Problems Of Diffraction Theory Research Articles

Generalized Inverse Gamma Distribution and its Application in Reliability

In this article, we introduce a new reliability model of inverse gamma distribution referred to as the generalized inverse gamma distribution (GIG). A generalization of inverse gamma distribution is defined based on the exact form of generalized gamma function of Kobayashi (1991). This function is useful in many problems of diffraction theory and corrosion problems in new machines. The new distribution has a number of lifetime special sub-models. For this model, some of its statistical properties are studied. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is derived. We also demonstrate the usefulness of this distribution on a real data set.

Method of integral equations for systems of difference equations in diffraction theory

A system of difference equations of the first order with meromorphic coefficients (not necessary periodic ones) subject to certain conditions of symmetry is analysed. These conditions arise in model problems of diffraction theory. A method for the constructive solution of the system of difference equations is proposed. It consists of three steps. First, it requires factorization of a certain function. Then, scalar integral equations with the same kernel and different right-hand sides should be solved. The kernel of the equations has a fixed singularity, and the solution belongs to a class of functions with a power singularity. Finally, arbitrary constants are fixed from some conditions which guarantee the equivalence of the original system of difference equations and the integral equations. The method is illustrated by solving a model electromagnetic problem of a plane wave diffracted from an anisotropic impedance half-plane at oblique incidence.

Guest Editorial

The 18 articles in this special issue, designed as a tribute to Professor Leopold B. Felsen, focus on electromagnetic waves in complex environments. Specific topics that are covered include: fundamental electromagnetic wave theory; canonical problems in diffraction theory; ray-based asymptotics; inhomogeneous media; random media; Green's functions; ray-mode hybrids; complex source points and beams; spectral representations; time-domain techniques and phenomena; phase-space parameterization; inverse scattering and target identification; network and fields; and propagation in and around periodic structures.

On the approximate factorization of scalar and matrix Wiener–Hopf kernels with applications in structural acoustics

The Wiener–Hopf technique has proved to be an extremely powerful aid to solving problems in diffraction theory, and in particular for acoustic wave scattering. The key step in the procedure is the factorization of the Wiener–Hopf kernel into a product of two functions with (overlapping) semi-infinite regions of analyticity. However, for complex problems, such as those concerned with the interaction between fluids and structures, the representation of the scalar factors can have technical difficulties which make their computation both slow and delicate. Further, many important models of this type give rise to matrix kernels, for which no exact factorization technique has yet been devised. In this paper, a new procedure is presented to obtain approximate but explicit factorizations of both scalar and matrix kernels. As well as being simple to employ both analytically and numerically, the accuracy of the component factors can be increased almost indefinitely with little increase in numerical effort. Further, rigorous bounds on the error of these approximations are easy to find. The method is demonstrated by way of example, and the particular relevance of the new scheme to fluid/structure interaction problems is discussed.

Quasicrystals and mathematics

Abstract Quasicrystals, and more generally aperiodic crystals, generate fundamental mathematical problems in diffraction theory and the theory of aperiodic tilings. Their solution requires a reformulation of the basic concepts of mathematical crystallography.

Some Results on the Invertibility of Wiener-Hopf-Hankel Operators

A study is presented on the invertibility properties of scalar operators defined as the sum of a Wiener-Hopf and a Hankel operator on L_2(\mathbb R^+) with symbols in L_{\infty}(\mathbb R) . This study is based on the properties of a vector Wiener-Hopf operator naturally associated with each of the operators mentioned above. The results obtained are applied to problems in Diffraction Theory.

Mixed boundary value problems of the type of Sommerfeld's half-plane problem

SynopsisVarious physical problems in diffraction theory lead us to study modifications of the Sommerfeld half-plane problem governed by two proper elliptic partial differential equations in complementary ℝ3 half-spaces Ω± and we allow different boundary or transmission conditions on two half-planes, which together form the common boundary of Ω±.

On the possibility of an analytic solution of some classical problems of diffraction theory

The two well-known approaches (the method of integral equations and the Jones method) do not exhaust all possibilities of reducing a boundary value problem to the Wiener-Hopf-Fock equation solvable in explicit form. Examples of these possibilities for two classical problems of electrodynamics are given.

Statistical problems in diffraction theory

Various formulations of problems in the statistical theory of diffraction and wave propagation are discussed: excitation of fields by random sources, diffraction of partially coherent waves, diffraction of waves by bodies having random shapes or positions, and diffraction and propagation of waves in a randomly inhomogeneous medium. For each of these types of problem, physical problems from acoustics, radio astronomy, radiophysics, optics, and other branches of physics are given as examples, and the methods (mostly approximate ones) most widely used for solving them are indicated. Among the problems discussed are those of the diffraction content of the radiation transport equation and the back scattering enhancement effect observed in the diffraction of waves by small bodies immersed in a randomly irregular medium. Examples of statistical problems of mixed type are also given.

On an Expansion Problem Occurring in the Theory of Diffraction

This paper considers an eigenfunction expansion arising in certain problems in diffraction theory involving an impedance-typeboundary condition. The expansion in question is usually convergent only in part of the domain of interest and involves an finite set of orthogonal functions which do not form ucomplete set, in the sense that it is not in general possible to expand ugiven function in terms of them no matter how well-behaved the function may be. A theorem is established which asserts the validity of the expansion whenever uconverges. Also u more general expansion formula is obtained which is valid for all sufficiently general functions and which reduces to the basic expansion whenever the latter converges.

Statisticheskie problemy v teorii difraktsii

Statisticheskie problemy v teorii difraktsii

Boundary layer problems in diffraction theory

Boundary layer problems in diffraction theory

The excitation of a perfectly conducting half-plane by a dipole field

Starting with the solution of two scalar problems in diffraction theory derived by MacDonald in 1915, it is shown that the following problem may be solved. An electric or magnetic dipole is situated in the presence of a semi-infinite, perfectly conducting, thin plane. This problem may be solved by appealing to an appropriate representation of the electromagnetic field. When the formulation is complete, we are left merely with a two-dimensional Poisson equation. The method serves to show why some orientations of the dipole are simpler to handle than others.

The edge conditions and field representation theorems in the theory of electromagnetic diffraction

In 1897 Rayleigh(14) pointed out that it is possible to obtain a family of solutions to certain problems in diffraction theory, notably those involving plane obstacles, by the simple operation of differentiation of some one solution. This, of course, alters the nature of the solution, especially so in the neighbourhood of a sharp edge. For, at an edge the parent solution may be finite and yet its derivative can become infinite. Further, such a differentiated solution may produce an infinite total energy in the neighbourhood of the edge. It is, therefore, natural to ask what conditions are required to make the solutions of these problems physically and mathematically acceptable and thus define the boundary-value problem uniquely.