithm to

resolve the modal pattern along the long dimension. As a consequence, convergence

is limited only by the algorithm's ability to resolve the transverse displacements

whose modest periodicity over a limited range is more readily approximated by

a finite power expansion. Their specific results are computed for GaAs, but as

Visscher et al. had emphasized, there is nothing in the nature of the numerical

algorithm which inherently limits it to any material or i i-I I1 structure.

The mode dispersions developed by Nishiguchi et al. lack comparison with

measurement. Directly measuring the eigenfrequencies of a resonator is straightfor-

ward, but directly measuring the long-wavelength subband dispersions of phonons

subject to a specific geometric boundary may not be. The technique of finding

bulk phonon dispersions using neutron scattering cannot be easily applied to a

geometrically confined low-wavelength environment-particularly if the sample is

microscopic. Also, full resonances are precise by definition and represent a severely

constrained computational problem. It is not obvious whether a set of linearly










independent, but not inherently orthogonal, trial functions remain similarly able

to converge strongly to a solution where the basis is constrained conceptually, but

not actually, along one direction. These concerns will be further addressed in the

results presented.

In general contemporary research, mesoscopic acoustic phonons have been

investigated theoretically, and to some extent experimentally, in primarily two

kinds of situations. The first is in respect of their interactions with phonons in

quantum wires and the second is their role as carriers of thermal energy. The

two topics have some overlap. In the next section, I will summarize the findings

with respect to electronic interactions in quantum wires and in Ch. 7 I will take

up the more emergent topic of kinetic heat transport by acoustic phonons in the

low-temperature high-confinement regime.



1.3 Acoustic Phonons in Quantum Wires


1.3.1 Confinement

From the end of the 1980's and continuing into the present, mesoscopic physics

has enjoi, d an obvious prominence driven largely by the relentless downscaling of

electronic devices. Commercial applications aside, the emergence of an ability

to fabricate devices within which quantum behavior becomes observable as a

consequence of dimensional confinement itself stimulates persistent and widespread

interests in the effects of small size and low temperatures. Certainly, tod i-, there

is no more ubiquitous device in the theoretical and experimental condensed matter

inventory as the "quantum wire."

In this context, so-called "phonon o. .-1ii in, ni has also become a phenomenon

of specific interest. Phonons, being a classical rather than quantum phenomenon










per se, are not subject to the sharp dimensional confinements possible for quantum

particles, like electrons. When characterizing phonons as "confined," authors are

only distinguishing phonon modes subject to quantization by physical boundaries

from bulk phonons quantized using periodic boundary conditions. There is no

inferred loss of dimensionality of the phonons, though, of course, the lowest sub-

bands typically have imaginary transverse wave numbers and higher subbands may

sometimes become energetically inaccessible.

While "(~.'1 ii il at drastically small scales can lead to actual dimen-

sional reduction of quantum particles, its generic effect is to render the physical

boundaries relevant. The signature feature of what is called i:: .-...' I remains

the commensurateness of the characteristic wavelength with the distance between

boundaries. For phonons, however, actual small scales and low temperatures tend

to magnify the departures from bulk behavior provoked by mesoscopic considera-

tions. Certainly, at any scale, mesoscopic phonons exist since some subset of the

phonon spectrum consists of phonons at wavelengths comparable to the size of the

physical environment. At small scales and low temperatures, however, mesoscopic

phonons are the dominant ones and their actual frequencies, which scale inversely

with overall size, become large enough to augment the probability of electronic

interactions.

Mesoscopic phonons are, by definition, of a wavelength large enough that

surface effects and geometry control their dispersion and wave structure. Periodic

boundary conditions and plane wave modeling give way to explicit boundary

solutions for specific geometries. Surface coupling endemic to the non-rigid bound-

ary conditions determines the eigenspectrum and mixes longitudinal and shear

contributions.










1.3.2 Mesoscopic Electron-Phonon Interactions

Absent an analytical solution for surface-coupled phonons, research into con-

fined phonon interactions progressed by using approximations or rough models

which were believed to mimic the most important features of actual phonons.

A brief survey of the results produced in this manner thus serves to identify

conclusions and calculations which should be revisited in the light of analytical

results from this investigation.

By the 1990's, papers explicitly calculating the interaction of electrons in

a quasi-one-dimensional gas (i.e., in a quantum wire) with "confined phonons"

were appearing regularly [27,28]. Initial investigations focused primarily on hot-

electron relaxation via emission of "confined" optical phonons. The phonon models

utilized during this time were typically uncoupled symmetric and antisymmetric

transverse resonances where the transverse components were taken either to be

zero (denominated "guided" modes) or a maxima at the boundaries (denominated

-I il" modes). The latter condition is actually similar to stress-free conditions for

elastic phonons. Results from the two flavors of boundary conditions are compared.

Among the interesting conclusions is that boundary conditions resembling stress-

free surfaces lead to an order of magnitude faster relaxation rate than obtained by

assuming zero displacement boundaries [27].

Longitudinal optical phonons in polar semiconductors will decay to acoustic

phonons (see Ref. [29] 6.2) within some average LO lifetime which can actu-

ally augment dissipation by virtue of precluding LO phonons in the process of

decay from being absorbed back by the carriers. However, at low temperatures

in quantum wires (e.g., below 30 K for a 100 A wide GaAs wire) hot-electron

scattering to LO phonons becomes exponentially weak and direct ... -1i c phonon










emission is the only important dissipative process" [30] though for quantum wire

wells, LO scattering may continue to dominate into lower temperature regimes [31].

Early in this period, theoretical interest in hot electron relaxation to acoustic

phonons in wires was encouraged by experimental developments. In 1992 Seyler

and Wybourne published a PRL [32] reporting on the detection of resonances with

presumably acoustic modes in small (approximately 20 nm thick and between

30 and 90 nm wide) Au-Pd wires over a broad range of low temperatures (1-

20 K). For the balance of the decade through to the present, this particular

experimental observation is among the most consistently cited as justification for

further theoretical exploration of the "importance of acoustic-phonon confinement

in reduced dimensional electronic structures" [4].

The essential role of acoustic phonons in energy dissipation within quantum

wires has been scrutinized in various v-i-, Senna and Das Sarma have investigated

a "Giant Many-Body Ei! i ii.'. iii, i, of electron-acoustic phonon coupling at low

temperatures by renormalizing the phonons in the presence of the electron gas [33].

They find that at low temperature (viz. 1 K), below which direct plasma resonance

cannot be significant, the quantum-mechanical (which they distinguish from ther-

mal) uncertainty in the phonon modes creates an interaction enhancement orders

of magnitude greater than for bare phonons. Mickevicius, Mitin, and Kochelap

used Monte Carlo calculations to investigate phonon radiation from a Quasi-1D

electron gas in rectangular GaAs quantum wires of 80x80 A in the neighborhood

of 4 K [34]. They report that virtually the entire power dissipation is due to the

transverse radiation of ballistic acoustic phonons.

All of the above cited theoretical explorations involve GaAs and/or AlAs as

the specific material of choice. While GaAs remained an almost predominant

material chosen for theoretical studies during and since this time, Si has also been










explored. In a comprehensive set of calculations, Sanders, Stanton, and C'!I 0 1; cal-

culated [35] a range of transport properties for a Si quantum wire-including defor-

mation potential scattering by "quantum confined phonons." Although the authors

explicitly reserve the term ...I i-I c" for the lowest subband, the phonon model

used corresponds exclusively to longitudinal elastic waves coupled to electrons via

a deformation potential. Possibly to avoid intimating that higher subbands of

these confined elastic modes should be called "optical" the authors describe them

as being "excited quantum-confined phonons." In any case, the focus in this paper

is on a second-quantization representation of the phonons without attempting to

mimic specific surface boundary conditions.

In a 1994 study, Yu, Kim, Stroscio, lafrate, and Ballato [4] (liberally cited else-

where herein) build upon the 1949 Ph.D. thesis work of Morse [36,3] to implement

a more realistic confined acoustic phonon model that incorporates some surface

coupling in a rectangular quantum wire environment. Upon reciting that "As is

well known, there are no exact solutions for the complete set of phonon modes

for a rectangular vii. Yu et al. adopt Morse's compromise strategy which was

expected to be adequate when the cross sectional aspect ratio was greater than

two. Morse had made progress solving the rectangular problem (which I investigate

more successfully herein) by treating the closest parallel surfaces as an independent

plate scenario (see section 4.4 herein) and finding what amounts to Rayleigh-Lamb

modes relative to those surfaces. He then observed that two of the three surface-

normal stress components become small at the .,.i i:ent surfaces as the aspect

ratio grows and tunes a free parameter in his plate solution to force one of the

two diminished stresses to be zero. Morse's transcendental equation derived from

considering the closer of the surfaces (equation (14) in Ref. [4] under discussion)

is very similar to, though less general than, my own intermediate equation (5.27)










which appears within my derivation of propagating coupled modes. His starting

assumptions are far less general, but at that corresponding point of the derivations

the situation is artificially similar. Restricting themselves to dilatational modes

(see comments below), Yu et al. proceed to compute a normalization of Morse's

separable solution and then compute electron-phonon scattering from deformation

potential interaction for a range of cross sections (principally 28x57 and 50x200

A) at 77 K. Their essential finding is that scattering rates are notably higher for

such confined coupled acoustic modes than for bulk acoustic modes and increase

dramatically as the overall scale is reduced. In fact, they compute that the

scattering rate for a 28x57 A wire are an order of magnitude larger than for a

100x200 A cross section.


1.3.3 Confined Acoustic Phonons and the Deformation Potential

Hereafter, my own use of the label *,ustic" will, absent qualification, encom-

pass all phonons that are a manifestation of elastic deformation. Consistent with

much of the literature being reviewed (see, e.g., Ref [4] discussed above), this

means that the subbands that arise from real boundary confinement of elastic

waves will continue to be referred to as acoustic despite the fact that in bulk

environments this term refers only to a dispersion branch that originates with

zero frequency at zero k. My convention is appropriate to this study where only

confined phonons are relevant, is consistent with texts and monographs devoted

to acoustics of solids [5,10,37,38]-and it appears consistent with emergent text

books devoted to phonons in nanostructures [29]. Unfortunately, past publications

are not ahv-- as inclusive, but in the worst case it will simply be important to

notice whether acoustic modes under discussion have been restricted to a specified

dispersion branch or explicitly stated to be bulk-like. As is consistent in the










historical literature of acoustics per se, I will distinguish modes whose dispersion

goes to zero as k -- 0 by calling them "fundamental."

Within the most inclusive use of the term, ui-I c" phonons are coupled to

carriers solely by the deformation potential which tracks the ion density fluctua-

tions. In each of the above-cited articles, only longitudinal (i.e., compressional)

phonons (see, especially, [39]) participate in this coupling. Since bulk shear mode

displacements can be generated by a vector potential, the divergence of their

displacements is zero and they cannot alter the ion potential field via fluctuations

in ion density.

There are, nevertheless, circumstances where a deformation potential can be

associated with non-compressional phonons. For materials with degeneracies in

band structure that can be broken by shear distortions, it is long been known that

deformation potential should be generalized to a tensor such that the potential

change due to deformation incorporates shear as well as divergence effects [40,41].

Even if such special circumstances do not apply, there remains a more fundamental

reason to revisit the exclusive focus on compressional modes once the phonons are

subject to boundary confinement.

In the continuum limit, acoustic phonons are merely elastic modes irrespective

of whether the environment is confined or bulk-like. This study is concerned,

however, exclusively with elastic modes confined to rectangular waveguides. As

will be developed in detail in subsequent chapters, elastic waveguide modes can

be partitioned with respect to the parity patterns of their displacements relative

to a coordinate systems aligned with the long axis and normal to the sides.

For the fundamental modes (i.e., those confined branches which do go to zero

as k -- 0) these patterns correspond to macroscopic motion of the waveguide

dilatational, torsional, and flexural. Of these, the dilatational is so named because










the displacement pattern of the fundamental modes resembles propagating density

fluctuations on the scale of the width of the waveguide. It is no doubt for this

reason that as phonon models in quantum wires take boundary confinement into

account, there is an explicit assumption made that only the part of the model

corresponding to dilatational modes need be considered. Torsional and flexural

modes correspond to twisting and bending and appear, on first impression, to be

dominated by shear and are assumed to have no significant deformation potential

(as assumed in Ref. [4]).
Transferring allegiance from longitudinal bulk modes to dilatational confined

modes promotes a significant oversight. In bulk media, torsion and flexural dis-

placement patterns do not even exist, but once the phonons are confined by actual

surfaces, these categories arise precisely because surface coupling mixes longitudi-

nal and shear contributions into the displacement pattern. Ignoring the torsion and

flexural modes-perhaps for lack of an analytical model for the surface coupling

comes at the cost of ignoring definite sources of density variation. A.r.vTi, tin- the

potential consequences of this oversight, it will later be seen that the branch shape

of some fundamental torsional, along with the fundamental flexural modes, shows

a high density of states at low frequencies.

