1、外文部分Chapter2Planewaves2/1IntroductionInthischapterwepresentthefoundationsofFourieracoustics-planewaveexpansions/Thismaterialispresentedindepthtoprovideafirmfoundationfortherestofthebook/introducingconceptslikewavenumberspaceandtheextrapolationofwavefieldsfromonesurfacetoanother/Fouriesacousticsisuse
2、dtoderivesomefamoustoolsfortheradiationfromplanarsources；theRayleighintegrals/theEwaldsphereconstructionoffarfieldradiation/thefirstproducttheoremforarrays/vibratingplateradiation/andradiationclassificationtheory/Finally/anewtoolcalledsupersonicintensityisintroducedwhichisusefulinlocatingacousticsou
3、rcesonvibratingstructures/Webeginthechapterwithareviewofsomefundamentals；thewaveequation/Eulersequation/andtheconceptofacousticintensity/2/2TheWaveEquationandEulersEquationLetp(x/y/z/t)beaninfinitesimalvariationofacousticpressurefromitsequilibriumvaluewhichsatisfiestheacousticwaveequation222210ppct(
4、2/1)forahomogeneousfluidwithnoviscosity/cisaconstantandreferstothespeedofsoundinthemedium/At020Cc=343m/sinairandc=1481m/sinwater/TherighthandsideofEq/(2/1)indicatesthattherearenosourcesinthevolumeinwhichtheequationisvalid/InCartesiancoordinates2222222xyzAsecondequationwhichwillbeusedthroughoutthisbo
5、okiscalledEulersequation/0vpt(2/2)Wherev(Greekletterupsilon)representsthevelocityvectorwithcomponentsu/v/w；vuivjwk(2/3)whereijandkaretheunitvectorsinthethex/y/andzdirections/respectively/andthegradientintermsoftheunitvectorsasijkxyz(2/4)Weusetheconventionofadotoveradisplacementsquantitytoindicatevel
6、ocityasisdoneinJungerandFeit/Thedisplacementsinthethreecoordinatedirectionsaregivenbyu/v/andw/ThederivationofEq/(2/2)isusefulindevelopingsomeunderstandingofthephysicalmeaningofpandv/Letusproceedinthisdirection/Figure2/1/InfinitesimalvolumeelementtoillustrateEulersequationFigure2/1showsaninfinitesima
7、lvolumeelementoffluidxyz/withthexaxisasshown/Allsixfacesexperienceforcesduetothepressurepinthefluid/Itisimportanttorealizethatpressureisascalarquantity/Thereisnodirectionassociatedwithit/Ithasunitsofforceperunitarea/2/NmorPascals/Thefollowingistheconventionforpressure/P0CompressionP0RarefactionAtasp
8、ecificpointinafluid/apositivepressureindicatesthataninfinitesimalvolumesurroundingthepointisundercompression/andforcesareexertedoutwardfromthisvolume/ItfollowsthatifthepressureattheleftfaceofthecubeinFig/2/1ispositive/thenaforcewillbeexertedinthepositivexdirectionofmagnitudep(x/y/z)yz/Thepressureatt
9、heoppositefacep(x+x/y/z)isexertedinthenegativexdirection/Weexpandp(x+x/y/z)inaTaylorseriestofirstorder/asshowninthefigure/Notethattheforcearrowsindicatethedirectionofforceforpositivepressure/Giventhedirectionsofforceshown/thetotalforceexertedonthevolumeinthexdirectionis(/)(/)ppxyzpxxyzyzxyzxNowweinv
10、okeNewtonsequation/f=ma=mut/wherefistheforce/0mxyzand0isthefluiddensity/yielding0uptxCarryingoutthesameanalysisintheyandzdirectionsyieldsthefollowingtwoequations/0uptyand0uptzWecombinetheabovethreeequationsintooneusingvectorsyieldingEq(2/2)above/EulersEquation/2/3InstantaneousAcousticIntensityItiscr
11、iticalinthestudyofacousticstounderstandcertainenergyrelationships/Mostimportantistheacousticintensityvector/Inthetimedomainitiscalledtheinstantaneousacousticandisdefinedas()()()Itptvt/(2/5)withunitsofenergyperunittime(power)perunitarea/measuredas(joules/s)/2morwatts/2m/Theacousticintensityisrelatedt
12、otheenergydensityethroughitsdivergence/eIt/(2/6)wherethedivergenceisyxzIIIIxyz(2/7)Theenergydensityisgivenby2211022|()|()evtpt(2/8)whereisthefluidcompressibility/201c(2/9)Equation(2/6)expressesthefactthatanincreaseintheenergydensityatsomepointinthefluidisindicatedbyanegativedivergenceoftheacousticin
13、tensityvector；theintensityvectorsarepointingintotheregionofincreaseinenergydensity/Figure2/2shouldmakethisclear/IfwereversethearrowsinFig/2/2/apositivedivergenceresultsandtheenergydensityatthecentermustdecrease/thatis/et0/Thiscaserepresentsanapparentsourceofenergyatthecenter/Figure2/2/Illustrationof
14、negativedivergenceofacousticintensity/Theregionatthecenterhasanincreasingenergydensitywithtime/thatis/anapparentsinkofenergy/2/4SteadyStateToconsiderphenomenainthefrequencydomain/weobtainthesteadythesteadystatesolutionthroughtransforms()1()2iwtptpwedw(2/10)leadingtothesteadystatesolution()()iwtpwpte
15、dt(2/11)Equation(2/10)canbedifferentiatedwithrespecttotimetoyieldtheimportantrelationship()1()2iwtptiwpwedwtsothat()()fptiwpwt(2/12)wherethecalligraphicletterfrepresentstheFouriertransformofthetimedomainwaveequation/Eq/(2/1)/yieldingtheHelmholtzequation220pkp(2/13)wheretheacousticwavenumberisk=w/c/thefrequencyisgivenby2f/