Impedance boundary conditions for statistically rough surfaces

It is shown that for an electromagnetic field incident on a perfectly conducting surface having small geometrical irregularities which are distributed at random but in a statistically uniform and isotropic manner, the boundary condition can be replaced by a generalized impedance condition applied at a neighbouring mean surface. The surface impedance is a tensor function of the direction at which the field is incident as well as of the statistical properties of the irregularities, but simplifies in certain particular cases. Although the detailed analysis is carried out for a mean surface which is flat, the boundarycondition is applicable to a curved surface providing the radii of curvature are large in comparison with the wavelength. It is believed that this approach is of value in studying the effect of minor surface roughnesses on the scattering of electromagnetic waves.