PHD Thesis: Existence, Uniqueness & Explicit Bounds for Scattering By Rough Surfaces

TL;DR

This thesis investigates mathematical models of wave scattering by unbounded rough surfaces, establishing existence, uniqueness, and explicit bounds for solutions in various boundary value problems using variational and integral equation methods.

Contribution

It provides new well-posedness results for scattering problems involving rough surfaces, including explicit bounds and analysis for Lipschitz surfaces at arbitrary frequencies.

Findings

Proved well-posedness of boundary value problems under Lipschitz surface conditions.

Established explicit bounds for solutions in scattering problems.

Demonstrated solution existence and uniqueness for various boundary conditions.

Abstract

We study 4 problems in the area of scattering of time harmonic acoustic or electromagnetic waves by unbounded rough surfaces/inhomogeneous layers. Specifically we study: i) a boundary value problem (BVP) for the Helmholtz equation, in both 2 and 3 dimensions, modelling scattering of time harmonic waves due to a source that lies within a finite distance of the boundary and which decays along the boundary, by a layer of spatially varying refractive index above an unbounded rough surface on which the field vanishes; ii) a BVP for the Helmholtz equation with an impedance boundary condition, in 2 and 3 dimensions, modelling the scattering of time harmonic acoustic waves due to a source that lies within a finite distance of the boundary and which decays along the boundary, by an unbounded rough impedance surface; iii) a problem of scattering of time harmonic waves by a layer of spatially varying refractive index at the interface between semi-infinite half-spaces of fixed positive refractive index (the waves arising due to a source that lies within a finite distance of the layer and which decays along the layer); iv) a BVP for the Helmholtz equation with a Dirichlet boundary condition, in 3 dimensions, modelling the scattering of time harmonic acoustic waves due to a point source, by an unbounded, rough, sound soft surface. We study i), ii) and iii) by variational methods; via analysis of equivalent variational formulations we prove these problems to be well-posed under various restrictions on the surface (must be Lipschitz), the frequency, the rate of change of the refractive index etc. We study iv) via a Brakhage-Werner type integral equation formulation, based on an ansatz for the solution as a combined single- and double-layer potential. We establish that when the rough surface is the graph of a Lipschitz function, the problem is well-posed for arbitrary frequency.