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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CHAPTER2PLANEWAVES21INTRODUCTIONINTHISCHAPTERWEPRESENTTHEFOUNDATIONSOFFOURIERACOUSTICSPLANEWAVEEXPANSIONSTHISMATERIALISPRESENTEDINDEPTHTOPROVIDEAFIRMFOUNDATIONFORTHERESTOFTHEBOOK,INTRODUCINGCONCEPTSLIKEWAVENUMBERSPACEANDTHEEXTRAPOLATIONOFWAVEFIELDSFROMONESURFACETOANOTHERFOURIESACOUSTICSISUSEDTODERIVESOMEFAMOUSTOOLSFORTHERADIATIONFROMPLANARSOURCES;THERAYLEIGHINTEGRALS,THEEWALDSPHERECONSTRUCTIONOFFARFIELDRADIATION,THEFIRSTPRODUCTTHEOREMFORARRAYS,VIBRATINGPLATERADIATION,ANDRADIATIONCLASSIFICATIONTHEORYFINALLY,ANEWTOOLCALLEDSUPERSONICINTENSITYISINTRODUCEDWHICHISUSEFULINLOCATINGACOUSTICSOURCESONVIBRATINGSTRUCTURESWEBEGINTHECHAPTERWITHAREVIEWOFSOMEFUNDAMENTALS;THEWAVEEQUATION,EULER’SEQUATION,ANDTHECONCEPTOFACOUSTICINTENSITY22THEWAVEEQUATIONANDEULER’SEQUATIONLETPX,Y,Z,TBEANINFINITESIMALVARIATIONOFACOUSTICPRESSUREFROMITSEQUILIBRIUMVALUEWHICHSATISFIESTHEACOUSTICWAVEEQUATION222210PPCT21FORAHOMOGENEOUSFLUIDWITHNOVISCOSITYCISACONSTANTANDREFERSTOTHESPEEDOFSOUNDINTHEMEDIUMAT020CC343M/SINAIRANDC1481M/SINWATERTHERIGHTHANDSIDEOFEQ21INDICATESTHATTHEREARENOSOURCESINTHEVOLUMEINWHICHTHEEQUATIONISVALIDINCARTESIANCOORDINATES2222222XYZASECONDEQUATIONWHICHWILLBEUSEDTHROUGHOUTTHISBOOKISCALLEDEULER’SEQUATION,0VPT22WHEREVGREEKLETTERUPSILONREPRESENTSTHEVELOCITYVECTORWITHCOMPONENTSU,V,W;VUIVJWK23WHEREIJANDKARETHEUNITVECTORSINTHETHEX,Y,ANDZDIRECTIONS,RESPECTIVELY,ANDTHEGRADIENTINTERMSOFTHEUNITVECTORSASIJKXYZ24WEUSETHECONVENTIONOFADOTOVERADISPLACEMENTSQUANTITYTOINDICATEVELOCITYASISDONEINJUNGERANDFEITTHEDISPLACEMENTSINTHETHREECOORDINATEDIRECTIONSAREGIVENBYU,V,ANDWTHEDERIVATIONOFEQ22ISUSEFULINDEVELOPINGSOMEUNDERSTANDINGOFTHEPHYSICALMEANINGOFPANDVLETUSPROCEEDINTHISDIRECTIONFIGURE21INFINITESIMALVOLUMEELEMENTTOILLUSTRATEEULER’SEQUATIONFIGURE21SHOWSANINFINITESIMALVOLUMEELEMENTOFFLUIDXYZ,WITHTHEXAXISASSHOWNALLSIXFACESEXPERIENCEFORCESDUETOTHEPRESSUREPINTHEFLUIDITISIMPORTANTTOREALIZETHATPRESSUREISASCALARQUANTITYTHEREISNODIRECTIONASSOCIATEDWITHITITHASUNITSOFFORCEPERUNITAREA,2/NMORPASCALSTHEFOLLOWINGISTHECONVENTIONFORPRESSURE,P﹥0→COMPRESSIONP﹤0→RAREFACTIONATASPECIFICPOINTINAFLUIDAPOSITIVEPRESSUREINDICATESTHATANINFINITESIMALVOLUMESURROUNDINGTHEPOINTISUNDERCOMPRESSION