Boundary element and transfer operator methods for multi-component wave systems

In this thesis, exact and semiclassical approaches are derived for predicting wave energy distributions in coupled cavities with variable material properties. These approaches are attractive because they can be extended to more complex built-up systems. For the exact treatment, we describe a multi-component boundary element method. We point out that depending on the boundary conditions and the number of interfaces between sub-components, it may be advantageous to use a normal derivative method to set up the integral kernels. We describe how the arising hypersingular integral kernels can be reduced to weakly singular integral and then using the piecewise constant collocation method. The normal derivative method can be used to minimise the number of weakly-singular integrals thus leading to BEM formulations which are easier to handle. The second component of this work concerns a novel approach for finding an exact formulation of the transfer operator. This approach is demonstrated successfully for a disc with boundary conditions changing discontinuously across the boundary. Such an operator captures the diffraction effects related to the change of boundary conditions. So it incorporates boundary effects such as diffraction and surface waves. A comparison between the exact results from the BEM against the exact transfer operator shows good agreement between both categories. Such an exact operator converges to the semiclassical Bogomolny transfer operator in the semiclassical limit. Having seen how the exact transfer operator behaves for a unit disc, a similar approach is adapted for the coupled cavity configuration resulting in the semiclassical transfer operator. Our formulation for the transfer operator is applicable not only for the quantization of a system, but also to recover the Green function