Homogenization of Helmholtz equation in a periodic layer to study Faraday cage-like shielding effects
S. Aiyappan, Georges Griso, Julia Orlik · Complex Variables and Elliptic Equations · 2023
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 δ≪1. The layer has a δ-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 δ 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 δ→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.