Boundary composed of small Helmholtz resonators: asymptotic approach

We consider the solution of the two-dimensional Neumann problem for the Helmholtz equation in a complex region composed of a square resonator with large number of smaller square resonators connected to it through small apertures along one side. The sizes of the apertures and distances between the neighbour apertures tend to zero. We use the method of matching of asymptotic expansions of solutions. By directing the number of attached small resonators to infinity, we obtain a problem for the Laplacian in the main square with energy-dependent boundary condition.