Barrier composed of perforated resonators and boundary conditions

Abstract

We consider the Laplace operator with the Neumann boundary condition in a two-dimensional domain divided by a barrier composed of many small Helmholtz resonators coupled with both parts of the domain through small windows of diameter \( 2a \). The main terms of the asymptotic expansions in \( a \) of the eigenvalues and eigenfunctions are considered in the case in which the number of the Helmholtz resonators tends to infinity. It is shown that such a homogenization procedure leads to some energy-dependent boundary condition in the limit. We use the method of matching the asymptotic expansions of boundary value problem solutions.