The theory of functions involving a single complex variable has proved to be a powerful tool in the study of diffraction problems, and many well established methods, such as the WienerHopf or SommerfeldMalyuzhinets technique, rely on it. However, some important diffraction problems seem to be out of reach of these classical methods, and remain unsolved in the sense, that no clear analytical solution or farfield expansion is available. One of these unsolved problems is the diffraction problem resulting from the interaction of a monochromatic planewave with a rightangled nocontrast penetrable wedge. Here, `nocontrast' means that either the densities of the materials inside and outside of the wedge, respectively, coincide (acoustic setting), or that these materials' electric permittivities or magnetic permeabilities, respectively, coincide (electromagnetic setting). The wavepropagation speeds, however, are different. This rightangled nocontrast penetrable wedge diffraction problem is the problem the majority of this thesis is dedicated to, and one of our main goals is to obtain closedform farfield asymptotics of the physical wavefields associated to it. To achieve this, we will use the theory of several complex variables. The thesis is divided into six Chapters. We introduce the nocontrast rightangled penetrable wedge diffraction problem in Chapter 1 and give an informal description of its physical farfield as predicted by the geometrical theory of diffraction. In Chapter 2, we outline aspects of the theory of functions of several complex variables which we will use throughout the remainder of this thesis. In Chapter 3, we derive the twocomplexvariable WienerHopf equation associated to the rightangled nocontrast penetrable wedge diffraction problem. By iteratively applying the onecomplexvariable WienerHopf technique to this equation, we obtain a system of two coupled integral equations which correct a previously published erroneous solution to the nocontrast rightangled penetrable wedge diffraction problem. In Chapter 4, we study the analyticity properties of the unknown twocomplexvariable spectral functions involved in the WienerHopf equation and unveil the spectral functions' singularities in $\mathbb{C}^2$. This allows for recovery of the physical farfield in Chapter 5. Particularly, we show that the diffraction coefficients associated with the cylindrical and lateral diffracted waves can be expressed in terms of these spectral functions evaluated at some given points. Finally, in Chapter 6, we give some ideas of how the framework developed throughout this thesis might be applied to the diffraction problem resulting from the interaction of a monochromatic planewave with a nocontrast penetrable wedge, that need not be rightangled.
Date of Award  1 Aug 2024 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  James Montaldi (Supervisor) & Raphael Assier (Supervisor) 

 Mathematical Physics
 Asymptotic Analysis
 Diffraction Theory
 Several Complex Variables
A twocomplexvariable approach to wave diffraction by a nocontrast penetrable wedge
Kunz, V. (Author). 1 Aug 2024
Student thesis: Phd