Impedance boundary conditions for the scattering of time-harmonic waves by rapidly varying surfaces

A method to build impedance boundary conditions incorporating the effect of rapid variations of a perfectly conducting surface oil the scattering of a scalar, E-polarized, time-harmonic electromagnetic wave is presented. The amplitude and the extent of the variations are assumed to be comparable to each other and small as compared to the wavelength. The derivation of the impedance boundary conditions is based on a decomposition of the field in two parts. The first part describes the overall behavior of the wave and the second one deals with its small scale variations. The effective boundary conditions are rigorously constructed for periodic surfaces presenting a large-scale global periodicity to suppress the boundary effects and a small local period to describe the rapid variations. Numerical examples prove that the method can even be heuristically extended to more general problems. In this respect, there are reported some results related to the numerical treatment of small details on a smooth surface and of rough surfaces without resorting to refined meshes.