Anyone who has ever wrung out a washrag realizes that fundamental torsional

modes involve patterns of change in local density. Bending a foam rubber object

creates a clear opportunity to observe the compressions and stretches which accom-

p .ni: that movement. Consistent with these real-life macroscopic observables, the

derivations presented in subsequent chapters reveal that the longitudinal potential

makes a necessary contribution to virtually every coupled mode. While it may

turn out that some subset of modes within a subband are dominated either by

shear or longitudinal patterns, there is no a priori justification for assuming that








18

any category of coupled modes can be safely ignored even if only the deformation

potential from density fluctuation remains relevant. Even Demarest took care to

point out that the dilatational, torsional, and flexural naming attached to parity

patterns should not be taken too seriously as each contained elements of shear and

dilatation [23].














CHAPTER 2
ASSUMPTIONS AND CONVENTIONS


2.1 Physical Model and Coordinate System


I assume that the waveguide is composed of an isotropic material formed

into an infinite bar whose rectangular cross section is invariant along its length.

The stiffness tensor for an isotropic material has only two independent elastic

constants and these, together with the material density, determine the bulk shear

and longitudinal velocities, denoted c, and cQ respectively [5]. To keep the results

completely general, I will reduce all derived observables to a dimensionless form

rescaled relative to the shear velocity, the smaller of the half-widths of the cross

section, h, and the aspect ratio a between the widths. Derived results will thus be

in terms of a single material characteristic-the ratio of velocities R = ce/c-and a

single geometric parameter-the cross sectional aspect ratio, a.

When computing definite eigenvalues to dipl!iw representative quantitative

results, R will be arbitrarily set to a value of -3. Some results will turn out to

be independent of R. For perspective, it is a well known result of linear elasticity

theory [42] that R cannot be less than /2 for isotropic materials. Commercial

aluminum, a modestly soft and nearly isotropic metal, has an R of roughly 2

whereas GaAs, a non-isotropic semiconductor, has an R averaging close to the v2

limit.

All phonons are contemplated in the i pi -'.-...Ipi regime, by which I mean

that their wavelengths are not less than an order of magnitude smaller than the

smallest cross sectional width. I also assume that treating the material as an elastic










continuum is justified by first assuming the waveguide material has a typical inter

lattice spacing much smaller than the smallest phonon wavelength considered. As

a practical matter, this would still permit important results to apply to quantum

wires with widths on the order of a few hundred atoms. The displacements will be

assumed sufficiently small that applied elasticity theory is well within the linear

regime. For phonons that are thermally excited or scattered from interactions with

itinerate electrons, magnons, and similar particles, this is a reasonable physical

assumption consistent with remaining within the phonons' own mesoscopic regime.

Calculations and representations will be rendered within a right-handed Cartesian









2ah




--2h-

Figure 2.1: Nomenclature of Elastic Waveguide. Frequency and wave number will
be rescaled relative to h (and c8). The smaller halfwidth will be denoted hz = h.
The only geometric factor in final results will be a, the cross-section aspect ratio.


coordinate system. The long axis of the bar will be considered the x axis to

facilitate easy comparison with publications of historical significance in which this

is the more common convention. The long coordinate will be embedded along the

geometrical center of the bar. This placement of the long axis symmetrically divides

the bar. Accordingly, the bar will be transversely bounded by -hy < y < hy and

-h, < z < hz. For convenience, hz will be consistently taken as the smaller half-










width should hy / h, and h without subscript will refer to this smaller quantity. As

already indicated, a will denote the cross sectional aspect ratio so that hy = ahz.


2.2 Symbolic Consistencies and Adopted Tensor Notation


In addition to foregoing nomenclature, the base symbol for all quantities related

to elastic displacements will be the letter u. Vectors, as opposed to their compo-

nents, will be bolded. When necessary to distinguish shear versus longitudinal

displacement contributions, a parenthesized superscript will be used, as in u()

and u(s). The displacement field will be decomposed into longitudinal and shear

parts generated by a scalar potential p and a vector potential H respectively. The

Greek letter TI will be consistently chosen as a base symbol for longitudinal wave

numbers and a for shear wave numbers so that these associations can be perceived

at first glance.

Einstein notation will be used in tensor equations. Repeated indices will,

unless otherwise noted, imply a summation over {x, y, z} coordinates. Commas

preceding one or more indices will abbreviate derivatives taken with respect to

them. However, the problem will be investigated within a Cartesian metric and

so there will be no distinction between covariant and contravariant vectors and

correspondingly there will be no methodical use of raised versus lowered indices.

Absent the ability to restrict summations to matching upper and lower indices,

summation may be assumed for all matching pairs of indices on any one side of an

equation-subject to contrary commentary.

The symmetric and antisymmetric parts of tensors with respect to any subset of

indices will sometimes be explicitly indicated using the standard notational devices

of enclosing the list of p indices that are symmetrized in parenthesis and any that










are to be antisymmetrized in square brackets.


A ...(j...) [A ...,j... + A ... j... + ...]


A [ij ...[ ] -L [A ... A... j... + .]


Hz,y Hy, = 2H[,y] = -2H[y,,]

The I .I !!y antisymmetric" tensor of any appropriate rank will be indicated using

an c (as opposed to E), as in


[V x H] = CijkHj,k



2.3 Linear Elasticity Theory


The phonons in this study are modeled as elastic vibrations. It is helpful

and important to identify and outline the origin of specific elements of elasticity

theory essential to the derivations developed herein. Besides clarifying notation

and connecting it to standard literature, this will serve the important purpose of

exposing implied and explicit assumptions which underlie my own derivations and

which could limit applicability of the results.

The basic results of the linear theory are straightforward, but elasticity the-

ory has many subtleties and complexities which will not be needed in what fol-

lows. For full development of the topic, monographs and texts range from com-

prehensive classics [43], to relatively recent standard texts which are cited by

most contemporary researchers who utilize an elasticity model of long-wavelength

phonons [37,38]. As a recommended supplement, a succinct development of ele-

mentary linear elasticity theory is provided within the theoretical physics series of










Landau and Lifshitz [42], but some lesser known texts and monographs are more

relevant to the study of elastic waveguides [5,10].

Linear elasticity can be viewed as a generalization of Hooke's Law applied

to a continuum whose elastic response need not be isotropic. Hence, the basic

stress-strain relation law takes the following general form:


oij = Cijk Uk (2.1)


The tensor Cijk is sometimes called the I Li- ;" tensor-in contradistinction to

the "compliance" tensor whose elements are those of the inverse of the stiffness

tensor's matrix representation. It is not uncommon to refer to the stiffness tensor as

the elastic tensor. The tensor Uke is a dimensionless strain defined to encapsulate

deformation in a form invariant under pure rotation while omitting non-linear

terms. This is accomplished by defining it to be the symmetric part of the gradient

of the displacement vector (sometimes denoted Vsu [38]).


[Vu](k,) = U = U(i,j) = 2(uk,e + u,k) (2.2)


The potential energy density then has the following form:


V = 2Uk Cijki utk (2.3)


Since uij is symmetric, directional invariance of V leads ultimately to essential

symmetries of the elastic tensor as follows:


Cijk = Cijk = Cjik = C .,, = Cjilk and Cijk = ,


(2.4)











These intrinsic symmetries substantially reduce the number of unique components

(81, before applying the symmetries) and give rise to an abbreviated notation,

ubiquitous in materials science, for denoting the, at most 21, unique elastic con-

stants which remain. Abbreviated indexing reflects the pattern of symmetries by

denoting pairs with single digits. The strain, stress, and elastic tensor components

are then expressed in terms of the correspondingly reduced number of indices. The

abbreviation scheme is simply


11 1 t 22-2 33 3

23, 32 4 13, 31 5 12, 21 6


This allows equation (2.1) to be expressed as the following two-dimensional matrix-

vector product relation:


91 C11 C12 C13 C14 C15 C16 U1

02 C22 C23 C24 C25 C26 U2

03 C33 C34 C35 C36 .
(2.5)
-4 C44 C45 C46 U4

05 C55 C56 U5

06 C66 U6


where omitted matrix elements are symmetric reflections. Various crystal symme-

tries will reduce further the number of unique elements of Cp,. Although I will

sometimes rely upon and refer to general properties of Cpq, the important novel

results depend upon the material being isotropic. Isotropy can be represented by

imposing rotational invariance on an existing cubic symmetry. The full elastic










matrix for a cubic material (in abbreviated notation) is


(2.6)


which can be rendered rotationally invariant by requiring that


C44 (Cll C12)


(2.7)


The result is that the three components can be parameterized using only two

constants (or moduli), and these can be chosen to be equivalent to the so-called

Lam6 constants which emerge naturally when the theory is derived starting from

an assumption of isotropy (see, e.g., [42]).


A = 12 and p = c44 with Cl = + 2p


(2.8)


For an isotropic material, the stress-strain relationship (2.1) can then be summa-

rized as


cij = Xij E Urr + 2/Uij


(2.9)


Notice that I have not been content to allow Ur, to imply summation because one

of the penalties of avoiding the use of upper indices is that, under the definition

(2.2), ui is proper notation for an individual diagonal component of u. Having

called attention to this problem, it can be mitigated by adopting a convention










that, absent clarifying comments or explicit summation, repeated indices in a rank

two tensor will imply summation (i.e., contraction) only when the usual letters

set aside as coordinate variables are not used. Under this rule, the use of r as a

subscript above in lieu of i, j, or k would have signaled that the ambiguity should

be resolved in favor of implying summation.

The free-body equation of motion within the media follows immediately from

Newton's law and substituting equation (2.1) into (2.3).


p ui,tt = i = aijj (2.10)


If the right side of equation (2.10) is expanded using equations (2.1), (2.2), and

the symmetries identified in (2.4), then the equation of motion can be expressed

in terms of displacements as


P ui,tt = Cijk Uk,j (2.11)


which, in the case of a normal mode (i.e., ei"t time dependency), immediately

yields a form of vector wave equation with possibly non-factorable i -


S/2Ui + Cijk Uk,j = 0 (2.12)


To exhibit the wave picture in the case of isotropy, I can use, instead, equation

(2.9) in expanding equation (2.10). This leads ultimately to a useful coordinate-free

equation of motion

U,tt ( ) V(V ) + V2u (2.13)

This expression is n,- I!i!" in the sense that an assumption that the displacement

field vanishes at infinity permits, via a well-known theorem of vector algebra, u










to be separated into divergenceless and solenoidal parts which, in turn, produces

separated wave equations with distinct velocities as follows:


u() (+ V2U)

u ) V2(V x H) (2.14)


Here I have begun to utilize ip as the scalar potential generating the longitudinal

displacement and H as the vector potential generating the shear displacement. In

general, these potential symbols may themselves represent superpositions.

The foregoing clearly identifies the two isotropic bulk velocities:


c, P/ 44 (2.15)

+ L (2.16)
P P

These can then be substituted back into equation (2.9) to begin utilizing the bulk

velocities as the primary characteristics of an isotropic material.


a = p (c2 2C2) .v. + 2p c2 (2.17)


2.4 Free Boundary Conditions

The problem investigated herein is defined by taking the stress normal to

the surfaces to be zero. I begin to transition toward a dimensionless form of

this situation by first dividing equation (2.17) by pc2. Recall that R has been

designated to be the isotropic material property defined by ce/c,. Then, in the

coordinate system of the problem, the assertion that stress normal to the ith










surface at xi = hxi is zero can be expressed in a form independent of p and c, as


(R2 2)6Jijrr + 2uij = 0 at x, = hx, (2.18)


Since ut,i = ui, the contraction urr in equation (2.18) is just V u. Since the

divergence of a curl is zero, there is no shear contribution to this term. Then, since

the longitudinal displacement is the gradient of the scalar potential 0, I can replace

V-u with V2( Since p is solution of a wave equation, we can finally replace V2s

with -k,2 where k = w/c at the eigenfrequency w.

Finally, since each u = 9 longitudinal strains are succinctly expressible in

terms of the scalar potential in the form u) = lPij.

Separating the shear and longitudinal parts and utilizing the foregoing, the

boundary conditions at either of .,Ii i.ent pairs of parallel surfaces can now be

restated as a set of qualities relating shear and longitudinal contributions. To

emphasize the essential structure independent of chosen surface, I will here use a

surface-oriented notational scheme as follows: Denote with s the coordinate axes

(i.e., y, or z) normal to these surfaces. With s as the starting coordinate, denote

the next cyclic coordinate (right-handed progression) as p and the next after that

as p. In terms of this notation, the boundary conditions for any transverse surface

can be written in the following form:


At s = h8 for all p c [-hp, h,] and all p c [-hp, hp]:


-, k2 + -2u') ( 9)



ys? ()- (2.19)
IOsp tsp











where 3 = R2 2.

The shear displacement vector components in

explicitly:


u(s) Vx


Hz,y

Hx,z

Hy,.


terms of H components are


Hy,

H,,,

H1,y


(2.20)


The shear strain tensors are then:


l(Hz,yy

1(Hz,zz

i(Hz,zy


Hz,xy Hy,zz

Hx,yz Hz,xy

Hy,zz H,iyz

Hy,yz + Hx,xzZ

Hz,zz + H, 1y

Hy,zz + Hy,xx


H ,,xx)

H 11v)


(2.21)


with the order of differentiation having been arranged to highlight patterns and

symmetries.

