## 4pPAa1. Nonscattering of sound by sound resulting from a head-on collision of two plane-wave pulses.

Peter J. Westervelt

Dept. of Phys., Brown Univ., Box 1843, Providence, RI 02906

Two arbitrary pulses p[sub 2](t-c[sub 0][sup -1]x) and p[sub 1](t+c[sub 0][sup -1]x) each of width c[sub 0](tau) pass through one another at x=0, generating a differential surface mass rate source density d(sigma)[sub s](t,x)=q[sub s] dx where q[sub s]=Ad(p[sub 1]p[sub 2])/dt and A= ((rho)[inf 0]c[inf 0][sup 4])[sup -1][2 + (rho)[inf 0]c[inf 0][sup -2](d[sup 2]p/d(rho)[sup 2])[inf (rho)[inf 0]]]. The equation for the scattered pressure p[sub s] is d[sup 2]p[sub s]/dx[sup 2]-c[sub 0][sup -2] d[sup 2]p[sub s]/dt[sup 2]=-q[sub s] and its solution is p[sub s](x',t)=(integral)dp[sub s]=(c[sub 0]/2)(integral)d(sigma)[sub s]=(c[sub 0]/2)(integral)q[sub s](x,t')dx, t' being the retarded time; thus, (p[inf s])[inf (plus or minus)] = (c[inf 0]/2)A(integral)[inf (plus or minus)c[inf 0](tau)/2][sup x'][d(p[inf 1]p[inf 2])[inf t=t'(plus or minus)]/dt]dx, where + is chosen when x'x, and t[inf (plus or minus)][sup '] = t (plus or minus) c[inf 0][sup -1](x'- x). Suppose x'>c[sub 0](tau)/2 then x' lies outside the interaction zone, we find (p[sub s])[sub -]=(c[sub 0][sup 2]A/4)(p[sub 2]p[sub 1]+p[sub 2](integral)[sub -(infinity)][sup t]p[sub 1]dt)=0 unless p[sub 1] has a nonzero average which is possible for plane waves but we assume this not to be the case [L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading, MA, 1959), p. 267]. In the event x'<-c[sub 0](tau)/2x' again lies outside the interaction zone and (p[sub s])[sub +]=(c[sub 0][sup 2]A/4)(p[sub 1]p[sub 2]+p[sub 1](integral)[sub -(infinity)][sup t]p[sub 2] dt)=0. Within the interaction region -c[sub 0][sup 2]/2