Diffraction of Harmonic Shear Waves on an Elliptical Cavity Located in a Viscoelastic Medium

The problem of diffraction of harmonic shear waves on an elliptical cylindrical cavity located in a viscoelastic medium is considered. The relationship between stresses and deformations is taken into account using the integral Boltzmann–Volterra hereditary relation. The problem of a dynamic stress-strain state around an elliptical cavity in an unbounded viscoelastic medium under the action of harmonic shear waves is reduced to a plane problem (plane deformation) of viscoelasticity. The Lame equation reduces to the solution of the Mathieu equation with complex arguments. Its solution is expressed in terms of Mathieu functions. Numerical results are obtained for different frequencies of incident waves, angles of incidence of the transverse wave and the ratio of the axes of the elliptical cavity.