Refinements to the relaxation model for sound propagation in porous media

Abstract

The relaxation formulation for sound propagation in porous media [D. K. Wilson, J. Acoust. Soc. Am. 94(2), 1136–1145 (1993)] was intended to provide a phenomenologically correct model with relatively simple equations. This paper presents several refinements, which connect the relaxation model more clearly to others in the literature, and clarify interpretation of the parameters. In particular, the relaxation function is applied to the dynamic tortuosity (rather than the inverse of the complex density), and the vorticity and thermal relaxation times are related directly to the length scales of Allard and Champoux [J.-F. Allard and Y. Champoux, J. Acoust. Soc. Am. 91(6), 3346–3353 (1992)]. When formulated in this manner, the relaxation and Allard-Champoux models coincide exactly in the low-frequency/small-pore and high-frequency/large-pore limits, although interpolation between these limits differs somewhat. The revised relaxation model can be readily transformed into a causal time-domain model, involving convolutions between the acoustic fields and a relaxation response function. The basic formulation has five parameters: porosity, static flow resistivity, tortuosity, and two relaxation times. A sequence of model reductions is described, leading finally to two-parameter (porosity and static flow resistivity) equations for the impedance and complex wavenumber, which provide a useful generalization of the Delany-Bazley equations.