scholarly journals Function-theoretic solution to a class of dual integral equations and an application to diffraction theory

1964 ◽  
Vol 21 (4) ◽  
pp. 269-283 ◽  
Author(s):  
Robert A. Schmeltzer ◽  
Myrna Lewin
Keyword(s):  
Integral Equations ◽  
Diffraction Theory ◽  
Dual Integral Equations ◽  
Theoretic Solution

A method for the exact solution of a class of two-dimensional diffraction problems

1951 ◽  
Vol 205 (1081) ◽  
pp. 286-308 ◽  
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Integral Equations ◽  
Scattered Field ◽  
Plane Waves ◽  
Angular Spectrum ◽  
Diffraction Theory ◽  
General Interest ◽  
Two Dimensional ◽  
Dual Integral Equations

In the last few years Copson, Schwinger and others have obtained exact solutions of a number of diffraction problems by expressing these problems in terms of an integral equation which can be solved by the method of Wiener and Hopf. A simpler approach is given, based on a representation of the scattered field as an angular spectrum of plane waves, such a representation leading directly to a pair of ‘dual’ integral equations, which replaces the single integral equation of Schwinger’s method. The unknown function in each of these dual integral equations is that defining the angular spectrum, and when this function is known the scattered field is presented in the form of a definite integral. As far as the ‘radiation’ field is concerned, this integral is of the type which may be approximately evaluated by the method of steepest descents, though it is necessary to generalize the usual procedure in certain circumstances. The method is appropriate to two-dimensional problems in which a plane wave (of arbitrary polarization) is incident on plane, perfectly conducting structures, and for certain configurations the dual integral equations can be solved by the application of Cauchy’s residue theorem. The technique was originally developed in connexion with the theory of radio propagation over a non-homogeneous earth, but this aspect is not discussed. The three problems considered are those for which the diffracting plates, situated in free space, are, respectively, a half-plane, two parallel half-planes and an infinite set of parallel half-planes; the second of these is illustrated by a numerical example. Several points of general interest in diffraction theory are discussed, including the question of the nature of the singularity at a sharp edge, and it is shown that the solution for an arbitrary (three-dimensional) incident field can be derived from the corresponding solution for a two-dimensional incident plane wave.

scholarly journals A Pair of Dual Integral Equations Occurring in Diffraction Theory

1962 ◽  
Vol 13 (2) ◽  
pp. 179-187 ◽  
Author(s):  
J. Burlak
Keyword(s):  
Integral Equations ◽  
Diffraction Theory ◽  
Dual Integral Equations ◽  
Special Cases ◽  
Image Position

Dual integral equations of the formwhere f(x) and g(x) are given functions, ψ(x) is unknown, k≧0, μ, v and α are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = −μ, α = 0 and v = μ = 0, 0<α2<1 were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case μ = v, k = 0.

scholarly journals A generalisation of the Erdélyi-Kober operators

1970 ◽  
Vol 17 (2) ◽  
pp. 139-148 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Integral Equations ◽  
Hankel Operator ◽  
Diffraction Theory ◽  
Dual Integral Equations

In this paper we introduce a generalised Hankel operator and generalised Erdélyi-Kober operators and deduce some relations between them. The operators are then applied to obtain solutions to some dual integral equations which have applications in diffraction theory.

scholarly journals DUAL INTEGRAL EQUATIONS TECHNIQUE IN ELECTROMAGNETIC WAVE SCATTERING BY A THIN DISK

2009 ◽  
Vol 16 ◽  
pp. 107-126 ◽  
Author(s):  
Mikhail V. Balaban ◽  
Ronan Sauleau ◽  
Trevor Mark Benson ◽  
Alexander I. Nosich
Keyword(s):  
Electromagnetic Wave ◽  
Integral Equations ◽  
Wave Scattering ◽  
Thin Disk ◽  
Dual Integral Equations ◽  
Electromagnetic Wave Scattering

scholarly journals The Approximate Solution of Dual Integral Equations by Variational Methods

1958 ◽  
Vol 11 (2) ◽  
pp. 115-126 ◽  
Author(s):  
B. Noble
Keyword(s):  
Approximate Solution ◽  
Variational Methods ◽  
Integral Equations ◽  
Separation Of Variables ◽  
Circular Disc ◽  
Cylindrical Coordinates ◽  
Dual Integral Equations ◽  
Particular Solution ◽  
Image Position

The classic application of dual integral equations occurs in connexion with the potential of a circular disc (e.g. Titchmarsh (9), p. 334). Suppose that the disc lies in z = 0, 0≤ρ≤1, where we use cylindrical coordinates (p, z). Then it is required to find a solution ofsuch that on z = 0Separation of variables in conjunction with the conditions that ø is finite on the axis and ø tends to zero as z tends to plus infinity yields the particular solution.

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