A coupled-mode theory for two-dimensional exterior Helmholtz problems based on the Neumann and Dirichlet normal mode expansion

TL;DR

This paper introduces a new coupled-mode theory for 2D exterior Helmholtz problems, utilizing a hybrid Neumann-Dirichlet mode expansion to improve convergence in wave field analysis around inhomogeneities.

Contribution

It develops a novel coupled-mode approach based on a hybrid normal mode expansion that enhances convergence for exterior Helmholtz problems involving inhomogeneities.

Findings

The hybrid Neumann-Dirichlet mode expansion improves convergence.

The method effectively models wave fields in exterior Helmholtz problems.

Numerical results verify the accuracy of the proposed theory.

Abstract

This study proposes a novel coupled-mode theory for two-dimensional exterior Helmholtz problems. The proposed approach is based on the separation of the entire space R2 into a fictitious disk and its exterior. The disk is allocated in such a way that it comprises all the inhomogeneity; therefore, the exterior supports cylindrical waves with a continuous spectrum. For the interior, we expand an unknown wave field using normal modes that satisfy some auxiliary boundary conditions on the surface of the disk. For the interior expansion, we propose combining the Neumann and Dirichlet normal modes. We show that the proposed expansion sacrifices L2 orthogonality but significantly improve the convergence. Finally, we present some numerical verifications of the proposed coupled-mode theory.