A wave equation for sound in unsteady, inhomogeneous moving media including thermoviscous and nonlinear effects.

An approximate wave equation is derived for sound propagation in unsteady, inhomogeneous, moving fluids, incorporating viscous and thermal-conduction effects. The formulation extends the work of Pierce [(1990). J. Acoust. Soc. Am. 87(6), 2292-2299] and relies on a separation of scales between acoustic and ambient fields. Acoustic wavelengths and periods are assumed to be small compared with the characteristic spatial and temporal scales of the background flow, and thermoviscous effects are assumed to remain weak over distances and times comparable to acoustic scales. Under these assumptions, propagation is locally equivalent to that in a homogeneous, steady medium, whereas cumulative effects due to unsteadiness, inhomogeneity, and thermoviscous processes emerge over many wavelengths and periods. The wave equation is further extended to include terms accounting for weak nonlinearities within the same asymptotic framework. In the particular case of steady, inhomogeneous low-Mach-number flows, the formulation is generalized to incorporate weak relaxation effects. In appropriate limiting regimes, the resulting models reduce to previously established approximate wave equations.