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外文翻译--平面波.doc

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外文翻译--平面波.doc

外文部分Chapter2Planewaves2.1IntroductionInthischapterwepresentthefoundationsofFourieracousticsplanewaveexpansions.Thismaterialispresentedindepthtoprovideafirmfoundationfortherestofthebook,introducingconceptslikewavenumberspaceandtheextrapolationofwavefieldsfromonesurfacetoanother.FouriesacousticsisusedtoderivesomefamoustoolsfortheradiationfromplanarsourcestheRayleighintegrals,theEwaldsphereconstructionoffarfieldradiation,thefirstproducttheoremforarrays,vibratingplateradiation,andradiationclassificationtheory.Finally,anewtoolcalledsupersonicintensityisintroducedwhichisusefulinlocatingacousticsourcesonvibratingstructures.Webeginthechapterwithareviewofsomefundamentalsthewaveequation,Eulersequation,andtheconceptofacousticintensity.2.2TheWaveEquationandEulersEquationLetpx,y,z,tbeaninfinitesimalvariationofacousticpressurefromitsequilibriumvaluewhichsatisfiestheacousticwaveequation222210ppct2.1forahomogeneousfluidwithnoviscosity.cisaconstantandreferstothespeedofsoundinthemedium.At020Cc343m/sinairandc1481m/sinwater.TherighthandsideofEq.2.1indicatesthattherearenosourcesinthevolumeinwhichtheequationisvalid.InCartesiancoordinates2222222xyzAsecondequationwhichwillbeusedthroughoutthisbookiscalledEulersequation,0vpt2.2WherevGreekletterupsilonrepresentsthevelocityvectorwithcomponentsu,v,wvuivjwk2.3whereijandkaretheunitvectorsinthethex,y,andzdirections,respectively,andthegradientintermsoftheunitvectorsasijkxyz2.4WeusetheconventionofadotoveradisplacementsquantitytoindicatevelocityasisdoneinJungerandFeit.Thedisplacementsinthethreecoordinatedirectionsaregivenbyu,v,andw.ThederivationofEq.2.2isusefulindevelopingsomeunderstandingofthephysicalmeaningofpandv.Letusproceedinthisdirection.Figure2.1InfinitesimalvolumeelementtoillustrateEulersequationFigure2.1showsaninfinitesimalvolumeelementoffluidxyz,withthexaxisasshown.Allsixfacesexperienceforcesduetothepressurepinthefluid.Itisimportanttorealizethatpressureisascalarquantity.Thereisnodirectionassociatedwithit.Ithasunitsofforceperunitarea,2/NmorPascals.Thefollowingistheconventionforpressure,P﹥0→CompressionP﹤0→RarefactionAtaspecificpointinafluid.apositivepressureindicatesthataninfinitesimalvolumesurroundingthepointisundercompression,andforcesareexertedoutwardfromthisvolume.ItfollowsthatifthepressureattheleftfaceofthecubeinFig.2.1ispositive,thenaforcewillbeexertedinthepositivexdirectionofmagnitudepx,y,zyz.Thepressureattheoppositefacepxx,y,zisexertedinthenegativexdirection.Weexpandpxx,y,zinaTaylorseriestofirstorder,asshowninthefigure.Notethattheforcearrowsindicatethedirectionofforceforpositivepressure.Giventhedirectionsofforceshown,thetotalforceexertedonthevolumeinthexdirectionis,,,,ppxyzpxxyzyzxyzxNowweinvokeNewtonsequation,fmamut,wherefistheforce,0mxyzand0isthefluiddensity,yielding0uptxCarryingoutthesameanalysisintheyandzdirectionsyieldsthefollowingtwoequations0uptyand0uptzWecombinetheabovethreeequationsintooneusingvectorsyieldingEq2.2above,EulersEquation.2.3InstantaneousAcousticIntensityItiscriticalinthestudyofacousticstounderstandcertainenergyrelationships.Mostimportantistheacousticintensityvector.InthetimedomainitiscalledtheinstantaneousacousticandisdefinedasItptvt,2.5withunitsofenergyperunittimepowerperunitarea,measuredasjoules/s/2morwatts/2m.Theacousticintensityisrelatedtotheenergydensityethroughitsdivergence,eIt,2.6wherethedivergenceisyxzIIIIxyz2.7Theenergydensityisgivenby2211022||evtpt2.8whereisthefluidcompressibility,201c2.9Equation2.6expressesthefactthatanincreaseintheenergydensityatsomepointinthefluidisindicatedbyanegativedivergenceoftheacousticintensityvectortheintensityvectorsarepointingintotheregionofincreaseinenergydensity.Figure2.2shouldmakethisclear.IfwereversethearrowsinFig.2.2,apositivedivergenceresultsandtheenergydensityatthecentermustdecrease,thatis,et﹤0.Thiscaserepresentsanapparentsourceofenergyatthecenter.Figure2.2Illustrationofnegativedivergenceofacousticintensity.Theregionatthecenterhasanincreasingenergydensitywithtime,thatis,anapparentsinkofenergy.2.4SteadyStateToconsiderphenomenainthefrequencydomain,weobtainthesteadythesteadystatesolutionthroughtransforms12iwtptpwedw2.10leadingtothesteadystatesolutioniwtpwptedt2.11Equation2.10canbedifferentiatedwithrespecttotimetoyieldtheimportantrelationship12iwtptiwpwedwtsothatfptiwpwt2.12wherethecalligraphicletterfrepresentstheFouriertransformofthetimedomainwaveequation,Eq,2.1,yieldingtheHelmholtzequation220pkp2.13wheretheacousticwavenumberiskw/c,thefrequencyisgivenby2f,

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