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

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 (Formula presented.). 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 (Formula presented.) 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.