Multiple scattering and visco-thermal effects

Abstract

For modeling sound propagation in a rigid-framed fluid-saturated porous material it is customary to use frequency-dependent density and compressibility functions. These functions, which describe ‘‘temporal’’ dispersion effects due to inertial/viscous and thermal effects, can be computed by FEM in simple geometries and give complete information about the long-wavelength properties of the medium. When the wavelength is reduced, new effects due to scattering must be considered. To study this, we consider solving the sound propagation problem in a 2-D ‘‘phononic crystal’’ made of an infinite square lattice of solid cylinders embedded in a fluid. An exact multiple-scattering solution is first developed for an ideal saturating fluid and then generalized to the case of visco-thermal fluid, by using the concept of visco-thermal admittances. The condition to use this concept is that the viscous and thermal penetration depths are small compared to the cylinder radius. We validate our results in the long-wavelength regime by direct comparisons with FEM data [A. Cortis, ‘‘Dynamic parameters of porous media,’’ Ph.D. dissertation (Delft U.P., Delft, (2002)]. When frequency increases, differences appear between the long-wavelength solution and the exact multiple-scattering solution, which could be interpreted in terms of ‘‘spatial’’ dispersion effects.