外文翻译--平面波(编辑修改稿)

外文翻译--平面波(编辑修改稿)内容摘要：

  so that ( ()) ( )f pt iw p wt  () where the calligraphic letter f represents the Fourier transform of the time domain wave equation,Eq,(), yielding the Helmholtz equation 220p k p  () where the acoustic wavenumber is k=w/c,the frequency is given by 2 f , and p is the function (x,y,z, ).For simplicity of notation we drop the bar above the variable. It will be clear from the context of the discussion if the quantity is in the frequency or in the time domain. The Fourier transform of Euler’s equation, Eq .(), bees, in the frequency domain 0i vp  () where Eq.() has been used again for the time derivative. Time Averaged Acoustic Intensity Now consider the intensity relationship for steady state fields .This is defined as the average of the instantaneous intensity over a period T, where T=1/f and f is the excitation frequency: 0() 1 ( ) ( )TI p t v t d tT   () Using plex variable notation this relationship bees 12() Re ( ( ) ( ) )I pv   () where stands for plex plex conjugate and Re for the real part .The onehalf results from the time average process .I is the average power over one period passing through unit area. For example , the x ponent of this flow xI represents the power passing through an element of area yz . Important in this chapter is the radiation from planar radiators .Of particular interest is the power flow crossing an infinite plane. For example, consider the total power crossing the corrdinate plane z=0, a quantity expressed is watts or joules persecond .We use the symbol () to represent the total power in watts crossing the boundary: () ( , , 0 )zI x y d x d y    () If there are no sources in the upper half space, then  is the total power radiated by 0zz has the same power passing through it, since is no absorption in the fluid and there are no sources above the boundary. The equation of continuity,Eq.(), bees 01 ( ) ( 0 ) ()T e e T ed t IT t T      () By the definition of stesdy state the energy density at time at time T is the same as the density at time 0, so that we have ( ) 0I   () This means that in a sourcefree field the divergence of the time averaged acoustic intensity must always be zero. The only way the intensity field can have a nonzero divergence is if there are sources or sinks of energy within the medium, or losses in the medium. Plane Wave Expansion We turn now to plane wave solutions of the wave equation in Cartesian coordinates .These solutions will be useful in the study of sources which are planar (or nearly planar) in geometry such as vibrating plates .We note that。