Derivation of a second-order acoustic wave equation in an air-saturated porous medium from a volume-averaged transport model

Abstract

For propagation of high-amplitude acoustic waves in a rigid porous frame saturated by air, significant departures from a constant flow resistivity model have been observed. This departure can be modeled as a quadratic modification to the linear Darcy’s drag law, known as Forchheimer’s correction. To study the physical cause and relative significance of this nonlinear correction with respect to convective nonlinearity, a second-order acoustic wave equation is derived from volume-averaged equations of mass, momentum, and entropy conservation. Due to the scale separation between the wavelength and the porous structure, the particle velocity is approximated as the sum of irrotational macroscopic and incompressible microscopic fields. The porosity, tortuosity, and permeability are defined and incorporated into the macroscopic conservation equations. Porous drag terms are also derived in the macroscopic momentum equation for Darcy’s law and Forchheimer’s correction. Conservation equations for fluid parameter perturbations are obtained, and a second-order wave equation is derived. Dimensional analysis, in terms of the acoustic Reynolds number and Darcy number, describes the criteria for significant Forchheimer and convective nonlinearity. A case is obtained for a Westervelt-like equation for approximately progressive plane waves. These results inform modeling of nonlinearities for sound propagation in porous media under practical conditions.