The radiation condition for Helmholtz equations above (locally perturbed) periodic surfaces

TL;DR

This paper introduces a new Floquet-Bloch transform-based method to establish a radiation condition for Helmholtz equations above periodic surfaces, improving the understanding of wave scattering in periodic media and enabling potential extension to 3D problems.

Contribution

The paper develops the first Floquet-Bloch transform method for radiation conditions in periodic media, enhancing the Sommerfeld condition and extending applicability to complex structures.

Findings

Improved radiation condition: d7 rac{\u2202 u}{d7 r}-i k u=O(r^{-3/2})

Method applicable to periodic media and extendable to 3D

Provides a new analytical tool for scattering problems in periodic structures.

Abstract

The radiation condition is the key question in the mathematical modelling for scattering problems in unbounded domains. Mathematically, it plays the role as the "boundary condition" at the infinity, which guarantees the well-posedness of the mathematical problem; physically, it describes the far-field asymptotic behaviour of the physical waves. In this paper, we focus on the radiation conditions for scattering problems above (locally perturbed) periodic surfaces. According to Hu et al. (2021), the radiating solution satisfies the Sommerfeld radiation condition: $$\frac{\partial u}{\partial r}-i k u=o(r^{-1/2}).$$ Although there are literature which have studied this problem, there is no specific method for dealing with periodic structures. Due to this reason, the important properties for the periodic structures may be ignored. Moreover, the existing method is not extendable to bi-periodic structures in three dimensional spaces. In this paper, we study the radiation condition for the time-harmonic scattering problem with periodic surfaces, which is modelled by the Helmholtz equation. We introduce a novel method based on the Floquet-Bloch transform, which, to the best of the author's knowledge, is the first method that works particularly for periodic media. With this method, we improve the Sommerfeld radiation condition for the scattered field from periodic media to: $$\frac{\partial u}{\partial r}-i k u=O(r^{-3/2}).$$ More importantly, the prospect of extending this method to 3D cases is optimistic.