Transient elastic wave scattering by a rigid spherical inclusion

The transient scattering of a stress wave by a rigid spherical inclusion embedded in an infinite elastic medium is considered. The Fourier synthesis technique is used to calculate the response of the inclusion and the interfacial stresses. This method of solution yields an exact solution for the inclusion response. The interfacial stresses are obtained in the form of an infinite series that is suitable for the long-time response but that does not converge rapidly for early time. Hence, an early time analysis is performed by reformulating the problem in the Laplace transform domain; by utilizing Watson's transformation, the early time stress response in the shadow region is obtained. Results in the shadow region using both techniques emphasize the importance of carrying out the early time analysis. The interfacial stresses in the illuminated zone using only the Fourier synthesis method are also given.