Wave diffraction by wedges having arbitrary aperture angle

The problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background. In fact, we may recognize significant research on this topic for more than one century. Despite this fact, up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate Sobolev space setting. In the present paper, we are considering the corresponding boundary value problems for the Helmholtz equation, with complex wave number, admitting combinations of Dirichlet and Neumann boundary conditions. The main ideas are based on a convenient combination of potential representation formulas associated with (weighted) Mellin pseudo-differential operators in appropriate Sobolev spaces, and a detailed Fredholm analysis. Thus, we prove that the problems have unique solutions (with continuous dependence on the data), which are represented by the single and double layer potentials, where the densities are solutions of derived pseudo-differential equations on the half-line

Semiclassical Estimates of Electromagnetic Casimir Self-Energies of Spherical and Cylindrical Metallic Shells

The leading semiclassical estimates of the electromagnetic Casimir stresses on a spherical and a cylindrical metallic shell are within 1% of the field theoretical values. The electromagnetic Casimir energy for both geometries is given by two decoupled massless scalars that satisfy conformally covariant boundary conditions. Surface contributions vanish for smooth metallic boundaries and the finite electromagnetic Casimir energy in leading semiclassical approximation is due to quadratic fluctuations about periodic rays in the interior of the cavity only. Semiclassically the non-vanishing Casimir energy of a metallic cylindrical shell is almost entirely due to Fresnel diffraction.Comment: 12 pages, 2 figure

Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System

We are concerned with rigorous mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the nonlinear wave system. This shock diffraction problem can be formulated as a boundary value problem for second-order nonlinear partial differential equations of mixed elliptic-hyperbolic type in an unbounded domain. It can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order. We establish a first global theory of existence and regularity for this shock diffraction problem. In particular, we establish that the optimal regularity for the solution is $C^{0,1}$ across the degenerate sonic boundary. To achieve this, we develop several mathematical ideas and techniques, which are also useful for other related problems involving similar analytical difficulties.Comment: 50 pages;7 figure

Spectral study of the Laplace-Beltrami operator arising in the problem of acoustic wave scattering by a quarter-plane

The Laplace-Beltrami operator on a sphere with a cut arises when considering the problem of wave scattering by a quarter-plane. Recent methods developed for sound-soft (Dirichlet) and sound-hard (Neumann) quarter-planes rely on an a priori knowledge of the spectrum of the Laplace-Beltrami operator. In this paper we consider this spectral problem for more general boundary conditions, including Dirichlet, Neumann, real and complex impedance, where the value of the impedance varies like $\textit{α/=r, r}$ being the distance from the vertex of the quarter-plane and α being constant, and any combination of these. We analyse the corresponding eigenvalues of the Laplace-Beltrami operator, both theoretically and numerically. We show in particular that when the operator stops being self-adjoint, its eigenvalues are complex and are contained within a sector of the complex plane, for which we provide analytical bounds. Moreover, for impedance of small enough modulus |α|, the complex eigenvalues approach the real eigenvalues of the Neumann case.R.C. Assier would like to acknowledge the support by UK EPSRC (EP/N013719/1).This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by Oxford University Press

Diffraction and scattering of high frequency waves

This thesis examines certain aspects of diffraction and scattering of high frequency waves, utilising and extending upon the Geometrical Theory of Diffraction (GTD). The first problem considered is that of scattering of electromagnetic plane waves by a perfectly conducting thin body, of aspect ratio O(k^1/2), where k is the dimensionless wavenumber. The edges of such a body have a radius of curvature which is comparable to the wavelength of the incident field, which lies inbetween the sharp and blunt cases traditionally treated by the GTD. The local problem of scattering by such an edge is that of a parabolic cylinder with the appropriate radius of curvature at the edge. The far field of the integral solution to this problem is examined using the method of steepest descents, extending the recent work of Tew [44]; in particular the behaviour of the field in the vicinity of the shadow boundaries is determined. These are fatter than those in the sharp or blunt cases, with a novel transition function. The second problem considered is that of scattering by thin shells of dielectric material. Under the assumption that the refractive index of the dielectric is large, approximate transition conditions for a layer of half a wavelength in thickness are formulated which account for the effects of curvature of the layer. Using these transition conditions the directivity of the fields scattered by a tightly curved tip region is determined, provided certain conditions are met by the tip curvature. In addition, creeping ray and whispering gallery modes outside such a curved layer are examined in the context of the GTD, and their initiation at a point of tangential incidence upon the layer is studied. The final problem considered concerns the scattering matrix of a closed convex body. A straightforward and explicit discussion of scattering theory is presented. Then the approximations of the GTD are used to find the first two terms in the asymptotic behaviour of the scattering phase, and the connection between the external scattering problem and the internal eigenvalue problem is discussed

Calculating conical diffraction coefficients

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Vertex diffracted edge waves on a perfectly conducting plane angular sector

The problem of electromagnetic scattering from a perfectly-conducting plane angular sector has been of interest for many years. An exact solution for this problem has been developed based on the separation of variables in sphero-conal coordinate system. In this solution, fields and currents are expressed in terms of scalar wave functions that are the solutions of a two parameter eigenvalue problem of two coupled spherical Lame differential equations and spherical Bessel functions. The resulting expressions are in the form of eigenfunction expansions. These expansions are slowly convergent and not suitable for high frequency scattering calculations. Despite their computational advantages, high-frequency modeling techniques fail to provide accurate results for many classes of problems. Lack of a vertex diffraction coefficient is a major factor that limits the accuracy of the high frequency diffraction techniques. Furthermore, the vertex-excited surface currents are strongly guided by the edges of the angular sector creating strong singularities on the current and charge densities. The singular behavior of the current density near the sharp edges of a scattering target is also known as the edge condition and should be modeled properly in numerical solutions to improve the accuracy. In this thesis, numerical diffraction coefficients are derived for vertex-diffracted edge waves induced on an infinitely-thin, perfectly conducting semi-infinite plane angular sector. The diffraction coefficients are formulated to be used in a purely high-frequency modeling of a scattering problem. The current density on the surface of the plane angular sector is modeled using the physical theory of diffraction (PTD). The vertex-diffracted currents are defined as the difference between the exact and PTD currents. The difference current is then modeled as a wave traveling away from the corner with unknown amplitude and phase factors. The unknown coefficients for the vertex-diffracted currents are calculated by using a least squares fit approximation. The vertex-diffracted currents are successfully modeled even for very narrow angular sectors for arbitrary directions of incidence. Illustrative examples are presented to demonstrate the substantial improvement provided by the vertex-diffracted currents to the accuracy of RCS patterns. Another aspect of the research in this thesis is the development of higher-order basis functions for the Method of Moments (MoM) solution. A set of divergence-conforming basis functions was developed to model the singular behavior of the surface currents near the edges and corners of an infinitely-thin, perfectly-conducting polygonal flat plate. The basis functions are derived by imposing the edge and corner conditions on the first order basis functions. It is demonstrated that, even though the new basis functions are not highest order complete, they provide accurate results without incurring complexity in the analysis or additional computational requirements