When tensors from equations (2.21) are substituted into surface manifestations

of equation (2.19), the overall cyclic pattern of the subscripts is fully apparent and

the boundary conditions can be resummarized completely in terms of potentials

as follows:

At s = h, for all p c [-hp, hp] and all p c [-hp, hp]:


-3 kp p + 29p,ss

'P,sp

P,sp


-2(Hp, -



+1 [(H,,pp


Hp,p) ,s

Hp,,ss) + (Hs,s

Hp,ss) + (Hs,s


Hp,p),p]

Hp,p),p]


(2.22)











2.5 Nonseparability of Boundary Solutions


Utilizing equation (2.19), I will directly demonstrate that assuming the exis-

tence of separable potential functions in this scenario leads to an essential contra-

diction. The z-boundary and y-boundary conditions are explicitly



at z = h for all y E [-h, h]:


-03 k2j + 2y


-,zz -u()


(2.23)


and at y = h for all z E [-h, h]:


kj2 o+ 2,yy=-2us

(8)
(P,yz --lyz)

,y -(8)
'PIXY "aY


(2.24)


Suppose that p is separable wave function representable as the product of wave

functions such as p(kxI,' (rqyy ,' (T1yz). Since longitudinal strains are symmetrical

in the derivatives taken to form them from o, separability of p will perpetuate to

the longitudinal strains. The x dependencies of the potentials will be assumed to

be common because that is the basis choice appropriate to propagating modes. I

can thus anticipate that the x-dependencies of each boundary condition will cancel

and will henceforth omit them from the remainder of this proof.

Assume also that H generates a u(s) such that shear strains formed by com-

bining their derivatives can be represented as separable products such as u(=)










,,,,,(yy),," (,-az). Again, since H must be a wave function satisfying the bound-

ary conditions, the separable products making up the strains it generates will

ultimately also be wave equations for each direction.

If the shear components were individually separable, the best possible case

would be that the shear strains which combine derivatives of these, nevertheless

produce individually separable functions. If they do not, the shear strains will

immediately be nonseparable and there will be no point in going further since the

case will have been made. It is sufficient, therefore, to assume this best possible

case.

Substituting these assumptions into equation (2.23) at +hz:


k(P3A; + r2T1 ,' (rTyy) '' (TI ) = -2k, (nayy) (7, h, )

ST) (lh) -,yii (ayy)i (a, h,)

k' U ( ) (9z)z) ( = -h (agy) uyz,(ah,) (2.25)


where the tilde's over functions conveys that they have been differentiated. It will

not matter to the proof precisely what the separable product or its derivative is

because the only facts essential to the result are that they are wave functions with

definite wave numbers fixing their directional dependencies.

For the foregoing equations to be valid, they must each be invariant in y. By

inspection, this will only be true if rq = ay including the possibility that they are

both zero.

Given the cyclic symmetry connecting the boundary conditions at .,.i i:ent

sides, substituting the same assumptions into equation (2.24) will similarly result

in the symmetrical requirement that rI, = z.








32

Separability of the potentials together with their satisfaction of wave equations

implies that




2 + 2 k2 2
+ a + +
Cs

If ry = ay and %, = cr, this will require that c = c,-which is impossible for real

isotropic materials since it has already been pointed out that c > V2 cs. This

calamity can be cured by (a) some superposition of solutions, (b) eliminating one

side by extending it to infinity (the infinite plate scenario as solved by Rayleigh), or

(c) special modal situations involving only shear (9 = 0). Each of these exceptions

will be encountered in the sequel.















CHAPTER 3
THEORETICAL ASPECTS OF RECENT NUMERICAL METHODS


3.1 Stationary Lagrangian


Holland [21], Demarest [23], and Visscher et al. [25] each construct a theoretical

justification for their numerical approximation methods by identifying solutions

with the stationary points of a Lagrangian. More precisely, the starting point is

stated to be Hamilton's principle, but the stationary points of the time-independent

Lagrangian were asserted to be equivalent. Whether or not this is a valid substitu-

tion, however, depends on whether the function space within which the stationary

functions are found is restricted to functions with a separable harmonic time

dependence. (Demarest makes some attempt at varying a more general class of

trial functions, but fails to properly factor the variation in his equation (4).)

This is not an inconsequential suspension of formality. Stationarity of the

Lagrangian within a function space of elastic normal modes follows immediately

from stationarity of the Hamiltonian precisely because the kinetic and potential

functions have the same harmonic time dependency. Integrating any such function

over time produces a factor of 1/2 so that stating variations to be zero is equivalent

to stating that variation of the time-independent part is zero.


2 L 2L = 6J L(r,t)dt -=6L(r)
at2 2


However, if the function space is not in some way restricted to those functions

whose spatial parts are at least consistent with their having harmonic time depen-

dencies, then it is elementary that stationarity of their spatial parts is not a










priori equivalent to the satisfaction of Hamilton's principle. The desired harmonic

dependency will be manifest by satisfaction of a wave equation. After reviewing,

below, the actual Lagrangian variational result, the point to be emphasized will

be that restricting the set of trial functions to those which satisfy wave equations

is indeed the most important prerequisite to validity of the result.

The Lagrangian variational analysis within each of the aforementioned numer-

ical approximation attempts is well represented by that contained in the paper by

Visscher et al. The reasoning is elegant and clear, and so it is helpful to reproduce

it while making relevant observations.

The derivation by Visscher et al. takes place within a closed domain of arbitrary

shape. It therefore is, in essence, a resonance problem. The same reasoning can

be extended to propagating modes by considering such a mode virtually bounded

within a rectangular waveguide by surfaces perpendicular to the sides and sepa-

rated by a phase difference of 27. If any surface integration is performed, reversal

of the outward normals of these virtual surfaces would cancel their contributions

to the overall surface integral insuring the same result as for a physically bounded

region.

Visscher et al. pre-incorporate harmonic time dependence by writing the kinetic

energy density as

P LUiUi (3.1)


They represent the potential energy density as


-CijkeUijUk,e (3.2)


which can be shown to be equivalent to equation (2.3) by expanding the defini-

tions of the strain tensors and exploiting symmetries of the elastic tensor. The










Lagrangian, then, is just the integral over the domain of the difference of these

densities.

L ~= J p Up2 Ui Cijki U, k,) dV (3.3)


I will assume, as do the authors, that the reader is able to complete the exercise

by doing the algebra and applying the divergence theorem to find the conditions

for the first variation of this Lagrangian to be zero. I will point out that their

particular choice of how to represent the potential energy density appears not to

have been accidental since it guides the algebra and makes the process rather more

transparent. The result, as is expected from any Lagrangian minimization, is the

equation of motion. However, the particular form of the surface term which must

be zero is important and so I will write the full statement of the result to include

it.


6L = 0 [= (p2 i- Cijke Uk,j)6ui dV + (cijW Uk,e)6ui rj dSj 0 (3.4)


where n is the outward normal surface vector and nj dSj is the magnitude of the

outward area vector. The parenthesized parts of the integrands are the elastic

wave equation and a representation of stress. These parenthesized terms should

be separately zero when the first variation of the Lagrangian is zero.

The same tensor algebra used to show that equation (2.3) is equivalent to

Visscher's choice of representation for the potential energy density can be reversed

to show that the integrand of the surface term in equation (3.4) can be replaced

with


Cijk Uk











which is just the right hand side of equation (2.1) by which stress is defined. Surface

normal stress, by definition, is just


oij nj


and so the surface term in equation (3.4) is zero if the surface normal stress is zero.

With respect to the wave equation aspect of this condition, the assumption of

harmonic time dependence guaranteed this in advance as can be seen from the

fact that in section 2.3 I derive the same wave equation from that assumption and

the application of Newton's Law. Lagrangian formalism is simply a substitute for

Newton, but leads to the same result.

What is new is the appearance of the stress-free boundary condition as a

prerequisite. There is nothing in section 2.3 that will directly permit me to infer

that modal functions formulated to satisfy the elastic wave equation correspond

to a stationary Lagrangian.

What Visscher et al. conclude from this outcome is that "...the displacement

functions u, which are solutions to the elastic wave equation with free boundary

conditions on S, are just those points in function space at which L is stationary."

I want to emphasize yet again that this result is partly self-fulfilled by starting

with an assumption of harmonic time dependence and that the satisfaction of the

wave equation is actually an implicit starting point rather than a condition on the

outcome.

In view of the foregoing, I draw a slightly different conclusion than that expressed

by Visscher's team nearly a decade ago. What I see in the result is the following

implication: Some superposition of functions which are guaranteed to satisfy

the elastic wave equation will also satisfy free boundary conditions when their

combination makes the Lagrangian stationary.










Having rephrased the conclusion it becomes easier to see how limitations can

arise from the basis choices employ, -1 in actual numerical calcualtion. Visscher et

al., like Demarest before them, choose a basis for computational convenience which

does not in fact consist of functions which satisfy any wave equation. Resonances

which involve a combination of components which are hyperbolic together with

some having low periodicity can be accurately determined. It becomes more

difficult to compute resonances for shapes, such as those having high aspect ratios,

which require more components having extended periodicities. It is harder to

assemble superpositions of waves from a basis devoid of wave solutions than it is

from a basis made up of wave solutions.

In any case the paper continues with a logical inversion. It is claimed that

finding a stationary point in some robust function space is paramount to finding

a function that is a good approximation to a normal mode. If, of course, the basis

is not made up of wave solutions, the stationary point must often simultaneously

produce a wave solution, but failure to constrain the basis means that more is

expected of the process than need be. It can only be said that concern for placing

extra burdens on the process is lost in the fact that resonance problems usually

converge nicely and give results which appear to be accurate for situations of

practical interest [26].

That convergence of some kind does occur in the case of resonance problems is

well documented in the literature cited. Indeed, the process has now been in used

to determine elastic constants for a decade. I have personally observed the process

and discussed it with Dr. Migliori. From his unpublished remarks, however, it is

clear that the numerical process will sometimes fail to converge when attempting to

match the actual eigenspectrum of objects with high aspect ratios against estimates

generated by the numerical process. This difficulty appears not to have been










the subject of published methodical analysis, but I have personally experimented

with rectilinear copper samples having roughly a 1x1x3 aspect and was unable

to find elastic constants using which the numerical algorithm could produce a

resonance spectrum that matched what was directly measurable. This author can

therefore confirm that whatever its success, there are clear indications that the

numerical algorithm premised on the Lagrangian formalism just reviewed is not

reliable outside of some convenient set of situations despite remarkable accuracy

when confined to that set of situations.

My purpose is not to bury the numerical techniques, nor even their theoretical

underpinnings. Rather, I wish to praise them since they are extremely helpful and

useful despite the limitations they operate under. It is important to repeat that

whatever the precise limits of the numerical algorithms turn out to be, the fact is

that they are demonstrably accurate within the domain they are typically deploy, 1

in. Moreover, the somewhat circular logic of the Lagrangian formulation carries

within it the inspiration for a renewed investigation of whether analytical solutions

are possible at least for the rectilinear geometry. Specifically, the Lagrangian

technique highlights that the most promising basis ought to be one made up of

functions which satisfy a consistent wave equation. By "<- i--1I-. i~i wave equation,

I simply mean any whose wave vector magnitude, and thus frequency, matches that

of the normal mode in a propagating problem.

It is interesting that my observations on the limitations of Lagrangian mini-

mization actually vindicates Holland's initial approach in that he purposely chose

products of sines and cosines as basis functions. If Holland had found a way

to implement his numerical approximation by separating his basis to expand the

longitudinal and shear contributions separately, his algorithm would have been an

appropriate numerical analog of the analytic approach which I develop herein.










3.2 Numerical Approximation


Equation (3.3) can be written as a difference of distinct functionals in the form


L = T[u] V[u] (3.5)


I am now using u in the sense of time-independent displacement functions which

inhabit a vector space equipped with the usual inner-product (i.e., vector dot

product). T and V are linear functionals operating on the space of u's and I invoke

a theorem of Functional Analysis (see Ref. [44] 30), usually taken for granted, that

permits me to assume the existence of operators such that I can rewrite (3.5) in

the form

L (ulTlu)- (ulVu) (3.6)

where the elements of a matrix representation of the operators are defined, as

usual, by the results of their operation on a basis for the u's. This meticulous

formality allows me to emphasize that this is a situation in which that basis is not

actually known and that the situation is thus far abstract.

Actual numerical computation is accomplished using a finite and definite basis.

Following Visscher et al., I will expand the displacements and estimate the oper-

ators using basis functions of the form xpyqzr. This is just a computationally

convenient realization of the more general use of products of functions of the

coordinates such that each element is a separable function. These are not, in

general, orthogonal, but they may be chosen to be linearly independent-as is the

case here where simple powers of the coordinates are used. It will follow that a

Rayleigh-Ritz like minimization of their combination is reducible to a generalized

eigenvalue problem. One distinct and useful feature of this particular basis will be

that its elements have definite parity.










Equation (3.2) which will form the basis for computing matrix elements of

the potential energy operator, involves a mixing of different components of the

displacement vector functions. I therefore expand the components of the displace-

ment vector functions in bases of their own-rather than expand the displacement

vectors in a basis of vector functions. Correspondingly, there are multiple (though

perhaps not distinct) basis sets {1 i)} (i = x, y, z) and these result in an expansion

of the form


Z X) (X)
Ux Zk Ck k

Ly J Ek "Ckk



In terms of these finite bases, approximations to the kinetic and potential operators

have the following representations:



T%(%(o) = Vo p 4 ) i dV (3.7)

VA(it(m) / 9 k) c aj 9 dV (sum over j & ) (3.8)

These matrix element definitions are in the same form as used by Visscher.

