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## ON THE SCATTERING OF SOUND BY SOUND

QIAN ZU-WEN
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• #### 摘要

关于传播方向不同的两有限束的相互作用问题,历年来曾存在着分歧,分歧的焦点是:在公共区之外有没有二阶散射场?Ingard用间断函数ρ={ej(wt-ky),|x|a 表示有限束(即所谓完全准直束),通过计算求得:在公共区外有二阶散射场。Westervelt讨论了两列平面波的相互作用,却得出否定的结论。实验上也同样出现分歧。AL-Temimi将空间分成内外两部分公共区,分别求解Westervelt方程,将所得到的解在边界上连接。结果表明,公共区之外有二阶散射场。此外,他还认为上述两种相反的结论能够相对地一致。本文讨论两束正交简谐波,将上述间断函数用二个阶跃函数之差表示,代入Westervelt方程求解。结果表明,由这种理想有限束所构成的二阶散射场不是真正的散射场,而是由于按界面分布的δ函数性质的偶极源与平面波相互作用所产生的场,它随着这种界面的消失而熄灭。而这种偶极面源如文献[3,4]所述是人为的,它是由于采用了不满足齐次波动方程的间断函数来表示一阶声场所带来的结果。本文进一步指出,从这种有限束出发求得的解却和文献[6]的结果相同。这就说明,上述两种相反的结论是不能相对地一致的。本文还对文献[6]的连接条件作了分析,并指出这些条件是不恰当的。根据本文的结果,作者认为用上述间断函数来表示有限束从而计算参量发射和接收阵也是有影响的。

#### Abstract

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.

#### 参考文献

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#### 施引文献

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##### 出版历程
• 收稿日期:  1974-03-22
• 修回日期:  1976-03-26
• 刊出日期:  1976-03-05

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