On the incoherent scattering of an acoustic or electromagnetic wave

A random configuration of objects in space, or a stochastically rough boundary, is considered to scatter an incident acoustic or electromagnetic wave having harmonic time dependencee^iwt. In the case of a stochastic surface, Beckmann has compared the Kirchhoff solution with his approach, which employs random walk. The latter approach is used to demonstrate the Rayleigh-distributed amplitude of a field scattered by a very rough surface. This demonstration requires the conjecture that large standard deviations in the random phases of the scattered elementary waves result in an incoherent scattered field. Beckmann's conjecture has not been rigorously proven. However, in this paper, incoherence of the scattered field and broad distributions, over many cycles, in the phases of the elementary waves are both shown to be implied by a third condition, which is defined. Furthermore, the random phase of an incoherent field is shown to be statistically independent of its amplitude and uniformly distributed on a 2-rad interval.