Influence of a Rough Thin Layer on the Potential

In this paper, we study the behavior of the steady-state potentials in a material composed by an interior medium surrounded by a rough thin layer and embedded in an ambient bounded medium. The roughness of the layer is supposed to be ε
^α
-periodic, ε being the small thickness of the layer and a is a parameter, that describes the roughness of the layer: if α <; 1 the layer is weakly rough, while if α > 1, the layer is very rough. Depending on the roughness of the membrane, the approximate transmission conditions are very different. We present and validate numerically the rigorous approximate transmissions proved by Poignard (Math. Meth. AppL Scu, vol. 32, pp. 435-453, 2009) and Ciuperca et al. (Research reports INRIA RR-6812 and RR-6975). This paper extends previous works of Perrussel and Poignard (IEEE Trans. Magn., vol. 44, pp. 1154-1157, 2008) in which the layer had a constant thickness.