A two-complex-variable approach to wave diffraction by a no-contrast penetrable wedge

  • Valentin Kunz

Student thesis: Phd

Abstract

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 Wiener-Hopf or Sommerfeld-Malyuzhinets 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 far-field expansion is available. One of these unsolved problems is the diffraction problem resulting from the interaction of a monochromatic plane-wave with a right-angled no-contrast penetrable wedge. Here, `no-contrast' 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 wave-propagation speeds, however, are different. This right-angled no-contrast penetrable wedge diffraction problem is the problem the majority of this thesis is dedicated to, and one of our main goals is to obtain closed-form far-field asymptotics of the physical wave-fields 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 no-contrast right-angled penetrable wedge diffraction problem in Chapter 1 and give an informal description of its physical far-field 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 two-complex-variable Wiener-Hopf equation associated to the right-angled no-contrast penetrable wedge diffraction problem. By iteratively applying the one-complex-variable Wiener-Hopf technique to this equation, we obtain a system of two coupled integral equations which correct a previously published erroneous solution to the no-contrast right-angled penetrable wedge diffraction problem. In Chapter 4, we study the analyticity properties of the unknown two-complex-variable spectral functions involved in the Wiener-Hopf equation and unveil the spectral functions' singularities in $\mathbb{C}^2$. This allows for recovery of the physical far-field 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 plane-wave with a no-contrast penetrable wedge, that need not be right-angled.
Date of Award1 Aug 2024
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorJames Montaldi (Supervisor) & Raphael Assier (Supervisor)

Keywords

  • Mathematical Physics
  • Asymptotic Analysis
  • Diffraction Theory
  • Several Complex Variables

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