### 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)].