Transmission conditions for the Helmholtzequation in perforated domains

Abstract

We study the Helmholtz equation in a perforated domain . The domain Ωε. The domain Ωε is obtained from an open set Ω⊂ℝ³ by removing small obstacles of typical size ε > 0, the obstacles are located along a 2-dimensional manifold Γ0 ⊂Ω, We derive effective transmission conditions across Γ0 that characterize solutions in the limit ε→0O. We obtain that, to leading order O(ε0)=O(1), the perforation is invisible. On the other hand, at order O(ε1)=O(ε), inhomogeneous jump conditions for the pressure and the flux appear. The jumps can be characterized without cell problems by elementary expressions that contain the ε0-order limiting pressure function and the volume of the obstacles.