Numerical solution of a singular integral equation in random rough surface scattering theory

Abstract A one-dimensional singular integral equation which appeared in a previous paper on random rough surface scattering theory ( J. Math. Phys. 13 , 1903 (1972) is solved numerically using quadratic splines. Its solution yields an approximation to the coherent (specular) scattered intensity for plane wave incidence on the surface. This approximate scattered intensity is plotted versus the Rayleigh factor Σ = κ 0 σ cos θ i , where κ i is the wavenumber of the incident plane wave, σ is the surface root mean square height, and θ i is the angle of the incident plane wave. For values of Σ > 1 this approximation yields more coherent intensity than the Kirchhoff approximation.