An equation for acoustic propagation in inhomogeneous media with relaxation losses

Abstract

An equation for acoustic propagation in an inhomogeneous liquid medium with relaxation loss is systematically derived from the classical dynamic equations together with the appropriate thermodynamic and constitutive relations. The derivation assumes small acoustic perturbations, but accommodates arbitrary spatial inhomogeneities in material compressibility, density, and parameters of relaxation. The linearized wave equation obtained for N relaxation mechanisms has order N+2, is causal, and yields the expected dependency of attenuation and phase velocity on frequency. A similar equation is also obtained for gases. The corresponding reduced wave equation simplifies to the classical form with a complex frequency-dependent compressibility for which the Kramer–Kronig conditions are verified and for which an explicit expression is obtained in terms of medium properties. Exact analytic expressions valid at all frequencies are given for the spatially varying attenuation coefficient as well as phase velocity. A Green’s function is found for the constant coefficient part of the equation. The results are used to illustrate the effect of relaxation on scattering and also may be employed to model scattering in applications such as image reconstruction.