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

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

**Peter J. Westervelt
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*Dept. of Phys., Brown Univ., Box 1843, Providence, RI 02912
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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.