On a Model Analogue of the Helmholtz Resonator in Homogenization

A boundary value problem for the Helmholtz equation in $\mathbb R^2$ with the Dirichlet boundary condition on a set of arcs is considered. This set is obtained from the circle by cutting out small-size openings that are arranged periodically and are close to each other. The relation between the size of the openings and the size of the boundary is established under which the boundary value problem considered is an analogue of the Helmholtz resonator with the Dirichlet boundary condition. The asymptotics of the poles with small imaginary parts is constructed for the analytic continuation of the solution to the perturbed boundary value problem.