Single and multiple acoustic scattering with applications to bioengineering

Abstract

The present doctoral thesis aims at presenting development and application of analytical and numerical techniques to the solution of multiple scattering problems in connection with biomedical applications. Mathematically, the exact (deterministic) multiple scattering problem is easily formulated: it is an exterior boundary-value problem (with a radiation conditions at infinity) where the boundary is not simply connected. However the problem is not easy to solve, due mainly to the complicated geometry. The framework of this work is inspired by the interesting biomedical problem of studying the response of a system consisted of a structure (simulating a human organ) hosting an inhomogeneity, to the incidence of a plane or point-source acoustic wave. Consequently we are restricted in the case of two scatterers, the one inside the other and we selected to handle the case of spheroidal shapes having in mind the ability of the spheroidal coordinate system to be easily adapted to several biomedical systems of interest. The effort in this thesis begins by exploiting classical methods concerning PDE’s solutions in spheroidal systems and progressively leads to a totally new analytical method permitting to the several physical fields involved in the scattering process to be represented through a very convenient novel set of eigensolutions. More precisely, our contributions are presented in the following four chapters v