外文部分Chapter2Planewaves2.1IntroductionInthischapterwepresentthefoundationsofFourieracoustics-planewaveexpansions.Thismaterialispresentedindepthtoprovideafirmfoundationfortherestofthebook,introducingconceptslikewavenumberspaceandtheextrapolationofwavefieldsfromonesurfacetoanother.Fouriesacousticsisusedtoderivesomefamoustoolsfortheradiationfromplanarsources；theRayleighintegrals,theEwaldsphereconstructionoffarfieldradiation,thefirstproducttheoremforarrays,vibratingplateradiation,andradiationclassificationtheory.Finally,anewtoolcalledsupersonicintensityisintroducedwhichisusefulinlocatingacousticsourcesonvibratingstructures.Webeginthechapterwithareviewofsomefundamentals；thewaveequation,Eulersequation,andtheconceptofacousticintensity.2.2TheWaveEquationandEulersEquationLetp(x,y,z,t)beaninfinitesimalvariationofacousticpressurefromitsequilibriumvaluewhichsatisfiestheacousticwaveequation222210ppct(2.1)forahomogeneousfluidwithnoviscosity.cisaconstantandreferstothespeedofsoundinthemedium.At020Cc=343m/sinairandc=1481m/sinwater.TherighthandsideofEq.(2.1)indicatesthattherearenosourcesinthevolumeinwhichtheequationisvalid.InCartesiancoordinates2222222xyzAsecondequationwhichwillbeusedthroughoutthisbookiscalledEulersequation,0vpt(2.2)Wherev(Greekletterupsilon)representsthevelocityvectorwithcomponentsu,v,w；vuivjwk(2.3)whereijandkaretheunitvectorsinthethex,y,andzdirections,respectively,andthegradientintermsoftheunitvectorsasijkxyz(2.4)WeusetheconventionofadotoveradisplacementsquantitytoindicatevelocityasisdoneinJungerandFeit.Thedisplacementsinthethreecoordinatedirectionsaregivenbyu,v,andw.ThederivationofEq.(2.2)isusefulindevelopingsomeunderstandingofthephysicalmeaningofpandv.Letusproceedinthisdirection.Figure2.1:InfinitesimalvolumeelementtoillustrateEulersequationFigure2.1showsaninfinitesimalvolumeelementoffluidxyz,withthexaxisasshown.Allsixfacesexperienceforcesduetothepressurepinthefluid.Itisimportanttorealizethatpressureisascalarquantity.Thereisnodirectionassociatedwithit.Ithasunitsofforceperunitarea,2/NmorPascals.Thefollowingistheconventionforpressure,P0CompressionP0RarefactionAtaspecificpointinafluid.apositivepressureindicatesthataninfinitesimalvolumesurroundingthepointisundercompression,andforcesareexertedoutwardfromthisvolume.ItfollowsthatifthepressureattheleftfaceofthecubeinFig.2.1ispositive,thenaforcewillbeexertedinthepositivexdirectionofmagnitudep(x,y,z)yz.Thepressureattheoppositefacep(x+x,y,z)isexertedinthenegativexdirection.Weexpandp(x+x,y,z)inaTaylorseriestofirstorder,asshowninthefigure.Notethattheforcearrowsindicatethedirectionofforceforpositivepressure.Giventhedirectionsofforceshown,thetotalforceexertedonthevolumeinthexdirectionis(,)(,)ppxyzpxxyzyzxyzxNowweinvokeNewtonsequation,f=ma=mut,wherefistheforce,0mxyzand0isthefluiddensity,yielding0uptxCarryingoutthesameanalysisintheyandzdirectionsyieldsthefollowingtwoequations:0uptyand0uptzWecombinetheabovethreeequationsintooneusingvectorsyieldingEq(2.2)above,EulersEquation.2.3InstantaneousAcousticIntensityItiscriticalinthestudyofacousticstounderstandcertainenergyrelationships.Mostimportantistheacousticintensityvector.Inthetimedomainitiscalledtheinstantaneousacousticandisdefinedas()()()Itptvt,(2.5)withunitsofenergyperunittime(power)perunitarea,measuredas(joules/s)/2morwatts/2m.Theacousticintensityisrelatedtotheenergydensityethroughitsdivergence,eIt,(2.6)wherethedivergenceisyxzIIIIxyz(2.7)Theenergydensityisgivenby2211022|()|()evtpt(2.8)whereisthefluidcompressibility,201c(2.9)Equation(2.6)expressesthefactthatanincreaseintheenergydensityatsomepointinthefluidisindicatedbyanegativedivergenceoftheacousticintensityvector；theintensityvectorsarepointingintotheregionofincreaseinenergydensity.Figure2.2shouldmakethisclear.IfwereversethearrowsinFig.2.2,apositivedivergenceresultsandtheenergydensityatthecentermustdecrease,thatis,et0.Thiscaserepresentsanapparentsourceofenergyatthecenter.Figure2.2:Illustrationofnegativedivergenceofacousticintensity.Theregionatthecenterhasanincreasingenergydensitywithtime,thatis,anapparentsinkofenergy.2.4SteadyStateToconsiderphenomenainthefrequencydomain,weobtainthesteadythesteadystatesolutionthroughtransforms()1()2iwtptpwedw(2.10)leadingtothesteadystatesolution()()iwtpwptedt(2.11)Equation(2.10)canbedifferentiatedwithrespecttotimetoyieldtheimportantrelationship()1()2iw