非线性声学谐波方程的特解及其在边值问题中的应用

在流体或固体介质中,微扰法求解非线性声波的反射和折射问题时,谐波场通常满足非齐次波动方程。应用分离变量法及拉格朗日变动参数法求它的特解,出现了待定的分离常数,给这类非线性声学边值问题带来困难。本文结果表明,为不使定解问题出现佯僇,其独立的特解只有两类,一类是沿边界法线方向有积累效应的解,另一类则是沿平行边界面方向上有积累效应的解,由定解条件来决定究竟选用哪一类解。应用这个结果研究了平面边界的反射和折射谐波,该理论对非线性声学中的平面边值问题有普遍的应用意义。

SPECIAL SOLUTION OF SECOND HARMONIC WAVE EQUATION IN NONLINEAR ACOUSTICS AND APPLICATIONS TO BOUNDARY-VALUE PROBLEM

Based on the perturbation theory, an investigation on reflection and refraction of nonlinear wave on a boundary in fluids or solids was carried out a familiar result was obtained that the second harmonic wave always satisfies an inhomogeneous wave equation. In order to find its special solution, the method of separated variables as well as Lagrange's method of variation parameters were invoked , and a trouble to the boundary-value problem of nonlinear acoustics will consequently result in that a separation constant is to be determined. In this paper , the constant was determined and the special solution was given uniquely. It is shown that a paradox will occur unless we select a special solution from the following two solutions, which are accumulation along the direction either parallel to or perpendicular to the boundary plane. Whether one can be selected depends on the boundary situation. By using the theory, the reflection and the refraction on a plane boundary were analysed. Furthermore, it is pointed out that this theory can deal with all of the boundary-value problems in nonlinear acoustics.