ASA 129th Meeting - Washington, DC - 1995 May 30 .. Jun 06

4pPA2. General proof of the nonscattering of sound by sound in the lossless dispersionless case.

Peter J. Westervelt

Dept. of Phys., Brown Univ., Box 1843, Providence, RI 02912

Introducing the variable (pi)=(rho)[sub s]c[sub 0][sup 2]+L+(Lambda)V into Eq. (14) of Westervelt [P. J. Westervelt, 200 (1957)], two equivalent dissipationless wave equations are obtained for the scattered variable (rho)[sub s]c[sub 0][sup 2]: Eq. (1) is (open square)[sup 2][(pi)+2((psi)[sup 2])[sub ,00]]=(Gamma)V[sub ,00] and Eq. (2) is (open square)[sup 2][(pi)-(Lambda)((psi)[sup 2])[sub ,00]]=(Gamma)T[sub ,00] in which (open square)[sup 2](psi)[sup 2]=L and (Gamma)=-(2+(Lambda)). Outside the region of interaction (pi)=(rho)[sub s]c[sub 0][sup 2] so the source for (rho)[sub s]c[sub 0][sup 2] can be either (Gamma)V[sub ,00] or (Gamma)T[sub ,00] and since, in general, T(not equal to)V, (rho)[sub s]c[sub 0][sup 2] must be zero (except for unidirectional waves when T=V in which case the well-known solution is still restricted to the interaction zone). As an example, consider two colliding waves for which the interaction energies satisfy T[sub 12]=-V[sub 12] requiring (rho)[sub s]c[sub 0][sup 2] from Eq. (1) to be the negative of that from Eq. (2) which demands that (rho)[sub s]c[sub 0][sup 2]=0. Within the interaction region the solution to either equation is (rho)[sub s]c[sub 0][sup 2]=((Lambda)-2)((phi)[sup +](phi)[sup -][sub ,00]+(phi)[sup -](phi)[sup +][sub ,00]-2(phi)[sup +][sub ,0](phi)[sup -][sub ,0]), where (psi)=(phi)[sup +]+(phi)[sup -] the sum of right and left traveling waves. The scattered pressure p[sub s]=(rho)[sub s]c[sub 0][sup 2]+(Lambda)V. For scattering to occur outside the interaction region, there must exist losses, as witnessed in the parametric array, or dispersion [see, e.g., P. J. Westervelt,``Scattering of Sound by Sound,'' in Finite Amplitude Wave Effects in Fluids, edited by L. Bjorno, Proc. 1973 Symp. Copenhagen (IPC Science and Technology, London, 1974), or interaction with real sources [P. J. Westervelt, Virtual Sources in the Presence of Real Sources in Nonlinear Acoustics, edited by T. G. Muir (Proc. of Univ. of Texas at Austin Conference, Nov. 1969)].