Scattering of elastic waves by an anisotropic sphere

Licentiate thesis, 2021

Scattering of a plane wave by a single spherical obstacle is the archetype of many scattering problems in physics and geophysics. Spherical objects can provide a good approximation for many real objects, and the analytic formulation for a single sphere can be used to investigate wave propagation in more complicated structures like particle composites or grainy materials, which may have application in non-destructive testing, material characterization, medical ultrasound, etc. The main direction of this thesis is to investigate an analytical solution for scattering of elastic waves by an anisotropic sphere in the special case with transverse isotropy. Throughout the thesis a systematic series expansion approach is used to derive displacement and traction fields outside and inside the sphere. For the surrounding isotropic medium such an expansion is made conveniently in terms of the traditional vector spherical wave functions. However, describing the fields inside the anisotropic sphere is more complicated since the classical methods are not applicable anymore. The first step is to describe the anisotropy in spherical coordinates, then the expansion inside the sphere is made in the vector spherical harmonics in the angular directions and power series in the radial direction. The governing equations inside the sphere provide recurrence relations among the unknown expansion coefficients. The remaining expansion coefficients outside and inside the sphere can be found using the boundary conditions on the sphere. Thus, this gives the scattered wave coefficients from which the transition T matrix can be found. This is convenient as the T matrix fully describes the sphere and is independent of the incident wave. The expressions of the general T matrix elements are complicated, but in the low frequency limit it is possible to obtain explicit expressions.

The T matrices may be used to solve more complicated problems like the wave propagation in polycrystalline materials. The attenuation and wave velocity in a polycrystalline material with randomly oriented transversely isotropic grains is thus investigated. These quantities are calculated analytically using the simple theory of Foldy and show a very good correspondence for low frequencies with previously published results and numerical computations with FEM.