Transient waves in two semi infinite viscoelastic media separated by a plane interface

We consider the reflection and transmission of a transient, cylindrical, longitudinal wave by a plane interface between two semi infinite linear viscoelastic media.The four relaxation functions are assumed to be synchronous, but no further restrictions are placed on the viscoelastic behaviour of the two materials.By means of Laplace and Fourier transformation together with Cagniard-de Hoop's way of integrating we obtain expressions for the displacement components in terms of one- or three-dimensional integrals. The integrands contain a function which in general has to be found numerically from its Laplace transform.Computations are carried out for elastic materials and for two different standard linear solids. The displacement of the incident wave is found for different radii, and at two observation points the displacement components of the reflected longitudinal wave with its faster head wave and of the reflected transverse wave are computed.