Scattering of elastic waves by a transversely isotropic sphere and ultrasonic attenuation in hexagonal polycrystalline materials

The scattering of elastic waves by a transversely isotropic sphere in an isotropic medium is considered. The elastodynamic equations inside the sphere are transformed to spherical coordinates and the displacement field is expanded in the vector spherical harmonics in the angular coordinates and powers in the radial coordinate. The governing equations inside the sphere then give recurrence relations among the expansion coefficients. Then all the remaining expansion coefficients for the fields outside and inside the sphere are found using the boundary conditions on the surface of the sphere. As a result, the transition (T) matrix elements are calculated and given explicitly for low frequencies. Using the T matrix and the theory of Foldy an explicit expression for the effective complex wave number of transversely isotropic (hexagonal) polycrystalline materials are presented for low frequencies. Numerical comparisons are made with previously published results and with recent FEM results and show a very good correspondence with FEM for low frequencies. As opposed to other published methods there is no limitation on the degree of anisotropy with the present approach