Scattering problem for a spherical inclusion in poroelastic media: Application of the asymptotic expansion method

Abstract

Scattering of longitudinal monochromatic waves on an isolated inclusion in an infinite poroelastic medium is considered. Wave propagation in both medium and inclusion is described by Biot’s equations of poroelasticity. For the material parameters typical for sedimentary rocks, the corresponding system of equations contains a second order differential operator with a small parameter. As the result, the wave field in the medium consists of a slowly changing part and boundary layer functions concentrated near the inclusion interface. Such a specific form of the wave field should be taken into account by the numerical solution of the problem since application of conventional numerical methods can result in substantial errors. In the present paper, the method of matched asymptotic expansions is applied to the solution of the scattering problem for a spherical inclusion. The systems of equations for a slowly changing part of the wave field and for boundary layer functions are derived. The solutions of these systems for the fields in the vicinity of the inclusion and for far wave fields are obtained and analyzed. Comparisons of the results of the straightforward solution and the asymptotic expansion are presented.