A Symmetric Riemann-Hilbert Problem for Order-4 Vectors in Diffraction Theory

Abstract

Summary A class of n × n matrices of the form G = R1 + a R2 (a is a scalar Hfunction defined on a contour L, R1 and R2 are rational matrices), which admit a closed-form Wiener-Hopf factorization is analyzed. It is shown that the diffraction problem ( E-polarization) for a penetrable right-angled wedge with an electrically resistive and a perfectly magnetically conductive sides can be formulated as a Riemann-Hilbert problem for two order-4 vectors with a 4 × 4 matrix coefficient G = R1 + a R2. This problem reduces to two scalar Riemann-Hilbert problems and one vector problem for two order-2 vectors with a certain constraint. The order-2 vector problem is solved by employing the theory of the scalar Riemann-Hilbert problem on a genus-1 Riemann surface. The component Ez of the electric field and the diffraction coefficient are determined.