The 36 forms of the acoustic wave equation derived in an earlier review (Campos, L. M. B. C., 2007, “On 36 Forms of the Acoustic Wave Equation in Potential Flows and Inhomogeneous Media,” Appl. Mech. Rev., 60, pp. 149–171) were grouped in four classes, of which the last (Class IV) concerned sheared mean flows; another type of vortical flow is swirling flow, and thus the present review completes the preceding by starting with Class V of linear, nondissipative acoustic wave equations in axisymmetric swirling, and also sheared, mean flow. These include general swirl and, in particular, rigid body and potential vortex swirl, combined or not with shear, for axisymmetric or general nonaxisymmetric acoustic modes, in two types of media: (i) inhomogeneous isentropic and (ii) homogeneous homentropic. Besides the 14 acoustic wave equations in sheared and swirling mean flows, the remaining ten acoustic wave equations derived in the present review all concern waves in homogeneous and steady media at rest, with dissipation or nonlinear effects to second-order or a combination of these two opposing effects, viz., (i) Class VI of linear, nondissipative wave equations with weak or strong thermoviscous dissipation in a homogeneous medium at rest; (ii) Class VIIA nonlinear one-dimensional wave equations in steady, homogeneous medium at rest without dissipation, or with viscous or thermoviscous dissipation, also in the case of a duct of varying cross section; (iii) Class VIIB of weakly nonlinear, three-dimensional waves or beams with thermoviscous dissipation in a homogeneous steady medium at rest. The 24 forms of the acoustic wave equation derived in the present review add to the 36 forms in the preceding review to form the set of 60 acoustic wave equations, whose interconnections are indicated in a family tree at the end. Numerous examples of the applications of the wave equations to the physical world are given at the end of each written section.

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