Axisymmetric diffraction of a cylindrical transverse wave by a viscoelastic spherical inclusion

In this paper, the scattering and diffraction of a cylindrical transverse shear wave in a viscoelastic isotropic medium by a spherical heterogeneity is analytically solved. The waves are generated by the harmonic longitudinal oscillations of the cylinder walls. The spherical inclusion is located at the radial center of the cylinder and differs from the cylindrical material only in its complex shear modulus. Small amplitude motion is assumed, such that linear system theory is valid. By employing multi-pole expansions, the incident and scattered wave fields are each defined in both cylindrical and spherical coordinates allowing for the satisfaction of the boundary conditions at the surfaces of these multiply connected bodies. The solution involves an infinite sum of improper integrals, which are evaluated numerically. The wave field is determined for a hydrogel (alginate) bead suspended in a different hydrogel (agarose) that fills a glass test tube. Numerical examples showing the effect on displacement fields of varying the stiffness of the inclusion are presented. This solution is further validated with a finite element simulation showing excellent agreement with the analytic results.