A note on the derivation of the wave equation in an inhomogeneous ocean

Abstract

Several derivations of the wave equation in an inhomogeneous medium appear in the literature. A term ∇p1⋅∇lnρ0 is found in all of them. Inhomogeneities in the sea are due to both gravity and other effects. The reference state characterized by ρ0(xyz) gives rise to two different limits, one due to gravity alone and the other due to inhomogeneities ignoring gravity. Both lead to the same form of the wave equation, but different arguments are required to demonstrate that ∇p1⋅∇lnρ0 can be ignored. A seldom cited paper by Bergmann [J. Acoust. Soc. Am. 17, 329–333 (1946)] is used.

Numerical investigation of transmission of low frequency sound through a smooth air-water interface

“…Formulations of acoustic propagation models generally begin with a three-dimensional time-dependent wave equation. Based on the governing assumptions and intended applications, the exact form of the wave equation can vary considerably (DeSanto, 1979;Goodman and Farwell, 1979). For most applications, a simplified linear, hyperbolic, second-order, and time-dependent partial differential equation is expressed as…”