### Abstract

Two arbitrary pulses *p* _{2}(*t*−*c* ^{−1} _{0} *x*) and *p* _{1}(*t*+*c* ^{−1} _{0} *x*) each of width *c* _{0}τ pass through one another at *x*=0, generating a differential surface mass rate source density *d*σ_{ s }(*t*,*x*)=*q* _{ s } *dx* where *q* _{ s }=*Ad*(*p* _{1} *p* _{2})/*dt* and *A*= (ρ_{0} *c* ^{4} _{0})^{−1}[2 + ρ_{0} *c* _{0} ^{−2}(*d* ^{2} *p*/*d*ρ^{2})_{ρ0 }]. The equation for the scattered pressure *p* _{ s } is *d* ^{2} *p* _{ s }/*dx* ^{2}−*c* ^{−2} _{0} *d* ^{2} *p* _{ s }/*dt* ^{2}= −*q*̇_{ s } and its solution is *p* _{ s }(*x*’,*t*)=∫*dp* _{ s }=(*c* _{0}/2)∫*d*σ_{ s }= (*c* _{0}/2)∫*q* _{ s }(*x*,*t*’)*dx*, *t*’ being the retarded time; thus, (*p* _{ s })_{±} = (*c* _{0}/2)*A*∫_{±c 0τ/2} ^{ x’}[*d*(*p* _{1} *p* _{2}) _{ t=t’±}/*dt*]*dx*, where + is chosen when *x*’<*x*, − when *x*’≳*x*, and *t* _{±} ^{’} = *t* ± *c* _{0} ^{−1}(*x*’− *x*).

Suppose *x*’≳*c* _{0}τ/2 then *x*’ lies outside the interaction zone, it is found that (*p* _{ s })_{−}= (*c* ^{2} _{0} *A*/4)(*p* _{2} *p* _{1}+*p*̇_{2}∫^{ t } _{−∞} *p* _{1} *dt*)=0 unless *p* _{1} has a nonzero average which is possible for plane waves but it is assumed that this is not the case [L. D. Landau and E. M. Lifshitz, *Fluid* *Mechanics* (Addison‐Wesley, Reading, MA, 1959), p. 267]. In the event *x*’<−*c* _{0}τ/2*x*’ again lies outside the interaction zone and (*p* _{ s })_{+}= (*c* ^{2} _{0} *A*/4)(*p* _{1} *p* _{2}+*p*̇_{1}∫^{ t } _{−∞} *p* _{2} *dt*)= 0. Within the interaction region −*c* ^{2} _{0}/2<*x*’<*c* _{0}τ/2 so that *p* _{ s }=(*p* _{ s })_{+} +(*p* _{ s })_{−}=(*c* ^{2} _{0} *A*/4)(2*p* _{1} *p* _{2} +*p*̇_{2}∫^{ t } _{−∞} *p* _{1} *dt*+*p*̇_{1}∫^{ t } _{−∞} *p* _{2} *dt*), which is precisely the result obtained previously [P. J. Westervelt, J. Acoust. Soc. Am. **94**, 1774 (A) (1993)], recalling that *p* _{1} is here a plane wave; thus in the earlier result *p* _{ m }=*p* _{1}/2. The zero scattering results obtained here conflict with nonzero results obtained by Trivett and Rogers [J. Acoust. Soc. Am. **71**, 1114 (1982)] in a related problem, which most likely stems from spurious sources arising from discontinuities similar to those that plagued Ingard and Pridmore‐Brown.

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