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

The starting point inthe formulation of most acoustic problems is the acoustic waveequation. Those most widely used, the classical and convectedwave equations, have significant restrictions, i.e., apply onlyto linear, nondissipative sound waves in a steady homogeneousmedium at rest or in uniform motion. There are many practicalsituations violating these severe restrictions. In the presentpaper 36 distinct forms of the acoustic wave equation are derived(and numbered W1$\sim$W36), extending the classical and convectedwave equations to include cases of propagation in inhomogeneousand/or unsteady media, either at rest or in potential or vorticalflows. The cases considered include: (1) linear waves, i.e., withsmall gradients, which imply small amplitudes, and (2) nonlinearwaves, 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 anddissipation by shear and bulk viscosity and thermal conduction.Consideration is given to propagation in homogeneous media andinhomogeneous media, which are homentropic (i.e., have uniformentropy) or isentropic (i.e., entropy is conserved alongstreamlines), excluding nonisentropic (e.g., dissipative);unsteady media are also considered. The medium may be at rest, inuniform motion, or it may be a nonuniform and/or unsteady meanflow, 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-dimensionalpropagation in ducts of varying cross section, including hornswithout mean flow and nozzles with low or high Mach number meanflow; or (3) unidirectional sheared mean flow, in the plane, inspace 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-upwork together with dissipative and other cases). The 36 waveequations are derived either by elimination among the generalequations of fluid mechanics or from an acoustic variationalprinciple, with both methods being used in a number of cases ascross-checks. Although the 36 forms of the acoustic wave equationdo 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 inisolation and a variety of combinations of multiple effects.Altogether they provide a useful variety of extensions of theclassical (and convected) wave equations, which are used widelyin 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 theclassical and convected wave equations are poor approximations,and more general forms of the acoustic wave equation provide moresatisfactory models. Numerous examples of these applications aregiven at the end of each written section. There are 240 referencescited in this review article.