Wiener-hopf technique and allied methods in a class of scattering problems

Abstract

This thesis is devoted to the study of a class of boundary value problems arising in the scattering of electromagnetic and acoustic waves by means of the Wiener–Hopf technique and related methods. The contents are organized into eight chapters. Chapter 1 provides a general introduction, surveying previous work relevant to the problems discussed. Chapter 2, titled Mathematical Preliminaries, contains derivations of various results and concepts, together with the notations used throughout the work. The remaining six chapters present the problems investigated. Chapter 3 considers a class of diffraction problems involving small-amplitude sinusoidal corrugations on the boundaries of the scatterers. This chapter is divided into three sections.

Section 3.1 examines the scattering of a plane wave by two unstaggered parallel half-planes with periodic wall perturbations. The solution uses the Wiener–Hopf technique together with a suitably designed perturbation scheme. The corresponding problem for a single half-plane with periodic wall perturbations is obtained as a limiting case by letting the spacing between the perturbed half-planes tend to zero. Section 3.2 addresses diffraction by a single periodically perturbed strip whose width is large compared to one wavelength. The problem is formulated as a three-part Wiener–Hopf equation, which is then reduced to integral equations of a special type. These integral equations are approximately solved under the assumption that the ratio of the strip width to the wavelength is large. Section 3.3 utilizes the scattered field produced by a single periodically perturbed half-plane under plane-wave excitation to investigate, in the high-frequency limit, the diffraction problem for a plane wave incident on two staggered parallel half-planes with periodic wall perturbations.

Chapter 4 is divided into two sections.

In Section 4.1, the field produced by the diffraction of a plane wave of short wavelength by two knife-edges is determined using the Wiener–Hopf technique, and the results are compared with the predictions of the uniform asymptotic theory of edge diffraction, in circumstances where mixed boundary conditions are prescribed on one of the scatterers. Two configurations are considered separately: (i) when the incident ray is not parallel to the line joining the two edges, and (ii) when the incident ray grazes the two edges. In both cases, a rigorous asymptotic expansion of the Wiener–Hopf solution is shown to be in complete agreement with the formal asymptotic solution provided by the uniform asymptotic theory of edge diffraction. Section 4.2 extends the analysis to the case where mixed boundary conditions—a Neumann condition on the bottom surface and a Dirichlet condition on the top surface (soft/hard)—are prescribed on both half-planes.

Chapter 5 considers the diffraction of a plane wave by two unstaggered, mixed-type parallel half-planes. A Dirichlet boundary condition is imposed on the facing surfaces of the half-planes, whereas a Neumann boundary condition is imposed on each of the two outer surfaces. The formulation leads to a system of 4×4 Wiener–Hopf equations, which is separated into two independent 2×2 Wiener–Hopf systems. The problem of solving these two systems is then reduced to solving two independent standard Wiener–Hopf equations involving unknown constants, by a procedure based on Wiener–Hopf arguments. The solutions to these Wiener–Hopf equations ultimately determine the solutions of the original systems explicitly, in terms of constants satisfying two independent linear algebraic systems. An analytic expression for the total field inside the waveguide formed by the half-planes—satisfying the desired boundary condition on the inner surfaces—is obtained in a computable form. Uniformly valid asymptotic forms of the total acoustic field outside the half-planes are also presented. Chapter 6 consists of two sections.

Section 6.1 re-examines the diffraction of a plane wave by three equidistant soft half-planes in light of the analysis employed in Chapter 5. The final section briefly examines the nature of the travelling wave in each of the two parallel-plate waveguides formed by three equidistant hard half-planes.

Chapter 7 addresses a special class of 2×2 matrix Wiener–Hopf problems, solved via a singular integral equation of a particular type. The analysis employs factorization of the matrix elements rather than factorization of the matrix as a whole. Two important and well-known particular cases of this class of 2×2 matrix Wiener–Hopf problems are considered at the end. The final chapter (Chapter 8) is devoted to the study of the scattering of a plane electromagnetic wave by a dielectric strip whose thickness is electrically small. The problem is formulated as an uncoupled system of three-part Wiener–Hopf equations by using a set of approximate boundary conditions. The resulting perturbed Wiener–Hopf equations are solved approximately for large electric width of the strip. An analytical formula is derived for the extinction cross-section of the strip. The radar cross-section is also computed, and variations of both physical quantities with respect to the angle of incidence are studied in special situations. The work reported in this thesis is partly based on the following papers and on papers that are being communicated for publication.