They are elements of approximate, not exact, operators.

There is no assumption that basis functions individually satisfy the bulk wave

equations of the elastic media, let alone the boundary conditions of the geometry.

There is no reliance upon orthogonality properties of the basis functions in any

geometry whatsoever.

One way to express that the Lagrangian is stationary, is to assert that the deriv-

atives of equation (3.6) with respect to components of u are all zero. When the u's

are discrete, this certainly is equivalent to the following (functional differentiation










will lead to the same result):


U2Tu) Vu) = 0 (3.9)


This is just a statement of a generalized eigenvalue problem for which computer

algorithms are widely available. The numerical algorithm generates a basis to

some limit on the powers of the coordinates, calculates the matrix elements of the

operators by integrating over the sample pursuant to equations (3.7) and (3.8),

and then invokes standard software to calculate the eigenvalues and, if desired the

eigenvectors.

This eigenvalue problem form of a minimization problem is expected to display

a convergence of eigenvalues toward some limit as the order of the operators

increases. It is common experience that this convergence is ah--,v- of a decreasing

nature so that the estimates are ah--,v- upper bounds of the true values. However,

when the basis functions used in the numerical procedures are merely products of

powers of the coordinates and when the physical reality being modeled does not

involve the converging eigenfunctions, or their derivatives, converging uniformally

to zero at the boundaries, the expectation that eigenvalues are upper bounds may

not be provable. A typical proof that a Rayleigh-Ritz procedure will consistently

produce upper bounds depends upon the trial functions themselves vanishing at

the boundaries [45]. In addition, although the numerical basis consists of elements

that are linearly independent, they are not in general orthogonal as is assumed, for

example, in a more elegant approach to a similar theorem [46] proposed by Peierls.

Of course, it may happen that some of the modes are well represented by

functions that vanish at the boundaries. In Ch. 6 I will derive all of the k = 0 modes

of a rectangular waveguide and a subset of these will be uncoupled and defined

explicitly by the boundary zeros of sines and cosines. Powers of the coordinates










certainly converge rapidly to isolated sine and cosine functions-a feature endemic

to numerical computations of their values. If the solutions consist of superpositions

of sines and cosines, however, and if the boundary values are not well-represented,

there are no such regularities to assist the rate of convergence and no vanishing

of the wave functions or their derivatives to converge toward. While it is beyond

the scope of this research to pursue these concerns further, I have attempted to

point out that the choice of a non-orthogonal basis coupled with the fact that

the boundary conditions do not generally involve vanishing of the basis elements,

creates inherent difficulties for the efficacy of prevailing numerical methods. This

strengthens the motivation for developing an analytical solution.


3.3 Partitioning the Problem into Parity Groups


Holland [21] carefully organizes his basis, made up of products of sines and

cosines, into parity groups related to a coordinate system centered in his paral-

lelepiped with axes normal to the faces. Demarest [23], and Visscher et al. [25]

adhere to the same system. The same parity pattern classification system p1 i, a

pivotal role in my own analytical derivations which follow.

The importance of this classification scheme derives from the interplay of

a rectilinear geometry with the symmetry properties of the elastic tensor (see

section 2.3) within the potential energy operator defined in equation (3.8). The

kinetic energy operator can be diagonalized independently, but careful examination

of equation (3.8) shows that the potential energy operator will block diagonalize

where the blocks are defined by parity patterns of any basis whose elements have

definite parity. Among the consequences of this fact is that the eigenvalue problem

stated in equation (3.9) can be divided into a distinct eigenvalue problem for each










block. From an analytical standpoint, however, the real value is that the symmetry

classifications are an organizing principle for the entire analytical approach.

Although the parity classification scheme was first identified in the context of

closed geometries, the potential energy operator does not change form in the case

of a rectangular waveguide and so the classification scheme remains equally valid

and applicable. The only distinction is that the parity along the long direction is

often arbitrary as will be so indicated. When turning a waveguide into a resonator,

the resonator modes symmetric along the former long axis will simply divide into

even and odd groups. For the sake of comprehensiveness, I therefore exhibit the

full parity pattern using the nomenclature of Holland.

Let E and O denote even and odd parity for functions relative to Cartesian

coordinates aligned normal to the surfaces and along the central axis of the wire.

An unspecified, but consistent, parity-as may be associated with the long axis of a

waveguide-is denoted with a P. Parity complementation of an otherwise unspecified

P (as, for example, resulting from differentiation) is denoted by P. Directional

dependencies are implied by juxtaposition in a product-in the order x, y, z.

Symmetry patterns of displacement functions are then limited to only the

families shown in Table 3.1-shown as column vectors of the x, y, z components.

The last two of these (flexural) are a degenerate pair in the case of square

rectangular cross sections, but as pointed out by Nishiguchi [1], they are distinct

for the general rectangular case.

Each of these parity patterns can be generated by a single product representing

the parity pattern of the scalar potential whose gradient produces the longitudinal

part of the displacement functions. The shear part must have a matching parity

pattern and so the pattern of the scalar potential fixes that of the mode generally.

In the case of Dilatation, for example, observe that the gradient of a product











Table 3.1: Mode
Parity Patterns


Families in Rectangular Geometry Defined by Displacement


EE
OE
EO


Dilatation group: D



^i .T, I
Torsion group: { T, t
IS1, I



Flexion of z-x plane: {




Flexion of y-x plane:
(


with parity pattern PEE automatically produces vectors with the parity pattern

of the Dilatation group. Similarly, the gradient of POO will generate the Torsion


group-and so on.


= E
= O


s
F3,



S3,
F2,


P=E
P=0















CHAPTER 4
MATHEMATICAL STRATEGY


4.1 Notation and Function Extension Issues


In section 3.3 I reviewed how, for rectangular geometries, the potential energy

operator block diagonalized to partition the problem into independent solution

families based on parity patterns of the basis. For this to be the case, it is

only necessary that the basis functions individually display a definite parity in

their separable parts. From this, any superposition of basis functions from the

same family will also exhibit the same definite parity pattern even though the

superposition becomes nonseparable.

My approach involves solving the boundary value problem in a way that takes

advantage of the fact that solutions are partitioned by parity family. However, this

does not imply that it is necessary to do a distinct derivation for each such family.

Rather, as much as possible, each derivation will encompass all parity families

in such a way that one algebraic result can be converted into a realization for

each distinct parity family by a straightforward substitution. It thus is a distinct

result that the solutions for each family are shown to be manifestations of a single

theoretical result.

In order to be able to transcend distinctions between parity families for each

derivation, it will be necessary to utilize a specialized notation. The notation

will not violate other conventions of mathematical notation. It will sharpen rather

than obscure important relationships. It will considerably shorten the expression of

individual relationships, dramatically shorten derivations, and preclude redundant










derivations. Results expressed in this notation will unify the manifestations among

parity families.

In its simplest characterization, the basis functions defined below will all be

products of functions that are coincident with sines and cosines inside of the

waveguide. It should be immediately pointed out that extending these functions

to infinity is a tempting, but inadvisable option. It is true that vanishing of

the density beyond the boundary would serve to keep the physical description

realistic even if the displacement per se were to be so extended. However, there

are distracting adverse mathematical consequences of indulging such an extension

that should be avoided. One of these is that the Helmholtz Theorem, which is

crucially relied upon to separate the displacement field into parts generated by a

scalar potential and a vector potential, becomes problematic when the fields do

not vanish. Either they should vanish totally at the boundary, or if extended,

some kind of convergence factor should be inserted. This, however, leads to other

complications. Ultimately, all such complications will be neatly avoided by a

judicious choice of transform, but it remains the case that I will need the freedom

to assume that the displacement fields have a behavior beyond the surfaces which

will not contravene assumptions of the Helmholtz Theorem and at the same time

I wish to avoid having to specify what that behavior is specifically.

Henceforth, the functions multiplied together to form displacement basis func-

tions will themselves be defined as cosine or sine functions only between the

surfaces. With respect to the stipulated coordinate system, these need to have

definite parity, and so they will never have constant offsets to their phase at least

in the transverse directions. It will prove an asset to intuition if I choose to simply










call such functions E and 0 in respect of their being either even or odd.

Scosqq q| < h
E, = (4.1)
undefined ql > h
Ssin q |q O0 = (4.2)
undefined |q| > h

where h is the boundary limit and q is y, z, or some coordinate value (such as h).

To minimize notation, subscripts and superscripts can be used to indicate wave

number and directional dependencies. Distinguishing x, y and z dependencies will

sometimes be inferred by position if there is no risk of ambiguity.


For example, E(qx)E(,y)E(lz) = EEY EE = ExEyEz = EEE.


To implement generality in the derivations, I will need to denote functions of

definite, but unspecified, parity by using a function variable. In general, the letter

P, with appropriate subscripting to distinguish variables, will be used for this

purpose. For example, PyPz could take on the specific function values of 00, EE,

OE, or EO.

Often, the mathematical structure of relationships will depend upon the rela-

tive parity of juxtaposed functions. I will accommodate this by denoting parity

complementation of a function variable by placing a bar over it. For example, PP

could be EO or OE.

Differentiation of E or 0 with respect to coordinate will be the dominant

operation. Because of the sign change introduced by differentiation of a E within

sample boundaries, it is not .i.-- li- correct to assume, for Pi as an abbreviation

of P(qTixi), that Pi,i = riPi. To nevertheless permit differentiations to be unam-

biguously specified at the highest level of abstraction, parity complementation










resulting from differentiation per se will be denoted by placing a tilde over a

function variable. Specifically, E= -0 while 0= E, and it is thus alv--,i correct

to write Pi,i = Ti Pi. Upon eventual substitution of E or O for Pi, the appropriate

sign changes can be made. However, the symmetries of the problem ultimately

result in the cancelation of sign distinctions and there are also cases of successive

differentiation which invoke the parity-invariant rule P= -P.

The derivations will eventually reveal that the sign distinctions inchoate in

notations like P are eliminated in the final results which can invariably be stated

strictly in terms of parity variables and their simple compliments. Solutions specific

to parity families will be realized simply by choosing E vs O assignments for at

most two function variables (viz. Py and Pz in the case of a waveguide with P, being

an additional variable only if the waveguide is capped to become a resonator). In

order for this to be resolved in the derivations, the following additional notational

device will be needed:


x\ x, P,=E
(4.3)
yP, Pi

Some simple examples that illustrate how this can be applied are:



'PY P4 P { (4.4)
-1 t1


Therefore, differentiation of any P or P can be expressed without recourse to the

tilde notation by using



Pi,i = ri Pi Pi,i /i Pi (4.5)
l1 --1
Pi Pi










4.2 Defining the Basis and Superpositions


My key physical strategy is to assume that some superposition of tractable

fundamental basis functions will assemble a tractable non-separable function that

manages to satisfy the boundary conditions. While this is clearly an obvious

approach to the predicament, I have been unable to find examples in the literature

that reveal any attempt to actually apply it to the analytical solution of this

problem. Though the assumption that some kind of superposition is needed can

be inferred from various discussions, a failure to even realize the possibility is

sometimes clearly evident (see, e.g., Ref. [3] IV). I can only speculate that the

unavailability of basis functions which themselves satisfy the boundary conditions

has been viewed as such a serious departure from typicality that it has been more

tempting to conclude unsolvability than to pursue superposition in spite of it.

A more courageous view has been exhibited by researchers looking for theoret-

ical underpinnings of numerical approximation attempts. I therefore give credit

for reinvigorating an analytic pursuit of superpositions to those who developed

numerical approximation methods. These have been reviewed in the prior chapter

in part because they create a framework in which superpositions and notions of

how their elements should be structured take concrete shape. In fairness, it should

thus be -r-.ii. -I. l that the results achieved herein are the result of adapting the

progress made in numerical methods to a revisit of the presumably intractable

analytic problem.

My own approach to formulating the basis functions is straightforward. I accept

that the longitudinal and shear contributions should be expanded in separate bases.

I depart from recent numerical approaches in that I require all basis functions

to at least satisfy the bulk wave equation with respect to longitudinal or shear

expansions. In fairness, it should be pointed out that Holland [21], by using










products of sine and cosine functions as trial functions, approached the problem

similarly, but he did not segregate the trial functions into longitudinal and shear

contributions. Neither did his successors, Demarest and the Visscher team, each of

whom abandoned any efforts to constrain trial functions to those which satisfied a

wave equation. Numerical approaches to date have sought an overall convergence

of displacement arrangements guided by a variational principle and so the trend

has been to impose increasing arbitrariness on the structure of the trial functions.

Analytically, this is fruitless-or worse. The reason is that the trial bases used by

Holland and his successors is not in an analytically "(~.i Il !. 1. representation.

The three key issues surrounding the construction of superpositions are (1)

which fundamental basis functions to use, (2) how to represent the superpositions of

these, and (3) how to transform the equations written in terms of the superpositions

so that they can be solved as a finite set or solved by some recursive process

operating on an infinite set. I have already indicated that I will expand both

longitudinal and shear contributions in basis functions which individually represent

wave solutions. What remains is to specify more concretely both the representa-

tion of these basis elements and the representation of their superpositions. In a

subsequent section of this chapter, I will take up the transformation issue-also

proposing a simple approach which has not appeared heretofore in the literature.

