On 36 Forms of the Acoustic Wave Equation in Potential Flows and Inhomogeneous Media

Abstract

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: (i) linear waves, i.e., with small gradients, which imply small amplitudes, and (ii) 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: (i) potential mean flow, of low Mach number (i.e., incompressible mean state) or of high-speed (i.e., inhomogeneous compressible mean flow); (ii) 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 (iii) 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 (i) nonlinearity in (ii) inhomogeneous and unsteady and (iii) 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 convected) 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.