A Non-Reciprocal Elliptic Spectral Solution of the Right-Angle Penetrable Wedge Transmission Problem

TL;DR

This paper develops an explicit elliptic spectral solution framework for a right-angle wedge transmission problem, analyzing reciprocity and identifying necessary modifications for physically consistent plane-wave solutions.

Contribution

It provides a novel explicit theta-function uniformization and Mittag-Leffler construction of elliptic RH solutions for the wedge problem, along with a reciprocity diagnostic.

Findings

Minimal one-point plane-wave forcing violates reciprocity.

Hardy pairing enforces reciprocity but nullifies scattered fields.

Additional forcing or pairing modifications are needed for physical reciprocity.

Abstract

We study the two-dimensional time-harmonic scalar transmission problem for an impedance-matched penetrable right-angle wedge: the exterior medium has wavenumber k_0 and the interior sector |theta| < pi/4 has wavenumber k_1 = nu*k_0 with nu > 1, with continuity of the total field and its normal derivative across each face. A Sommerfeld-Malyuzhinets reduction leads to a 2x2 matrix Riemann-Hilbert (RH) problem on the Snell surface Sigma_nu. For the right angle the surface has genus one, and we give an explicit theta-function uniformization and a closed-form Mittag-Leffler construction of the full family of elliptic RH solutions with finite forcing (prescribed poles and residues), subject to a single residue-sum constraint encoding the Meixner edge condition. We then consider the additional forcing data required to model plane-wave incidence. Numerical reciprocity tests show that the minimal one-point plane-wave prescription does not yield a physically closed solution: the natural sheet-swap pairing u# = u + omega_2 produces nontrivial scattered densities but violates the far-field reciprocity benchmark, whereas the Hardy pairing u# = omega_1 - u enforces reciprocity but collapses the scattered field to zero. The paper therefore provides an exact elliptic spectral framework and a reproducible reciprocity diagnostic, while identifying what must be added (multi-point forcing and/or modified pairing incorporating both wedge faces) to obtain a reciprocity-consistent plane-wave solution.