In derivations which follow, fundamental basis elements will .1.-- 'i- take the

form of a separable product in the form PPyPz where the P's (see preceding

section on notation) stand for particular even or odd functions which, within the

boundaries of the sample, are coincident with cosine and sine functions, respec-

tively. The eight parity patterns corresponding to the eight mode families defined

in table 3.1 correspond to the eight possible values of PPyP,. Note, however,

that as long as each component of a superposition has a common parity pattern,










any superposition of them will exhibit the same parity pattern even though it

is itself not a separable function. Also, since I am concerned with propagating

modes characterized by translational invariance, P, will naturally cancel out among

the relations. (For a rectangular resonator, mode families could be generated by

superimposing propagating modes of varying x-dependencies.)

Since the mode family is defined by a parity pattern of the overall displacement,

the parity pattern of the shear components is constrained by the necessity that

the parity pattern of V x H be identical to the parity pattern of V(p. It is easily

checked that for

X P P,P,

this constraint will be satisfied so long as the components of H are made up of

fundamental elements which have parity patterns in terms of these as follows:


H, ~ PPy P,

H, ~ PPy P

H, ~ P PyP,


Fundamental basis elements (either longitudinal or shear) are linearly inde-

pendent, in the intervals defined by the medium, for the obvious reason that

sines and cosines with distinct wave numbers along any coordinate form a linearly

independent set. Any linear combination of the basis elements must satisfy the

bulk wave equation and this will be achieved so long as the elements individually

satisfy "a" wave equation for the same magnitude wave number-which then factors.

This is accomplished by requiring the wave numbers of terms in each basis element

to satisfy the bulk dispersion relations:

k2
k2 + 2 + I = for components of o (4.6)
c2










^2
k2+ a + a -= for components of each Hi (4.7)
Cs

This constraint reduces the degrees of freedom by one. I contemplate a mode with

a fixed k, and choose one transverse direction as representing a degree of freedom

while constraining the second according to equations (4.6, 4.7).

These functions can be treated as orthogonal with respect to each coordinate,

but this will only be exploited obliquely in the sense that this property is deeply

buried in the nature of the transform that will be applied to their superpositions.

A superposition can be modeled as discrete or continuous. On physical grounds,

however, the discrete distribution is the correct one in this case. The physical

reality being modeled consists of elastic waves in bounded media. Although the

boundary conditions will be viewed through the lens of mathematical abstraction,

the reality is that elastic waves refract at the boundaries and any superposition

models the summation process that superimposes all of their reflections. Since

there are a finite set of surfaces, each refraction is a discrete event. The superpo-

sition is thus a discrete summation.

Accordingly, the scalar potential will have the form


o = P1(k x) diPy(li y)P,(rl z) (4.8)


where the superscript denotes a "longitudinal conjugation" defined by


Tr = /2/c,-k2- positive root (4.9)


For the shear superpositions, the expansions take the form


H, = P,(k x) aP,(a y)P (j z)
J










Hy P1(kx)ZbPy(,ajy)P, (fz)

H, P((kx)ZlcPy(ay)P, (, z) (4.10)


where the + superscript denotes a "shear conjugation" defined by


j+ = w/c2-k2-2 positive root (4.11)


Note that, in the foregoing, both li's and ai's can range over real and imaginary

values within the same superposition.

There is a further, important, observation. The foregoing sums contain weighted

terms that are solely products of sines and cosines-at least within their boundary

domains. Since each such term has definite parity with respect to the sign of the

wave number in its arguments, the effect of the sign of the wave number is ahl--,v-

factorable, in the sense of


-1
Pi (-k x) = P}(k xi)
Pi

A series of terms which differ only in the combination of the signs of the wave

numbers in the arguments will ah-- i-< factor into a single term with positive

wave number arguments multiplying a sum and difference of coefficients. The

combination of coefficients can ah--bi- be absorbed into a single coefficient.

The result is that in each of the foregoing sums, I ah--bi-i choose a representation

in which only positive (albeit real or imaginary) wave numbers are summed over,

but for which some of the coefficients may be negative.










4.3 Dimensionless Representations


In section 2.1 the groundwork was laid for reporting the results of derivations

in a dimensionless form. The specific scale factors and notation for this are as

follows:

First, recall that of the rectangular half-widths, the h, will be arbitrarily

notated as the smallest one whenever hy / hz. The appearance of an unsubscripted

h will refer to h,. The dimensionless aspect ratio a will be hy/h,. Of the two

isotropic velocities, c, will used to rescale results and the ratio ce/c8 is, as noted,

denoted R.

Frequency will be rescaled by the rule


Q -2 where 0 = (4.12)
Lo, h


This is a common rescaling in the historical literature of the problem and, in

addition, it is common to report f in units of 7/2.

Consistent with this scheme, wave numbers will be put into compatible dimen-

sionless units by multiplying them by h. There is no absolute rule on the choice of

symbols for longitudinal versus shear wave numbers in dimensionless units, but the

general attempt will be to show dimensionless forms by converting Latin letters

from lower to upper case and choosing distinct Greek symbols to convert existing

Greek symbols to dimensionless form. For example, K = kh, a = a h and 3 = h

are common choices. The bulk wave dispersion relations in dimensionless terms

would then take forms as follows:


a2 + a2 + K2 2

a+ +K2 R2
/3;+/3+^2-^(4.13)










In some derivations, a free wave-number variable, A, will be used. In its dimen-

sionless form, it will be denoted A.

The ease with which dimensionless forms of results can be written from inspec-

tion of dimensional ones will become apparent as examples appear.


4.4 Derivation of Rayleigh-Lamb Equation


In the Introduction, it was pointed out that in 1889 Lord Rayleigh was able

to derive a transcendental "frequency equation" which defines the propagating

modes of an isotropic infinite plate. This equation is now universally referred to as

the "Rayleigh-Lamb Equation" and the dispersion patterns it generates are often

called "Lamb \\ i. Derivations, and introductions designed to promote reader-

completed derivations, appear often in the literature, but most of these expositions

are considerably more cumbersome than the derivation about to be demonstrated

(see Rayleigh [8], Miklowitz [10], and Ch. 10 of Auld [38]). Nevertheless, the

form of this equation is of fundamental importance in the results to follow and

it will be helpful to demonstrate how it can be re-derived succinctly. Besides

producing a needed result, this exercise will provide a clarifying example of the

devised specialized notation at the same time it introduces the basic pattern for

novel derivations which follow.

The physical scenario consists of an isotropic elastic material sandwiched between

infinite planes at z = h. The surfaces are stress free. A plane-wave system

propagates along the x direction. At any x position, there are no variations along

the y directions.

Referring to boundary conditions (2.22), s -- z, p -- x, p -- y:


At z = hz for all y and all x:











-k3 k p + 2 ,z2

PwZ

,OX


-2(Hy, --

i [(H/,y -

-2 [(HYzXX


Hx,y ),z

Hx,zz) + (HZ,z

- Hyz,) + (H,,


(4.14)


However, since Oy -- 0, the foregoing will simplify dramatically. In addition, the

simplest vector potential that will generate a shear wave with no y displacement,

is just:


and so I can set H= H,


0

H H

0

0. The boundary conditions above now collapse to


-3 k2 p + 2o,


-2Hyxz

(Hy,xxzz Hyzz)


(4.15)


Assuming that I will not need superpositions, individual basis elements that pro-

duce potentials able to satisfy the wave equation are (according to my already

reasoned basis characterization above) simply


p = DP,(kx) Py(ry) Pz,(l*z)

H, = AP,(kx) Py(ay) Pz,(*z)


With D and A as unknown constants. But if Oy -- 0, it must be that r = a = 0,

and Py(O) E(O) = 1 and so the potentials can be further simplified to


D P,(kx) Pz(r*z)

A P,(k x) Pz(a+z)


(4.16)


Hy,y),z]

- Hx,x),,]










Substituting equations (4.16) into (4.15) and performing the differentiations at

z = h produces the following simultaneous equations:


D (p/3k + 2(T*)2) P1(k x) P,(T* h)


1
Pz (T* h)
1


2Ak


J+ YPz (,h)
-1
t p


A -(k2 (+)2) P1(kx)P,(~+ h) (4.17)
2


At this point it is trivial to divide one equation into the other to eliminate the

unknown coefficients as well as all of the x-dependencies. The result is


(pk/ + 2(q*)2) P (T* h) 4 ka+ P (a+h)
kl* P,(* h) (k2 (7+)2) P,(7+ h)


(4.18)


It is gratifying to notice that the two conflicting sign contingencies for P, simply

each resolve to an assured minus sign-and the Pz sign contingencies neutralize each

other. This is a pattern that will repeat itself in the more involved derivations.

The result is already in the form of a "frequency equation," but besides some

rearrangement, there is a final analytical step to be performed. Since the wave

systems are made pII ii. "by virtue of rl = a 0, the conjugations are:


(T*)2


L2/c2 k2

L2/c2 k2


(4.19)


Recall that 3 = R2 2 with R = ce/cs. Applying these to the parenthetical on the

left of equation (4.18), it can be restated more usefully.


(k + 2(*) (2 2) + 2 -
(jOk + 2(q/*)2) (R2 -2) R' +2(R' k2)
R2 C 2a(, R2(,


(k2 (+)2)


Dk


2k -
SC 2I2
\ s










The details of such steps are indulged here because they are prototypical of rearrange-

ments that recur in subsequent derivations, but which will not be hereafter pre-

sented in detail. With this particular re-expression substituted, and following some

rearrangement of terms, equation (4.18) can be written in the form


P (* h) P (a h) 4k2 a+
(4.20)
P, (* h) P (a+h) (k2 (+)2)2

Equation (4.20) is, in fact, the Rayleigh-Lamb frequency equation in a repre-

sentation which encapsulates both its alternative forms. The so-called symmet-

ric form follows from setting Pz = E = cos in which case the left hand side

becomes tan(q*h)/tan(a+h). If Pz = O = sin, the left hand side becomes

tan(a7+h)/tan(q*h) and the equation is said to be in its antisymmetric form.

Accordingly, the Rayleigh-Lamb equation is often written in the form

tan(qr*h) 4 k2 a+ -I
tan(7+h) (k2 (7+)2> (4.21)

In this representation, the symmetric and antisymmetric forms correspond to the

exponent on the right being positive or negative respectively. The names given

these forms obviously match the parity of Pz in my specialized notation, but that

is not why they were so-named. If the displacement patterns corresponding to these

equations are mapped, it is readily seen that the symmetric form corresponds to

dilatationss" in which the plate surfaces are either extended or indented together

at each x position. In the antisymmetric case, the sides of the plate are alternately

extending or indenting-giving rise to a ripple effect. Indeed, the l i-in., i,"

solutions are flexural in nature.













Finally, the Rayleigh-Lamb equation

would be

P,() P(a)
P 3) P (a)


in dimensionless form (see section 4.3)


4K2f 3
(K2 a2)2


6.



5.



4.4 4.
o

-rl

a/
r1





a
1.
R2 =3
Pz =E

0.
I I I 1 1 1 1 1 1 1 1 1 I I II I
5i 4i 3i 2i ii 0 1 2 3 4 5 6
K [hkx]

Figure 4.1: Rayleigh-Lamb Dilatational Modes. In the historical development of
the solution, these are referred to as the "symmetric" modes.


The roots of this equation can be mapped as dispersion curves using contour

plotting. There will be a set of branches for P, = E and another for P, = O. A

rearrangement of the equations is necessary to preclude fatal divergences in the

numerical computation involved in the plotting. In terms of the dimensionless

representation, numerical plotting is based on finding the roots of


[(K2 a212 +4 K2 P ) P(a) =0

PI P,


(4.22)


(4.23)














Figure 4.1 shows branches of the dilatational or iii,,. I i c" modes resulting from

substituting P, = E = cos into dimensionless Rayleigh-Lamb equation. Figure

4.2 shows the branches of the flexural or il I-i1in,. i inc" modes resulting from

substituting P = O = sin into the equation.




6. -



5. -



44 4.
0

r1
3.


r1

*% 2.



1.
R2 =3

Pz =0
0.
I I I I I I I I I I I I
6i 5i 4i 3i 2i li 0 1 2 3 4 5 6
K [hkx]

Figure 4.2: Rayleigh-Lamb Flexural Modes. In the historical development of the
solution, these are referred to as the ii:i -iiiii ii." modes.



The Rayleigh-Lamb equation defines coupled modes of the infinite plate, but

there are also a set of uncoupled modes that can propagate in this geometry. Recall

that surface coupling is an interaction between longitudinal and that polarization

of shear waves having displacement components perpendicular to the surface. I can

set p = 0 in the boundary equations (4.14) and contemplate plane waves that only

have displacement parallel to the z = h surfaces. This can be realized simply by













setting H, = Hy= 0. The boundary conditions then reduce simply to


0 Hzy


0 = Hz,,


at z = h


(4.24)


Shear waves with polarization resulting in displacements solely parallel to encoun-

tered surfaces are often called SH waves (for "shear, with displacements horizontal

to the suin ) in contradistinction to SV waves (for -!h. ir, with dispacements

vertical to the sui .. -. ) that are coupled to longitudinal waves at surfaces.


4.



3.
44
0
.4J-
-H
2.

-H
m


0 1 2 3 4
K [hkx]

Figure 4.3: Infinite Plate SH Modes. Shear waves with displacements parallel to
the surfaces, and which vanish there, form an uncoupled propagating system in an
infinite plate. The lowest even and odd subbands are shown together.



