Dispersion analysis of numerical approximations to plane wave motions in an isotropic elastic solid

It is well known that isotropic, nondispersive continuous hyperbolic problems become dispersive and anisotropic upon discretization. The purpose of this paper is to conduct a dispersion analysis of the nondissipative numerical approximations to plane wave motions in isotropic elastic solids. The discrete formulations considered are: an explicit, second-order accurate finite difference scheme, a consistent mass matrix formulation with linear quadrilateral elements and the corresponding lumped mass matrix formulation. Dispersion relation is derived for each of these formulations. In the context of the finite difference scheme, expressions for group velocity for both the shear and longitudinal waves are derived and the effect of using meshes of unequal size in x and y directions is studied. Results from numerical experiments confirming the predictions of analysis are also presented.