Cylindrical and Spherical Shear Waves in Nonhomogeneous Isotropic Viscoelastic Media

Abstract

This paper is concerned with the propagation of viscoelastic shear waves in nonhomogeneous isotropic media. Herein we develop formal methods of solving the linearized equations of viscoelastodynamics in two and three dimensions for nonhomogeneous Maxwell solids whose properties depend continuously on a single radial coordinate. These methods are developed for the linearized equations of motion formulated in terms of shear stresses, and are based on Cooper's and Reiss' extension to linear homogeneous viscoelastic media of the Karal–Keller technique. Shearing stesses are applied to the boundaries of cylindrical and spherical openings in the viscoelastic media, and formal asymptotic wave front expansions of the solutions are obtained. In both cases a modulated progressive wave that propagates with variable velocity is obtained. The modulation depends on the moduli of rigidity and viscosity, whereas the velocity depends only on the modulus of rigidity. When the viscosity parameter in our Maxwell element tends to infinity, the results reduce to the known results for nonhomogeneous elastic solids.