Now, Hz, by my chosen basis representation, must have the form


Hz = B P(kx)P,(y) P (a+z)


Even & Odd SH Modes
Independent of R
. . . . . .










Thus, it is easy to see that the nontrivial solutions for this SH system follow simply

by setting the z-derivative of Pz(a+z) at z = h to zero.



P,(ah)= 0 sin(v=22 -K2)= 0 (4.25)
cos(V 2 -K2


I will make comparison to these SH solutions in the sequel. Subbands of this

solutions set are shown in Figure 4.3. They are analagous to torsional modes

of a waveguide whose dominant displacement pattern are also characterized by

displacements parallel to the surfaces.



4.5 How to Transform Superpositions


The re-derivation of the Rayleigh-Lamb equation was an exercise in organizing

the problem into an algebraic form amenable to the simple elimination of unknown

constants. The three independent boundary value equations constitute at most

three constraints. Absent superpositions, the scaling constants for the potentials

will constitute one degree of freedom for the scalar potential and as many additional

degrees of freedom as there are distinct components of the vector potential to

resolve. However, at each surface only one directional coordinate will be fixed and

so there is possibly one additional degree of freedom to be resolved with respect

to the other. The Rayleigh-Lamb scenario reduces enough degrees of freedom

to balance the constraints. Specifically, restricting the scenario to plane waves

eliminates the directional degree of freedom at each surface and reduces the number

of constants from a maximum of four to a manageable two. With the nontrivial

constraints reduced to the same number, a solution follows.

In the rectangular waveguide case, the directional degree of freedom at each sur-

face persists by the fact that plane waves no longer suffice. Barring some fortuity,










the constraint equations will go to six-three for each surface-with no eliminations.

A question arises: how many independent vector potential components must there

be and thus how many degrees of freedom due to them? Two directional degrees

of freedom plus the scalar plus at least one vector potential component makes

the minimum degrees of freedom to be four. If I couple the .,.1] i:ent sides, the

independent constraints may be reduced and if I include additional vector potential

components I can increase the degrees of freedom. But as I have demonstrated

(section 2.5) abstractly, nonseparability compels the introduction of superpositions

and a new problem arises of how to resolve their expansion coefficients. The goal

of devising a transformation is to deal with this latter complication.

Once transforms are applied to a superposition substituted into a rectangular

system, it will be seen that there are elementary Rayleigh-Lamb relationships

between components of the coupled potentials. This result might be anticipated

qualitatively by noticing that many dispersion subbands revealed by numerical

approximations of the rectangular case bear uncanny resemblance to Rayleigh-

Lamb dispersions of a plate. (This was noticed somewhat by Nishiguchi [1] 3).

Given that I have explicitly chosen a representation of the potentials as sums

of products of sines and cosines (over only positive wave numbers), the boundary

conditions are readily visualized as qualities among sums of exponentials. A

Fourier transform is then naturally -,-'1; -1. .1 The domain of the transformed

terms is, however, finite, and so it is necessary to either periodically extend the

functions beyond the boundary or to modify the definition of the functions so that

they vanish or converge towards zero beyond the boundary. A simple periodic

extension would have the desirable consequence of producing delta functions under

a Fourier transform corresponding to each term-but only if the wave numbers

were real and the coordinate was being mapped to real transform variables-or










imaginary and being mapped to imaginary transform variables. However, it is to

be expected that a given potential is represented by a sum containing both real and

imaginary wave numbers with the result that a standard Fourier transform would

diverge. The inescapability of this expectation comes from examining Rayleigh-

Lamb solutions for which the fundamental modes distinctly involve transverse wave

numbers which are pure imaginary. In addition, as anticipated in section 4.1, a

periodic extension would, at least abstractly, render the assumptions supporting

application of the Helmholtz Theorem invalid.

Preservation of the ability to transform into delta functions, despite divergence

of the Fourier transform and other issues, can be secured by a simple expedient.

Instead of the general Fourier transform, I use a -IiIl!!., '1" transform which is

valid only on exponentials (though without regard to whether the wave numbers are

real or imaginary) and which avoids the divergence problem by virtue of the details

of its defined domain. Moreover, since the domain of functions to be transformed

is explicitly constrained by the physical boundaries of the sample, there is nothing

illogical or restrictive in defining the applicable domain for the transform to include

only exponentials defined on the coordinate intervals that measure the sample and

thus there need not be any concern over periodic extension. The transform to be

used then has a simple definition determined by the element mapping


aeV 2a6(7 r) (4.26)


-hq < q < hq (q = x, y, z) 7,i TE R U 9 a E C


The factor of 2 is a convenience to dispose of the factor I in the exponential

representation of the sine and cosine.










It is almost self-evident that the transform is 1-1 between the set of exponentials

and set of delta functions so restricted. Since a can be zero, an additive identity

exists on each side and we have an isomorphism between two groups. Because the

range of r] and 7 is defined to encompass both real and pure imaginary values, the

transform operates without difficulty on any combination of exponentials with real

or imaginary wave numbers.

It is troublesome to write down an integral form of this transform which

smoothly adapts to whether the argument is real versus imaginary and which

limits itself to the coordinate boundaries. Of course, the underlying mechanism is

a trivial Fourier transform. Fortunately, since the domain of functions is strictly

limited and the element mapping from that domain to the transform domain is

clear and unambiguous, the transform can be performed without difficulty. It is

inil i. -I i1- therefore, that the inverse transform can be trivially written down in

an integral form that is not troublesome and which applies adaptively to transforms

involving imaginary delta arguments as well as real ones. One example would be


1 rX=+o \ =+ioo
f (y) {2A f (A) e CAYdA + e AdA
2 VX-oo J-ioo


Of course, this is only valid for f(A) that are produced by the -:iiIlIII. ,I" trans-

form in the first place.

The left or right hand side of any boundary value constraint will involve one or

more sums in the following general form (which omits common factors of P (k x)

which are also subject to differentiation):


Saj f (p, pj) P, (pj qi)P2 (p q2) (4.27)
3


Here, I have generalized the various cases:










ql, q2 stand for distinct coordinates y or z;

P1, P2 are distinct function variables, or derivatives of them: Py or P,;

p will be an r or a for representations of the scalar or component of the vector

potential respectively;

pt is an abbreviation for the conjugate wave number based upon the applicable
velocity: p* = /z2/c-k2-p2 p+ = /2/c2-k2-p2;

f (pj, pt) will be a prefactor resulting from one or more derivatives taken. These

will ahlv--, be a single product or sums of products of pj and/or p>. To

characterize the effect of transforms on the boundary expressions, it will be

sufficient to contemplate f(pj, p1) as being a single such product since a sum

of such terms can be distributed to produce sums of summations.


By summarizing below how the ilphi.d" transform affects boundary value

terms generically defined by equation (4.27), it will be possible to immediately

write down the transforms of the actual boundary conditions without elaboration.

First, with y -- A chosen to make the example concrete, consider the general effect

of the transform on a function variable.


1 1
y{P( y)} 6(A ) + 6(A + ) (4.28)
-i\ -1


I can now write down the transform of equation (4.27) with respect to ql 7.

Note that 7 stands for whichever transform dimension variable is matched to ql.

My convention henceforth will be that y -- A and z -- In the rendering of

transforms, ql and q2 could be either y or z, though alv--iv- distinct in a given case.










While the reader is presumed capable of writing down Fourier transforms of
sines and cosines on his/her own, the usual results are somewhat simplified and
adjusted in this case by virtue of the stipulation that superpositions will alv--
be chosen to involve only positive (albeit possibly imaginary) values of whatever
wave number variable p designates. The result, therefore, is that only one of the
two delta terms survive in each case and the specific form of the result depends
upon the sign of the transform variable in a way that can be neatly summarized.
So, for convenience I list the results in detail:

u-fl { Ej aj f (pj, pj) K(pji qi)Pqp (p q2)}



a f (, 7t) 1 ( ) P (7t q2) for 7 > 0 (4.29)
Pq1


aj f(O, Ot) 1 2 6p) (t q2) for 0 (4.30)
-i p 0


ajf( 7) 6 (171- ) (7q2) for/71<10 (4.31 )
-i -1

Similarly, I can now write down the transform of equation (4.27) with respect to
q2. There are again the same three cases depending upon the sign or 7.

q2-7 { Ej a f (pj, pj) Pq (pj qi)lPq(p q2)}



Sa f (Qt) 1 ( -P ) q 1(7t ) for7>0 (4.32)
{f2


a f(0t,0) 1 2 6(0o-p) P,(0tqi) for7 0 (4.33)
I P2 IP2












Zaj f(t7 7{) } 1 } P p) (7t qi) for 7 < 0 (4.34)
S-i -1 2

Having been meticulous in motivating, then ju -I iii:- and now demonstrating

the effects of this -iiiiphjii' i transform. It can be drastically simplified in practice

with the following observations:

The prior stipulation that expansions will only be over positive, though possibly

including imaginary, wave numbers has allowed the results of transforms in the

context of the problem to be more easily summarized in terms of single delta

functions. One additional fact can now literally trivialize the use of this transform

in practice. Namely, the fact that any given derivation takes place in the context of

a specific parity pattern guarantees that within any derivation the parity pattern

on each side of a boundary equation will be identical. This parity agreement

guarantees that when either coordinate is transformed, the P function transformed

on each side will be the same. That being the case, all of the prefactors which

depend on the parity of P will cancel between the sides in all cases where that P

function is common to all additive terms. In general, this condition nearly alv-

fulfilled. Moreover, with those distinctions gone, an examination of equations

(4.29-4.34) will reveal that replacing 7 with |7| throughout is sufficient to cover all

cases. Therefore, as a practical matter, the only rule that will be needed is just

Replace every Pq(pq q) to be transformed with 6(171 pq).

It will not matter whether pq is a wave number or its conjugate.

It will not matter whether pq is real or imaginary (it is guaranteed to

be positive).

The triviality of this rule is a direct consequence of having imposed a meticulous

series of specific choices. It is neither fortuitous nor could it have been readily

anticipated that it would reduce to this.








69

Once a boundary equation that has had a superposition substituted into it

becomes transformed in this way, it becomes an equality in terms of the transform

variable. Given the behavior of delta functions, each value of the transform variable

will, on each side of the summation, select out either a specific term of the sum,

or be identically zero. This collapses the equality of functions of sums into a

constraint between components of sums from each side. Correlating transforms

over y with those over z is a remaining issue, but how this must be done will be

developed in the derivations which follow.















CHAPTER 5
DERIVING NORMAL MODES OF PROPAGATION


5.1 General Considerations


The full boundary conditions revealed generically by equations (2.22) infer that

all three components of the shear vector potential are mixed together in satisfying

the stress-free surface boundary constraint. If, however, I restrict attention to just

those modes which propagate, it is not immediately clear whether all three vector

components are needed. Some inspiration can be drawn from the Rayleigh-Lamb

derivation of section 4.4. That derivation involves an infinite plate bounded in

the z directions and only requires the Hy component of the vector potential. The

intimation is that bounding also in the y directions might simply invoke the need

for the H, component, but there is no a priori reason to expect to need an H,

component as well.

The foregoing motivates an attempt to find the essential relationship between

H, and the other vector components which participate in satisfying the boundary

conditions along the surface of a waveguide in which normal modes propagate in

the x direction. Unless some constraint can be found that eliminates components or

establishes some dependency among them, there will be more degrees of freedom in

the propagating problem then constraints available to resolve them. I thus proceed

to investigate this relationship among components in a way that is independent of

the boundary conditions per se so as to confidently narrow the approaches used in

solving the boundary problems.










5.2 Acoustic Poynting Vector of a Normal Mode


Propagating modes carry energy. In analogy with electrodynamics, there will

be a vector that indicates both the direction and magnitude of the energy flux. This

vector, the acoustic Poynting vector, describes a physical reality whose invariances

under manipulation of the coordinate system constrain its mathematical form. The

components of H participate in the construction of a mode via antisymmetries

arising from the fact that shear displacement is V x H. Moreover, satisfying the

boundary conditions ostensibly involves mixing up the components of H. This

-i --_ -I I look for constraints on the relationship between components of H that

might be required to maintain invariances of the Poynting vector while preserving

the antisymmetries built into the shear contributions to the mode. Some familiarity

with the acoustic Poynting Vector reveals it to be a complicated object when

expressed in terms of strain and this provokes a curiosity over whether it may

harbor such constraints. It is difficult to articulate further what is, in the end, an

intuition that this might be so. The intuition will ultimately be justified by the

result.

A representation of the acoustic Poynting Vector itself can be readily derived.

The Poynting Vector will be denoted herein as J since P's have been extensively

used for another purpose. It its defined by the property that integrating it over a

surface S produces the energy flux through that surface.


E,t Jinids (5.1)


where n is the surface normal vector. Components of the force density at a point

on the surface are


- ij ni


(5.2)










These are the internal forces which are a response to strain. Though inconsequen-

tial in what follows, the minus sign which appears above propagates to one in the

expression for J where it is likely to seem counterintuitive.

Force density times displacement is just work density and thus the measure

of energy transport per unit volume. Taking the force density as constant over

infinitesimal displacements, the time rate of change of Fjuj is just Fjujt. The result

is that power density is simply the scalar product of force density and velocity-a

result familiar from elementary mechanics. Applying this to equations (5.1) and

(5.2), the expression for flux and Poynting vector components it infers are


Et /(-Jijnj')utdS ? Ji =- ij uj,t (5.3)


I shall be concerned only with invariance of directionality and symmetry prop-

erties and, since only normal modes are relevant, I will dispense with the time

derivative and ignore the minus sign.

