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arXiv · 2212.04671

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

S Aiyappan·Georges Griso·Julia Orlik
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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.

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S Aiyappan, Georges Griso, Julia Orlik. 2022-12-09. Homogenization of Helmholtz equation in a periodic layer to study Faraday cage-like shielding effects. https://doi.org/10.1080/17476933.2022.2155637

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