Electromagnetic scattering from an anisotropic impedance half-plane at oblique incidence: the exact solution

Abstract

SummaryScattering of a plane electromagnetic wave from an anisotropic impedance half-plane at skew incidence is considered. The two matrix surface impedances involved are assumed to be complex and different. The problem is solved in closed form. The boundary-value problem reduces to a system of two first-order difference equations with periodic coefficients subject to a symmetry condition. The main idea of the method developed is to convert the system of difference equations into a scalar Riemann-Hilbert problem on a finite contour of a hyperelliptic surface of genus 3. A constructive procedure for its solution and the solution of the associated Jacobi inversion problem is proposed and described in detail. Numerical results for the edge diffraction coefficients are reported.

The Problem of Linear Conjugation on a Closed Riemann Surface

“…This is a homogeneous problem "adjoint" to problem (2), and its solutions are searched for in a so-called "adjoint" class. Its solution can be constructed in the same way as a solution of homogeneous problem (12).…”

“…., dw h (p) is a basis of Abelian differentials of the first kind. If the right-hand side of (77) is multiple of the integer divisor (dΨ0) J D , then from (77) we can obtain a particular solution of non-homogeneous problem (2). This leads to a system of linear equations for finding the numbers c 1 , .…”

“…is the theorem on the index of problem(2). If h = 0, then one of the numbers l and l is equal to zero and the other can be calculated using (40).If h ≥, it is possible that the both numbers l and l are positive.…”

Riemann–Hilbert Problem for Automorphic Functions and the Schottky–Klein Prime Function

“…The solution to the classical inversion problem (4.27) always exists [18] and can be expressed through the M zeros of the associated Riemann theta-function of the surface R [1][2][3]. Notice that since (4.29) does not depend on the integers n ν , they can be chosen arbitrarily say, n 1 = · · · = n M = 0, provided the points q ν and the integers m ν solve the problem (4.27).…”

Diffraction of a skew incident plane electromagnetic wave by a wedge with axially anisotropic impedance faces

“…Furthermore, just like the Wiener‐Hopf technique, which was regarded for many decades as not being suitable for wedge problems but now has outperformed the Sommerfeld‐Malyuzhinets technique in certain aspects, the Sommerfeld‐Malyuzhinets technique could one day get the upper hand again. In addition, a closed‐form explicit solution to the problem under study ought to be found yet (for such solutions in special cases see, for instance, Lyalinov and Zhu [1999, 2003a] and Antipov and Silvestrov [2006]). Hence this solution procedure deserves to be presented to the electromagnetics community.…”