For an isotropic material, dividing equation (2.17) by material constants c, and

p expresses the proportionality of stress to strain independent of the material.


ij ~ -36ij 1 u~r + 2uij (5.4)
r

where 3 = R2 2. From the discussions following equation (2.18), I can replace

the invariant sum with V2( = -k2p. rij is symmetric in its indices. Substituting

this result into the representation for J in equations (5.3), I can express the pro-

portionality of the time-independent part of Ji to a function of the undifferentiated

scalar potential, total strain, and displacement.


Ji ~ P3 k2 O Ui 2uj uij


(5.5)










uij separates into shear and longitudinal parts and the longitudinal part is

immediately expressible in terms of the scalar potential using


4j)= ,ij (5.6)


whereas applying definition (2.2) to u?) and replacing the displacements thereof

by curls of the shear vector potential requires the more involved substitution


Uij = 2(ia33,arj + EjcrH33,ci) (5.7)


Correspondingly, components of the remaining displacement term can be expanded

in terms of potentials by

Uj = CjabHb,a + (~,j (5.8)

With the foregoing substitutions, the ujuij on the right side of equation (5.5)

expands to


UjUij = (CejabCiapHb,aH/,ac + CjabCja/pHb,aH3,ai)

+-(CiabHb,aj + CjabHb,ai)L,j

+CjabHb,a'P,ij + (,j(,ij (5.9)


After a tedious amount of tensor algebra, the first term on the right of equation

(5.9) reduces to

2H HI ,, + 2H[a,b]Ha,bi (5.10)

The remaining terms do not simplify in useful v--, at this level of expression.

Collecting the foregoing results, equation (5.5) can now be fully expressed in terms










of the scalar potential and components of the vector potential by


Ji ~ (P/kf)(CiabHb,a + P,i)

-4(H H ,, + H[a,b]Ha,b)

-(CiabHb,aj + EjabHb,ai)P,j

-2CjabHb,aiP,ij 2o,j' ,ij (5.11)


The actual physical direction of energy propagation should be invariant under

exchange of the transverse coordinates and so, using equation (5.11), I show that

in order for this to be true, the y and z components of H cannot be allowed to

mix with the x component.

Specifically, since the Poynting vector is composed, in part, of curls of H, any

exchange of y and z will interact with the handedness of the coordinate system to

require an appropriate antisymmetry. To be consistent with this, the wave number

in the x direction must also be taken to change sign with any y +- z interchange

and so all derivatives with respect to x will also be required to change sign. In view

of the translational invariance of the modes, this is simply a reflection of the need

to change k, the common wave number in the x direction, to -k to accommodate

inversion of the x direction.

The full expansion of the Poynting vector reveals a complex mixing of x versus

y, z components of H, but I shall show that failure to separate solutions to avoid

this mixing results in failures in the y +- z symmetries of some of the terms and

therefore implies that solutions must be built distinctly from cases in which H1 = 0

versus those in which, alternatively Hy = Hz = 0. It should also be noted that this

will turn out to be consistent with an analysis of the k = 0 case to follow. In the

k = 0 case all derivatives with respect to x vanish and the boundary conditions










naturally take a form which reflects a decoupling of the x versus y, z components

of H.

The first additive term on the right hand side of equation (5.11) contributes a

term to the x component of the Poynting vector that is proportionate to


6xabHb,a + ,x = 2H[z,y] + ,x (5.12)


which is fully antisymmetric under y <- z once we incorporate the rule that p,9 -

-yp,x. Moreover, this antisymmetry is preserved even if either one of the terms

vanishes.

The second term on the right hand side of equation (5.11) contributes a term

to the x component of the Poynting vector equal to


-4 H [.,,(y](H.,y],. + H[yz],z)

+H[x,z](H[x,,],x + H[zy],v)

+2H[y,z]H[y,],x,} (5.13)


and before sign changes due to differentiation by x, this expression is totally

symmetric under y z. Expansion of the antisymmetric parts entails production

of the pair

... Hy,H,wx... H,, H,zx ~... (5.14)

and thus does not uniformally assemble odd numbers of x differentiations with

symmetric yz terms and so the needed antisymmetry is not fully realized unless

some terms vanish. If H, is set to zero, then it is easily checked that antisymmetry

will be realized, to wit:


[Hy,xHyx Hy,H[],z + H,,Hx H ,,H[,,y],y + 8H[y, ]H[y ,]] (5.15)










Similarly, if H, does not vanish, but Hy and Hz vanish simultaneously, then,

again, the resulting expansion will become antisymmetric under y <- z once the

x-derivatives are considered, to wit:


[H,,yH,,yx + Hx,zHx,zx] (5.16)


The third term on the right of equation (5.11) contributes the following term

to the x component of the Poynting vector:


-2 [H[z,y],xu,l + H[z,y],xJ,y + H[z,y],~',z

+H[z,y],x45,x + (H[x,z],\,x,y + H[1,y],15, z)] (5.17)


Under y + z this is totally antisymmetric when the derivatives of x are consid-

ered. Moreover, if Hx vanishes or, alternatively Hy and Hz vanish together, the

antisymmetry of the result is preserved.

Finally, the last two terms on the right of equation (5.11) contribute the

following terms to the x component of the Poynting vector:


-4 [H ],,i + (H[x,,]p,xy H, .,xz)]

-2 [p,xp,,x + (pP,yP,xy + P,zPz)] (5.18)


It is easily checked that the desired antisymmetry is preserved and that, again, the

vanishing either of Hx alone or Hy and Hz together does not change this result.

The conclusion is that, when assembling a propagating normal mode, the shear

contribution must be made out of components for which all the Hx's are zero, or

for which all the Hy and Hz parts are zero. In considering how to represent the

shear superpositions of a propagating normal mode, there is no case in which a











superposition for H, will be mixed with ones for Hy and H,. This removes at least

one degree of freedom from the problem.



5.3 Propagating Modes Involving Hy, H, Shear


5.3.1 Deriving the Frequency Equations

The ease with which the Rayleigh-Lamb solution is derived inspires a deriva-

tion that follows the same pattern. Armed with the conclusion that H, cannot

even be accommodated in a normal mode solution that also includes Hy and H,

components, I proceed to derive the spectrum of propagating modes with H, = 0.

Accordingly, from equations (2.22), the boundary conditions at z = h, with

S- z, p -- x, p -+ y, and Hx = 0 become


3 k2j p + 2,p,,


I,zy


- [(Hy,zz Hy,zz) + Hz,zy]

+ (Hz,zz Hyy)


(5.19)


As a reminder, under the basis rules devised for this problem (see section 4.2), the

representations of potentials will have the following forms:


S= Px(k x) di Py(li y) P,(T* z) with T1*
i


0

H = P,(k x) E, aP,( y-)P(a z) with

Ej bjP,(y7 y)P(aj+ z)


2/ _2 ck2 i 2


o( = w2/c2 2 _- 2
a+ S


2Hy,x










Substituting into the first boundary condition of equations (5.19) I obtain


di (Ok} + 2(l)2)P (riy) P,(Tl h,)


2k {+1 P,(Gj y) Pz (j h,) (5.20)
-1 3 -1


The -iip!.-" transform devised in section 4.5 is then applied so that Py(rTiy)

6(IA\ ,y) and Py(7j y) -) 6(IA oj). By choosing a value Ao of the transform

variable A such that Ao E {(r} i {nj}, the sums on both sides collapse leaving the
following equality:


do (Ok2 + 2(A )2) P(A h ) 2k { o oA Pz(A, h,) (5.21)
-1 I-1


This provides one constraint on possible combinations of (w, Ao) at a given value
of k. Here and in subsequent steps, Ao =- Ao .

Repeating this process for the second boundary condition in equations (5.19),
the transformed version of the second constraint becomes


do k A: Pz(A h,)
P P

-P,(Af h) -ao(k2 (A 2) bo Ao(A) (5.22)
-1 1
Py P_









Substituting into the third boundary condition in equations (5.19), the trans-
formed result is


do -AoA 1 P}(A h)


-k } bo A -ao Ao P (A}+ h,) (5.23)
2 -1 1 1
P. Pz Py

Equations (5.22) and (5.23) can be reconciled into one constraint by finding a
relationship between ao and bo that renders them equivalent. Dividing the two
equations will eliminate the transcendental terms and some common factors, leav-
ing

ao ({ k2 (+A)2) + bo A, A+

-k--2 P (5.24)

bo A+ ao Ao


Solving this for the required relationship between coefficients yields


bo -o A[2 (5.25)



Substitution of equation (5.25) into either equation (5.22) or (5.23) to eliminate bo
will produce the following result:


( 1 1 F(2 A2) A )2
do \A: P,(A h,) -a, k ( 2 P (A+ h,) (5.26)
1 2 k2 2 A 0
SP.










Having reduced the system to two equations with two unknown coefficients, those

coefficients can be eliminated by dividing equation (5.26) into equation (5.21).

First, however, inspired by a similar step in the Rayleigh-Lamb derivation, the

parenthetical on the left side of equation (5.21) can be manipulated and found to

have a convenient equivalent representation which coincides with a subexpression

in equation (5.26).


(/k + 2(A )2) -((k2 A)_ (A+)2)


Substituting this and then performing the division followed by the usual rearrange-

ment produces the following frequency equation:

P,(A* h) P,(A+ h,) -4 AA+ (k2 + A2)
(5.27)
P, (A: h) P (At h) [(k2 + A) (A]+)]2

Comparing with equation (4.20) this is seen to be a Rayleigh-Lamb equation with

the quantity k2 + A 2 1 i ving the role of the magnitude of the propagation vector.

This equation is parameterized by Ao and is valid, at a given k, only for certain

combinations of Ao and u. Consider, nevertheless, what happens when k > Ao (or,

effectively Ao -- 0 for finite k).


A* = /2/c2-k2- --A k* = k2 = (in equation (4.21))

A, = /w2/c~-k2-A2 k =i k2 = a (in equation (4.21))

(5.28)


Therefore, at large k, it is expected that equation (5.27) becomes more like the

classic Rayleigh-Lamb equation (4.21) for plane waves in an infinite plate. At this

intermediate point, the constraint of boundary conditions at the y = hy surface










has not yet been imposed, but the implication nevertheless is that for large k

the system looks substantially like a Rayleigh-Lamb system. This implies that

as k > 0 the effect from the sides tend to decouple and the propagation looks

increasingly like simple decoupled Rayleigh-Lamb plane-wave propagation.

It now becomes important to focus on a simple observation. The foregoing step

produced an equation in terms of the transform variable A whose permissible values

(denoted Ao) come from the set of values in {qi} and {jrj} (what mathematicians

would call the "support" for p and Hy, H,). There could be more elements of

those support sets than possible values of Ao and these additional values may be

uniquely connected to the transform variable p applied to boundary conditions at

the .,.i i:ent surface. However, connecting the y = hy and z = hz surfaces will

be possible only to the extent that some of the values which the two transform

variables take on are indeed shared selections from {T1} and {crj}.

From equations (2.22), the boundary conditions at y = hy with s -- y, p -- z,

p -- x, and H1 = 0 will be


-3 k2 P + 29,yy = 2H ,y

,yx i [(Hzxx Hzvy) + Hy,H z

Izy -(Hy, Hz,zz) (5.29)


The transformed first boundary condition from equations (5.29), before appli-

cation of the delta functions, will be


ZdQ 2k + 2T/)P(/i ah,) 6( ) =


2k E-j Py (j ah,) (af+ -/) (5.30)
1 3 -1










To link the .,11i i:ent surface conditions, I rely upon the fact that each root of

equation (5.27) necessarily corresponds to the existence of specific elements of the

support sets {T1} and {oaj}. In fact, if Ao is, with some value of w, a root of equation

(5.27), it is solely because 3Iro E {ql} and 3o0 E {aj} such that Ao = ro = ao.

Now, the left sum in equation (5.30) a will necessarily encounter Ao = To.

Suppose, then, that I contemplate the value of p corresponding to To = Ao for

which the delta function on the left of equation (5.30) is nonzero. Obviously, it will

be A*. If, for that value of p, the right hand side of equations (5.30) is not trivially

zero for the same value of p, there must exist some ao such that a = = A .

In the alternative, I could first contemplate a value of p for which the argument

of the delta function on the right of equation (5.30) is zero. Obviously, it will be

A+. If, for that value of p, the left hand side of equations (5.30) is not trivially

zero for the same value of p, there must exist some fll such that Tl = p = A+.

The upshot of this reasoning is that I can connect the transforms of the two

sets of boundary conditions by contemplating simultaneous (w, Ao) roots of both

of them. To write the transformed boundary conditions for the .,1i ,i:ent side in

terms of Ao, I require either that

p --> A


so that, by operation the delta function, the wave number variables take on values:


rli Tlo Ao

di do which will be eliminated



aoj o- i.e. presumed to exist

ai al unknown, but to be eliminated










J+ A

S- (A)+ (5.31)


or I require that

P--> A+

so that, by operation the delta function, the wave number variables alternatively

take on values:


Jj g o = Ao

bj bo which will be eliminated

+
cry -- Ao

Tli ll i.e. presumed to exist

ai al unknown, but to be eliminated



i (A)* (5.32)


I will name the first of these alternatives (i.e., equations (5.31 )) "L-Conjugation"

since it is premised on equating the longitudinal conjugation of the z-surface

solution with the y-surface solution. The second alternative (i.e., equations (5.32))

"S-Conjugation" since it is premised on equating the shear conjugation of the

z-surface solution with the y-surface solution.