,ANDFORCESAREEXERTEDOUTWARDFROMTHISVOLUMEITFOLLOWSTHATIFTHEPRESSUREATTHELEFTFACEOFTHECUBEINFIG21ISPOSITIVE,THENAFORCEWILLBEEXERTEDINTHEPOSITIVEXDIRECTIONOFMAGNITUDEPX,Y,ZYZTHEPRESSUREATTHEOPPOSITEFACEPXX,Y,ZISEXERTEDINTHENEGATIVEXDIRECTIONWEEXPANDPXX,Y,ZINATAYLORSERIESTOFIRSTORDER,ASSHOWNINTHEFIGURENOTETHATTHEFORCEARROWSINDICATETHEDIRECTIONOFFORCEFORPOSITIVEPRESSUREGIVENTHEDIRECTIONSOFFORCESHOWN,THETOTALFORCEEXERTEDONTHEVOLUMEINTHEXDIRECTIONIS,,,,PPXYZPXXYZYZXYZXNOWWEINVOKENEWTON’SEQUATION,FMAMUT,WHEREFISTHEFORCE,0MXYZAND0ISTHEFLUIDDENSITY,YIELDING0UPTXCARRYINGOUTTHESAMEANALYSISINTHEYANDZDIRECTIONSYIELDSTHEFOLLOWINGTWOEQUATIONS0UPTYAND0UPTZWECOMBINETHEABOVETHREEEQUATIONSINTOONEUSINGVECTORSYIELDINGEQ22ABOVE,EULER’SEQUATION23INSTANTANEOUSACOUSTICINTENSITYITISCRITICALINTHESTUDYOFACOUSTICSTOUNDERSTANDCERTAINENERGYRELATIONSHIPSMOSTIMPORTANTISTHEACOUSTICINTENSITYVECTORINTHETIMEDOMAINITISCALLEDTHEINSTANTANEOUSACOUSTICANDISDEFINEDASITPTVT,25WITHUNITSOFENERGYPERUNITTIMEPOWERPERUNITAREA,MEASUREDASJOULES/S/2MORWATTS/2MTHEACOUSTICINTENSITYISRELATEDTOTHEENERGYDENSITYETHROUGHITSDIVERGENCE,EIT,26WHERETHEDIVERGENCEISYXZIIIIXYZ27THEENERGYDENSITYISGIVENBY2211022||EVTPT28WHEREISTHEFLUIDCOMPRESSIBILITY,201C29EQUATION26EXPRESSESTHEFACTTHATANINCREASEINTHEENERGYDENSITYATSOMEPOINTINTHEFLUIDISINDICATEDBYANEGATIVEDIVERGENCEOFTHEACOUSTICINTENSITYVECTOR;THEINTENSITYVECTORSAREPOINTINGINTOTHEREGIONOFINCREASEINENERGYDENSITYFIGURE22SHOULDMAKETHISCLEARIFWEREVERSETHEARROWSINFIG22,APOSITIVEDIVERGENCERESULTSANDTHEENERGYDENSITYATTHECENTERMUSTDECREASE,THATIS,ET﹤0THISCASEREPRESENTSANAPPARENTSOURCEOFENERGYATTHECENTERFIGURE22ILLUSTRATIONOFNEGATIVEDIVERGENCEOFACOUSTICINTENSITYTHEREGIONATTHECENTERHASANINCREASINGENERGYDENSITYWITHTIME,THATIS,ANAPPARENTSINKOFENERGY24STEADYSTATETOCONSIDERPHENOMENAINTHEFREQUENCYDOMAIN,WEOBTAINTHESTEADYTHESTEADYSTATESOLUTIONTHROUGHTRANSFORMS12IWTPTPWEDW210LEADINGTOTHESTEADYSTATESOLUTIONIWTPWPTEDT211EQUATION210CANBEDIFFERENTIATEDWITHRESPECTTOTIMETOYIELDTHEIMPORTANTRELATIONSHIP12IWTPTIWPWEDWTSOTHATFPTIWPWT212WHERETHECALLIGRAPHICLETTERFREPRESENTSTHEFOURIERTRANSFORMOFTHETIMEDOMAINWAVEEQUATION,EQ,21,YIELDINGTHEHELMHOLTZEQUATION220PKP213WHERETHEACOUSTICWAVENUMBERISKW/C,THEFREQUENCYISGIVENBY2F,

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