ON THE SCATTERING OF SOUND BY SOUND

The problem of the non-linear interaction between two fully collimated plane-wave beams travelling in different directions has given rise to much of the controversy to date as to whether the secondary scattered radiation exists outside the interaction region. Ingard et al. expressed the primary beams with a type of discontinuous function ρ={ej(w-ky),|x|a. Through calculations, they claimed that a scattered radiation is shown to exist outside the region of interaction. Assuming primary fields are plane waves of infinite extent, Westervelt studied the same problem, but a negative conclusion was obtained. By dividing co-ordinate space into the inside and outside of the common volume, Al-Temimi solved Westervelt's equation for both cases and concluded that the two conflicting results could relatively be brought together.Although in this paper only ideal beams interacting at right angles are discussed, the author suggests that this type of discontinuity can be more adequately described with a certain combination of unit-step functions. By applying and solving Westervelt's equation, the author obtains an interesting result, i.e., the secondary scattered radiations outside the common volume originate not from a volume source as claimed by Al-Temimi, but from a δ-function surface-dipole. However, this surface source is. artificial, because discontinuous functions which do not satisfy the homogeneous wave equation have been used to describe the primary waves. It is shown that the solution obtained by the author is the same as that of Al-Temimi, therefore, a relative agreement cannot be reached between the two conflicting results. A comment is also made on the latter's paper concerning the inappropriateness of the continuous conditions assumed at the boundaries. Based on the above discussions, the author predicts that if the primary beams are to be described by discontinuous functions, then the theories of the parametric transmitting and recieving arrays will be similarly affected.