## 4pPA3. Scattering of sound by sound within the interaction zone: Approximate solutions.

Peter J. Westervelt

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

The exact second-order transient solution to the interaction of an arbitrary wave with a plane wave is given by Westervelt [P. J. Westervelt, 3320 (1994)]. Let the arbitrary wave be (chi)(x[sup 0]-m(centered dot)r), a plane wave traveling in the m direction. In this case (sigma)=(1-n(centered dot)m)[sup -1](chi) and the solution to Eckart's equation becomes (rho)[sub s]c[sub 0][sup 2]=E[sub 12] +(1-n(centered dot)m)[sup -1]((Lambda)+2n(centered dot)m)=cos (theta) (del)[sup 2](psi)[sup 2], which is identical to Eq. (10) of Westervelt [P. J. Westervelt, 934 (1957)] provided the substitutions (psi)[sup 2]=1/2(w[sub 1]w[sub 2])[sup -1]W[sub 12] and n(centered dot)m=cos (theta) are made. It is asserted that this exact solution serves as an approximate solution to the far-field interaction of arbitrary sources. This is done by allowing m and n to be space dependent. As an example, the exact solution for the cardioid wave (chi)=(4(pi)r)[sup -1][n(centered dot)m(G[sub ,0]+Gr[sup -1])-G[sub ,0]] interactions with a plane wave is obtained from (sigma)=-(4(pi)r)[sup -1]G[sub ,0], where G=G(x[sup 0]-r) and m=rr[sup -1]. In the far field of the cardioid source, (sigma)=(1-n(centered dot)m)[sup -1](chi), as in the plane wave--plane wave interaction, thus demonstrating the assertion.