Because I have stipulated that all members of the support sets are positive

and because only positive square roots are used, all of the preceding mappings are

guaranteed to be unambiguous. The reader is invited to review the definitions of

conjugation denoted with and + superscripts which were introduced in connec-

tion with equations (4.8) and (4.10). From those definitions, it can be noted that









these conjugations have the property


(A*)- A (A+)+ A


A detailed expansion of the final relation in equations (5.31) is


(A)+ (A*)2 \A+2 A2 ()2 (5.33)


and of the final relation in equations (5.32) is


(A) -) -2 (A+)2 )2 + ( 2 (5.34)


These expansions illustrate the general rule that L-Conjugation and S-Conju-
gation solutions are related by straightforward substitutions of variables. It will
thus be sufficient to complete details of the ongoing derivation for the L-Conjugate
case and then state the analogous results for the S-Conjugate case. Applying
L-Conjugation to equation (5.30), and after collapsing the sums, the result is


-do (pk + 2A ) P,(Aoah,) = 2k b1 (A*)+ P1[(A)+ ahz] (5.35)


Similarly, the second and third boundary conditions (5.29) under L-Conjugation
can eventually be put into the following forms:



1 -1
01k \ \\ }Py(Aoah )


iP,[(A/)+ ah] -bi[k2 ((A )+)2]+ a1 A (A)+ (5.36)
1 -1
2 ~L\/o \'\










do0AAo{ }{ Py(Aoahz)
1t 1
[y bl P

Sk at (A *) b1 A:* P[(A*)+ ahz] (5.37)
-1 1 1


Dividing equations (5.36) and (5.37) to put one al in terms of b1, I obtain

-1 -1 t (A)+Ao*
a = bi )(5.38)
k2 + 2
ht t 0iM^

It may be noted that equation (5.38) is not identical to (5.25). This highlights the
fact that al and bl are expected to be distinct from ao and bo. If equation (5.38)
is substituted into either equation (5.36) or equation (5.36) to eliminate al, the
result is identical, to wit:

-1} 1 t (k2 + (A*)2) ((A*)+)2
do Ao Py(Aoaho ) b k k2 Pm[(A) ah (5.39)
12 k 2 + (A* )20
P.

In what is by now a ritual, the parenthetical on the left of equation (5.35) can
be manipulated to show its relation to the numerator in the fraction on the right
hand side of (5.39).


(p3k + 2A ) = -[(k 2 + (A)2) ((A )2] (5.40)


Having reduced the unknown coefficients to only do and bl, equation (5.39) can be
divided into equation (5.35) to eliminate them and produce the frequency equation










associated with the y = hy side.


Py(Ao ah,) P,[(A:)+ ah,] -4 A, (A)+ ( (k2 + (A)2)
(5.4t)
Py(Ao ah,) Py[(A\)+ ah,] [(k2 + (A)2) ((+A)22 (2

Once again, this is a Rayleigh-Lamb equation. Here, the quantity (k2 + (A2)2)

1p i'l the role of the square of the magnitude of propagation.

Equation (5.41) is manifested in the classic Rayleigh-Lamb structure which

shows the consistency of the result with preceding derivations. However, in this

case, that consistent manifestation masks an interesting feature. Namely, all k

dependency in equation (5.41) cancels internally. To see this, expand the terms

which ostensibly appear to show a k dependency and observe that k is eliminated.


S 2 [2 2 k2 A] ( A
( 2 2 c2 2 C2 R 2


(k + ) k2 + k2 a A2

Equations (5.27) and (5.41) define (w, Ao) root systems for independent equa-

tions. The coincidences of these simultaneous constraints define values of w that

constitute the eigenspectrum with respect to the L-Conjugation case. There is a

set of common roots for each distinct value of k. Plotting the k versus w dispersions

requires methodical plotting at different values of k, but, in principle, the spectrum

(for L-Conjugation) has been analytically and precisely specified.

The entire derivation is also invariant under y z exchange-though this might

not be immediately apparent. The key to realizing that it must be so is to realize

that relabeling the directions must also be applied to the representations of the

potentials as superpositions and that with directions relabeled the aspect ratio is










defined by h, = ahy. The reader can easily verify that the results of derivation

will be equivalent to the foregoing.

Repeating the same sequence of steps from equation (5.30) to the present

point, using S-Conjugation defined by equations (5.32), the S-Conjugate analog

to equation (5.41) is


P,((A )* ah ) P [A, ah -4 Ao (A,+)* (k2 + (A+)2)
P,((A)* ah,) Py[Ao ah] [(k2 + ()2) ]2


5.3.2 Interpretation

Had anyone been insightful enough to anticipate that coupled modes of a

rectangular waveguide could be characterized by a coincidence of Rayleigh-Lamb

solutions, proving that it was so would have remained as elusive as history shows

the main problem to have been. Moreover, there are aspects of the result which,

had it been somehow forseen as possible, would have argued against believing it.

The main impediment would have been that there is an intrinsic interference built

into the result which precludes boundary satisfaction at both surfaces without

the rest of the superposition. Although the elegant-looking result involves a

coincidence with the full superposition-it is still not the full superposition. What

is intriguing is that I do not need to have a full description of the superposition in

order to find the eigenspectrum.

The derivation tells us that any superposition of shear and longitudinal compo-

nents that satisfy boundary conditions at .,.i i,:ent sides must include a particular

combination of components that, in a partial sense, mimic a Rayleigh-Lamb wave

system. The L-Conjugation and S-Conjugation cases are merely two different v-i-,

of realizing this. Figures 5.1 and 5.2 illustrate the essence of these alternatives.

They imply two recipes for building the superpositions.









88


z=h

r7 =r 17



long shear


..+
II
r +
C b
II
S on,= (7*)+



shear L-Conj


II





Figure 5.1: Illustrating the L-Conjugation Case for Modal Solutions. Shown are the
relationships among longitudinal and shear wave vector components of the defining
physical waves which must be among those making up the total superposition
needed for a solution. The common x directional component (k) is normal to the
page.


The recipe implied by L-Conjugation begins with a longitudinal wave at a

desired k value. Conceptually, one could imagine starting out with some r] and

some u close to a modal solution. Add a shear wave with the same fixed k.

The polarization of this initial shear contribution is such that shear displacement

is not parallel to the sides. Now adjust u until the Rayleigh-Lamb equation is

satisfied with respect to the z = h, sides. There will be a range of o's for which

Rayleigh-Lamb can be satisfied at these parallel sides. For each possible U the bulk

dispersion relations will fix if* and a+ wave vector components pointing along the

z directions. Now add a second ("conj- i, i. ) shear wave at the same k and close









89


z= h

cr I rj = cr


long
shear *
b
II

+
b


II

7conj= ("+)* S-
S-Conj

long

b
II
11F
0




Figure 5.2: Illustrating the S-Conjugation Case for Modal Solutions. Shown are the
relationships among longitudinal and shear wave vector components of the defining
physical waves which must be among those making up the total superposition
needed for a solution. The common x directional component (k) is normal to the
page.


to the w implied by the process so far. The polarization of this shear wave should

also result in a displacement not parallel to the sides. The initial a+cj is set equal

to *. Just as the first shear wave had a wave vector component common with the

foundation longitudinal wave along the y direction, this conjugate shear wave has

a wave vector component common with the foundation longitudinal wave along the

z direction. Bulk dispersion will fix the value of ocomj. Now adjust w over the range

of values that continue to satisfy Rayleigh-Lamb at the z = h, surfaces until one

is found for which Rayleigh-Lamb is also satisfied for the foundation longitudinal

wave and the just-added conjugate shear wave at the y = hy surfaces. When










such an u is found, it defines an eigenfrequency of a propagating system at the set

value of k. There will be as irn Ii such o/s as there are subbands.

The recipe implied by S-Conjugation is procedurally the same as the recipe

for L-Conjugation except that the foundation wave is a shear wave instead of a

longitudinal one and two longitudinal ones are added instead of two shear waves.

The recipes each focus on combining waves in total disregard of their reflections

at .Ii] ,' ent surfaces. Satisfaction of Rayleigh-Lamb incorporates reflections only at

parallel surfaces. The reflections from .ili ,i'ent surfaces being ignored constitute

the rest of the superpositions. What the derivation shows is that to find the

eigenspectrum, one can ignore these reflections. To build a complete description of

the wave function, however, these reflections must be included. Incorporating the

ignored reflections is not trivial. There are two principle complications. The first

is that shear and longitudinal waves scatter into each other upon reflection and the

scattering amplitude ratios are non trivial transcendental relations even when the

surface environment is free of other interactions-which they are not. Secondly, from

plotting the roots of the equation (5.27) paired alternately with equation (5.41) or

equation (5.42) there appear to be cases where, for a given eigenfrequency, there

are multiple root coincidences and it is not clear whether each is an independent

foundation from which the reflections can be taken as El i i, i lii.- or whether the

multiple root coincidences are, in some sense, resonances of each other and building

reflections from only one of them is sufficient.

Since the amplitude of reflection into either shear or longitudinal is bounded

by unity, successive reflections must progressively dissipate in amplitude and the

implication is that each foundation set constitutes a first-order characterization of

the entire wave function.










In spite of the practical difficulties involved in building a complete description

of the wave function that fully describes the superposition, the implication of the

derivation remains a strong one. Namely, no matter how complicated the details

of the wave function are, the dispersion is precisely defined by the behavior of only

one overlapping set of components which must exist as a dominant part of the full

superposition.


5.3.3 Mode Dispersions

The most important physical feature which can now be exhibited are the

mode bands and particularly the dispersions. As the terms L-Conjugation and

S-Conjugation will now appear more frequently and together, the contractions

"L-Conj" and "S-Conj" will begin to be used routinely.

Equation (5.27) paired alternately with equation (5.41) or equation (5.42)

defines simultaneous transcendental relationships which determine the propagat-

ing normal modes of an elastic isotropic rectangular waveguide. Extracting the

actual subband dispersions is accomplished by substituting successive K values

into a dimensionless form of equation (5.27), contour plotting its root system, and

superimposing that root system on top of the root system plotted for equation

(5.41) or (5.42). It is of some help that the latter equations are K-independent

and need only be plotted once in dimensionless form.

The subbands resulting from L-Conjugation (viz. equations (5.27) and (5.41)

combined) have a strong similarity to the standard Rayleigh-Lamb bands. This

can be seen by methodically plotting the lowest subbands over a range of K values

and superimposing them on the lowest subbands of the standard Rayleigh-Lamb

solution. This is shown for dilatational modes in Fig. 5.3 where the dispersion for

the first three subbands of these propagating waveguide modes are shown against
















3. -
4.












PY = E R2 = 3
-PZ = E
0
0 1 2 3 4
K [hkx]

Figure 5.3: Plotted K-Dispersion for Propagating Dilatational Modes L-
Conjugate Case. Heavy lines correspond to waveguide modes (solid lines are
square cross section, dashed lines are 1x2 cross section) and thin reference lines
are Rayleigh-Lamb infinite plate "symmetric" solutions.


the background of the first three Rayleigh-Lamb branches. Waveguide modes for

the square cross section (solid lines) are shown together with modes corresponding

to a 1:2 cross-sectional aspect ratio (dashed lines). (L-Conj and S-Conj dilatational

modes in combination are compared with numerical mode computations in Fig.

6.4.)

The dilatational subbands resulting from S-Conjugation are not intrinsically

similar to infinite plate modes of the same parity pattern. Figure 5.4 shows the

dispersion of the lower S-Conj subbands for the dilatational family against the

background of the first three Rayleigh-Lamb branches. Waveguide modes for the

square cross section (solid lines) are shown together with modes corresponding to

a 1:2 cross-sectional aspect ratio (dashed lines).











4.



'44















P =E
3.














0

0 1 2 3 4
K [hkx]

Figure 5.4: Plotted K-Dispersion for Propagating Dilatational Modes -S-
Conjugate Case. Heavy lines correspond to waveguide modes (solid lines are
square cross section, dashed lines are 1x2 cross section) and thin reference lines
are Rayleigh-Lamb infinite plate "symmetric" solutions. For combined L-Conj and
S-Conj dilatational modes compared to numerical results, please see Fig. 6.4.


Figure 5.5 shows the dispersion for L-Conjugated flexural solutions against a

background of Rayleigh-Lamb flexural modes. The dashed lines correspond to a 1:2

cross-sectional aspect ratio where the flexing motion of the fundamental modes is

of the same plane (y x) in both the Rayleigh-Lamb infinite plate and rectangular

waveguide scenarios.

Although no torsion of an infinite plate is possible, the displacement pattern of

low-order torsion modes (viz. dominantly parallel to the surfaces) is analogous to

the uncoupled SH modes (viz. also parallel to the surfaces, but vanishing there)

of the infinite plate derived at the end of section 4.4. Figure 5.6 shows torsional

modes of a square waveguide against a background of infinite plate SH modes.