A Genus-3 Riemann-Hilbert Problem and Diffraction of a Wave by Two Orthogonal Resistive Half-Planes

Abstract

Diffraction of a plane electromagnetic wave (E-polarization) by two orthogonal electrically resistive half-planes is analyzed. The physical problem reduces to a Riemann-Hilbert problem in the real axis for four pairs of analytic functions \(\Phi^+_j(\eta)(\eta\ \in\ \rm C^+)$$ and $$\Phi^-_j(\eta) = \Phi^+_j (-\eta)(\eta\ \in\ {\rm C}^-),j = 1,2,3,4,\) where ℂ+ and ℂ are the upper and lower half-planes. It is shown that the problem is equivalent to two scalar Riemann-Hilbert problems on a plane and a Riemann-Hilbert problem on a genus-3 hyperelliptic surface subject to a certain symmetry condition. A closed-form solution is derived in terms of singular integrals and the genus-3 Riemann Theta function.