Sound absorption by perforated walls along boundaries

We analyze the Helmholtz equation in a complex domain. A sound absorbing structure at a part of the boundary is modelled by a periodic geometry with periodicity $\epsilon >0$. A resonator volume of thickness $\epsilon$ is connected with thin channels (opening ${\epsilon }^3$) with the main part of the macroscopic domain. For this problem with three different scales we analyze solutions in the limit $\epsilon \to 0$ and find that the effective system can describe sound absorption.