A radiation condition for the 2-D Helmholtz equation in stratified media

TL;DR

This paper establishes a new radiation condition for the 2-D Helmholtz equation in stratified media, ensuring unique and bounded solutions even with guided waves, expanding understanding of wave behavior in complex media.

Contribution

It introduces a Sommerfeld-Rellich radiation condition for the 2-D Helmholtz equation in stratified media and proves both uniqueness and existence of solutions under this condition.

Findings

Proved uniqueness of solutions with the new radiation condition.

Established existence of bounded solutions satisfying the radiation condition.

Extended analysis to media with guided waves.

Abstract

We study the 2-D Helmholtz equation in perturbed stratified media, allowing the existence of guided waves. Our assumptions on the perturbing and source terms are not too restrictive. We prove two results. Firstly, we introduce a Sommerfeld-Rellich radiation condition and prove the uniqueness of the solution for the studied equation. Then, by careful asymptotic estimates, we prove the existence of a bounded solution satisfying our radiation condition.