Scattering of a scalar time-harmonic wave by a penetrable obstacle with a thin layer

This work looks at the asymptotic behaviour of the solution to the Helmholtz equation in a penetrable domain of 3 with a thin layer of thickness δ which tends to 0. We use the method of multi-scale expansion to derive and justify an asymptotic expansion of the solution with respect to the thickness δ up to any order. We then provide approximate transmission conditions of order two defined on an interface located inside the thin layer, with accuracy up to O(δ2), which allow one to take into account the influence of the thin layer.