FUNDAMENTALS OF ACOUSTICS(10)

FUNDAMENTALSOFACOUSTICS(10)3-6Harmonicplanewavetravelinginonemedium•Ifalltheacousticvariablesarefunctionsofonlyonespatialcoordinate,thephaseofanyvariableisaconstantonanyplane,suchawaveiscalledaplanewave.•Thesimplestsolutionstothewaveequation(3-4)arethosethatdependononlyoneofthethreespatialcoordinates.2222(34)pcptWemayaswellcallthatonex.theequation(3-4)reducestofollowequation0zpyp222022ppctxThesolutionofthewaveequationWealreadyknowthenatureofitssolutions•Weintroducetwonewvariablequantities00,,xctxct02p2()pf1200()()xxpftftcc)()(21ffp222022ppctx•Thesumofthesetwofunctionsisthecompletegeneralsolutionofthewaveequation.•Thefunctionf1(t-x/c0)representsawavetravelingintherightataconstantspeedc0•Similarly,f2(t+x/c0)representsawavemovinginthe–xdirectionwithspeedc0.1200()()xxpftftccThecomplexformoftheharmonicsolutionfortheacousticpressureofaplanewaveis•Wherethewavenumberkisdefinedby00()()12xxjtjtccpAeAe)(2)(1kxtikxtieAeAp20ckA1,A2aretwoarbitraryconstantsWheretheamplitudespmisaconstant,itdoesnotchangewiththedistance.•Ifthereisnotreflectedwaves,A2=0soweobtain:()()1jtkxjtkxmpAepeTherelationshipbetweenthevelocityandpressure•Formtheequationofmotion01udtxpudtxpu1()()00jtkxjtkxmmpueuec00puc()()1jtkxjtkxmpAepe()jtkxmuudtej00()()22jkxjtkxjtmmueeeDisplacementis3-6-3Theacousticimpedanceandthecharacteristicimpedanceofthemedium•TheratioofacousticpressureinamediumtotheassociatedparticlespeedistheacousticimpedanceupZaForplanewavesthisratiois00aZcAlthoughtheacousticimpedanceofthemediumisarealquantityforplanewave,thisisnottrueforstandingplanewavesorfordivergingwave.Ingeneral,ZawillbefoundtobecomplexaaaZrjxWhereraiscalledtheacousticresistanceandxatheacousticreactanceofthemediumfortheparticularwavebeingconsidered.•TheMKSunitofacousticimpedanceisPa.s/m•TheproductOftenhasgratersignificanceasacharacteristicpropertyofthemediumthandoeseitherp0orc0individually.•Forthisreasonp0c0iscalledthecharacteristicimpedanceofthemedium.00c•Atatemperatureof200candatmosphericpressurethedensityofairis1.21kg/m3andthespeedofsoundis343m/s,givingthestandardcharacteristicimpedanceofair00415./cPasm•Atatemperatureof200candoneatmosphericpressure,resultinginacharacteristicimpedanceofwateris1.5*106Pa.s/m•3-6-4AnalogiesbetweenelectricalandacousticalsystemTheunitofacousticimpedanceisoftengivenasRayl.(CGSg/cm2.s=Rayl)Where1Pa.s/m=1MKSRayl•ElectricalquantityapuZeVIZeLZCAcousticalquantitySaKZ0Theenergyrelationshipoftheplanewaves2222000222000011112222ppuucccos()muutkx)(cos220kxtumTdtT01Fromtheenergydensityequation(3-4-1)00pucForaplaneharmonicwavetravelinginthexdirection20uAverageenergydensity0000222220000000011cos()cos()2TTmmmeeeIpudtTpptkxtkxdtTcppcuccc2222002001122emmpucAcousticintensity20aesWIdSIScuSHerewehaveidealconstantwavefrontpropagation,i.e.intensityremainsconstantforanydistancefromthesourcebecauseofplaneacousticwave.Thisisnottrueforsphericalacousticwavepropagation•Foraplanewavetravelinginsomearbitrarydirection,itisplausibletotryasolutionoftheform(,,)xyz0xyznr波阵面(,,)xyz0xyznr波阵面()0,jtkrprtpeExample•Forharmonicplaneacousticwavepropagationinthepositivexdirection,showthatparticlevelocityleadsparticledisplacementby900.Whatisthephaserelationshipbetweenacousticpressureandparticledisplacementwhenthewavesaretravelinginthenegativexdirection?•Forharmonicplaneacousticwavepropagationinthepositivexdirection,particledisplacementisexpressedas()(,)(,)cos()jtkxxtAeorxtAtkxParticlevelocity()0sin()cos(90)jtkxdjAejdtdorAtkxAtkxdtThustheparticlevelocityleadstheparticledisplacementby900•Forharmonicacousticwavepropagationinthenegativexdirection()(,)jtkxxtAeNowacousticpressure()000000jtkxpcujcAejcThereforetheacousticpressurelegstheparticledisplacementby900HOMEWORK•TextbookP2763-73-11