(麻省理工)波传导-二维波动

《(麻省理工)波传导-二维波动》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、3.1.Reflectionofsoundbyaninterface11.138J/2.062J,WAVEPROPAGATIONFall,2000MITNotesbyC.C.MeiCHAPTERTHREETWODIMENSIONALWAVES1Reflectionandtranmissionofsoundataninter-faceReference:BrekhovskikhandGodin§.2.2.Thegoverningequationforsoundinahonmogeneousfluidis

2、givenby(7.31)and(7.32)inChapterOne.Intermofthetheveloctiypotentialdefinedbyu=∇φitis1∂2φ2=∇φ(1.1)c2∂t2wherecdenotesthesoundspeed.Recallthatthefluidpressurep=−ρo∂φ/∂talsosatisfiesthesameequation.Wefirstgeneralizetheplanesinusoidalwaveinthreedimensionalspac

3、ei(k·x−ωt)i(kn·x−ωt)φ(x,t)=φoe=φoe(1.2)wherenistheunitvectorinthedirectionofk.Herethephasefunctionisθ(x,t)=k·x−ωt(1.3)Theequationofconstantphaseθ(x,t)=θodescribesamovingsurface.Thewavenumbervectork=knisdefinedtobek=kn=∇θ(1.4)henceisorthogonaltothesurf

4、aceofconstantphase,andrepresensthedirectionofwavepropagation.Thefrequencyisdefinedtobe∂θω=−(1.5)∂t3.1.Reflectionofsoundbyaninterface2Is(1.2)asolution?Letuscheck(1.7).Ã!∂∂∂∇φ=,,φ=ikφ∂x∂y∂z22∇φ=∇·∇φ=ik·ikφ=−kφ∂2φ2=−ωφ∂t2Hence(1.1)issatisfiedifω=kc(1.6)Con

5、sidertwosemi-infinitefluidsseparatedbytheplaneinterfacealongz=0.Thesoundspeedsintheupperandlowerfluidsarecandc1respectively.Letaplaneincidentwavearivefromz>0attheincidentangleofθwithrespecttothezaxis,pi=exp[ik(xsinθ−zcosθ)](1.7)implyingthatiiik=(kx,kz)=

6、k(sinθ,−cosθ)(1.8)Themotionisconfinedinthex,zplane.Onthesame(incidence)sideoftheinterfacewehavethereflectedwavepr=Rexp[ik(xsinθ+zcosθ)](1.9)whereRdenotesthereflectioncoefficient.Thewavenumbervectorisrrrk=(kx,kz)=k(sinθ,cosθ)(1.10)Inthelowermediumz<0thetra

7、nsmittedwavehasthepressurept=Texp[ik1(xsinθ1−zcosθ1)](1.11)whereTisthetransmissioncoefficient.Alongtheinterfacez=0werequirethecontinutiyofpressureandnormalvelocity,i.e.,[p]=0,z=0(1.12)and[w]=0z=0,(1.13)3.1.Reflectionofsoundbyaninterface3wherethesquarebr

8、acketssignifythejumpacrosstheinterface:[f]≡f(z=0+)−f(z=0−)(1.14)WedefinetheimpedanceofasimpleharmonicwavesbypZ=−(1.15)wwherewistheverticalcomponentofthefluidvelocity.Because∂w∂pρ=−iωρw=−,(1.16)∂t∂zp−iωρp=−(1.17)w∂p∂zItfollowsfromthetwocontinuityrequire