Surface waves on a half space with depth-dependent properties

The dispersive properties of surface waves on an isotropic elastic body with elastic moduli and mass density that depend on depth have been analyzed in the high frequency range, for the case of axially symmetric surface waves, which are of interest for point loading of a body. The method of approach requires some simplifications, but the final analysis yields simple expressions for the displacements, for the case that the two elastic moduli and the mass density each have different dependencies on depth. In a high-frequency approximation expressions are obtained for the displacements and the stresses. The condition that the surface tractions vanish at the free surface yields the dispersion equation which relates the surface wave velocity to the wavenumber. Conditions have been derived for a class of examples for which this equation yields a real valued surface wave velocity, and the displacement amplitudes decay exponentially with depth. Results for the surface wave velocity as a function of the wavenumber have been compared with numerical results which were obtained when the continuous inhomogeneity with depth is replaced by an equivalent layering. For some typical cases of increasing and decreasing material properties with depth, excellent agreement has been obtained between analytical and numerical results.