位势流动和非均匀介质中声波方程的36种形式

ON 36 FORMS OF THE ACOUSTIC WAVE EQUATION IN POTENTIAL FLOWS AND INHOMOGENEOUS MEDIA

声波方程是对大多数声学问题进行数学描述的出发点。那些得到广泛应用的经典波动方程及对流波动方程都存在苛刻的适用条件,即仅适用于描述处于静态或匀速运动状态的定常均匀介质中的线性无耗散声波。然而,很多实际场合并不满足这些严格的适用条件。本文对经典声波方程和对流声波方程进行推广,导出了编号为W1～W36的36种不同形式的声波方程,涵盖了处于静止、势流或旋涡流状态下的非均匀和/或非定常介质中的声波传播问题。所考虑的声波传播情形包括:(1)线性波,即具有小梯度(小振幅)性质;(2)非线性波,即具有陡峭梯度性质,包括"波纹"(小振幅大梯度)或者大振幅波。本文仅考虑非耗散声波,即排除了由剪切、体积黏度及热传导所引起的耗散。对具有匀熵或等熵(熵沿流线守恒)性质的均匀介质和非均匀介质中的声传播进行了研究但非等熵(即耗散)情况除外;另外,对非定常介质中的声波问题也进行了分析。所涉及的介质可以处于静止、匀速运动状态,或者是非匀速的和/或非定常的平均流动,包括:(1)低Mach数的势平均流(即不可压缩的平均态),或高速势平均流(即非均匀可压缩的平均流);②变截面管道中的准一维传播,包括无平均流的号管和具有低或高Mach数平均流的喷管;或③平面的、空间的、或轴对称的单向剪切平均流。本文没有探讨其他类型的旋涡平均流(将与耗散及其他情形一起留待下一步研究),例如,可能与剪切效应相结合的轴对称旋转平均流。通过对流体力学的一般方程进行消元处理或根据声学变分原理,导出了36种波动方程,对一些波动方程还采用这两种方法进行相互校验。尽管声波方程的36种形式没有涵盖非线性、非均匀与非定常及非匀速运动介质这3个效应的所有可能的组合情形,但它们的确包括了孤立状态下的各种效应,并包括了多种多重效应组合的情形。虽然经典波动方程和对流波动方程仅适用于处于静止(或匀速运动)的均匀定常介质中的线性无耗散声波,但它们在相关文献中已被广泛采用;本文给出的36种声波方程提供了它们多种有用的推广形式。在许多实际应用中,经典波动方程和对流波动方程仅是粗略的近似,声波方程的更一般形式可提供更令人满意的理论模型。本文每节末尾给出了这些应用的众多范例。在这篇评论文章中引用了240篇参考文献。

The starting point in the formulation of most acoustic problems is the acoustic wave equation. Those most widely used, the classical and convected wave equations, have significant restrictions, i.e., apply only to linear, nondissipative sound waves in a steady homogeneous medium at rest or in uniform motion. There are many practical situations violating these severe restrictions. In the present paper 36 distinct forms of the acoustic wave equation are derived （and numbered W1-W36）, extending the classical and convected wave equations to include cases of propagation in inhomogeneous and/or unsteady media, either at rest or in potential or vortical flows. The cases considered include： （1） linear waves, i.e., with small gradients, which imply small amplitudes, and （2） nonlinear waves, i.e., with steep gradients, which include ＂ripples＂ （large gradients with small amplitude） or large amplitude waves. Only nondissipative waves are considered, i.e., excluding and dissipation by shear and bulk viscosity and thermal conduction. Consideration is given to propagation in homogeneous media and inhomogeneous media, which are homentropic （i.e., have uniform entropy） or isentropic （i.e., entropy is conserved along streamlines）, excluding nonisentropic （e.g., dissipative）; unsteady media are also considered. The medium may be at rest, in uniform motion, or it may be a nonuniform and/or unsteady mean flow, including： （1） potential mean flow, of low Mach number （i.e., incompressible mean state） or of high-speed （i.e., inhomogeneous compressible mean flow）; （2） quasi-one-dimensional propagation in ducts of varying cross section, including horns without mean flow and nozzles with low or high Mach number mean flow; or （3） unidirectional sheared mean flow, in the plane, in space or axisymmetric. Other types of vortical mean flows, e.g., axisymmetric swirling mean flow, possibly combined with shear, are not considered in the present paper （and are left to follow-up work together with dissipative and other cases）. The 36 wave equations are derived either by elimination among the general equations of fluid mechanics or from an acoustic variational principle, with both methods being used in a number of cases as cross-checks. Although the 36 forms of the acoustic wave equation do not cover all possible combinations of the three effects of （1） nonlinearity in （2） inhomogeneous and unsteady and （3） nonuniformly moving media, they do include each effect in isolation and a variety of combinations of multiple effects. Altogether they provide a useful variety of extensions of the classical （and eonvected） wave equations, which are used widely in the literature, in spite of being restricted to linear, nondissipative sound waves in an homogeneous steady medium at rest （or in uniform motion）. There are many applications for which the classical and convected wave equations are poor approximations, and more general forms of the acoustic wave equation provide more satisfactory models. Numerous examples of these applications are given at the end of each written section. There are 240 references cited in this review article.