Homogenization of Helmholtz equation in a periodic layer to study Faraday cage-like shielding effects

TL;DR

This paper analyzes the Helmholtz equation in a thin, periodic layered structure to understand Faraday cage-like shielding effects, deriving an effective limit problem as the layer thickness approaches zero.

Contribution

It introduces an asymptotic analysis of the Helmholtz equation in a complex layered medium with contrasting coefficients, using periodic unfolding to derive the limit problem.

Findings

Derived the limit Helmholtz problem with Dirichlet boundary conditions

Analyzed the effects of coefficient contrast in layered structures

Provided asymptotic formulas for thin periodic layers

Abstract

The work is motivated by the Faraday cage effect. We consider the Helmholtz equation over a 3D-domain containing a thin heterogeneous interface of thickness $\delta \ll 1$. The layer has a $\delta-$periodic structure in the in-plane directions and is cylindrical in the third direction. The periodic layer has one connected component and a collection of isolated regions. The isolated region in the thin layer represents air or liquid, and the connected component represents a solid metal grid with a $\delta$ thickness. The main issue is created by the contrast of the coefficients in the air and in the grid and that the zero-order term has a complex-valued coefficient in the connected faze while a real-valued in the complement. An asymptotic analysis with respect to $\delta \to 0$ is provided, and the limit Helmholtz problem is obtained with the Dirichlet condition on the interface. The periodic unfolding method is used to find the limit.