Analogues of the Helmholtz resonator in homogenization theory

Perturbed two-dimensional boundary-value problems are considered for Helmholtz's equation with Dirichlet and Neumann boundary conditions on a family of arcs obtained from the boundary of a bounded domain $\Omega$ by cutting out a large number of small holes distributed almost periodically and close to one another. Relations between the sizes of the openings and of the boundary ensuring that the solution of the perturbed problem converges to the solutions of the Dirichlet or the Neumann problem in $\Omega$ and outside $\overline\Omega$ are established. In the case when $\Omega$ is a disc, the holes are periodically distributed and the homogenized problems are Dirichlet problems, asymptotic formulae with respect to a small parameter $\varepsilon$ (characterizing the sizes of the openings and the distance between them) are constructed for the poles with small imaginary parts of the analytic continuation of the solution of the perturbed problem and their resonance behaviour is demonstrated.