Scalar wave scattering by a rough sphere: Bourret´s approximation for the coherent field of a point source

The average field of a point scalar source over a statistically rough sphere with a small surface impedance is analysed. The isotropic and statistically uniform inhomogeneities are assumed to be small and smooth. These assumptions allow the use of perturbation theory. All the values depending on spherical co-ordinates are represented as expansions of spherical functions. If substituted into the equivalent boundary conditions on the average surface, these expansions yield effective reflection coefficients for the spherical waves in the expansion of the coherent field. These coefficients depend on the effective impedance, the calculation of which is simple in the case of a large sphere.