Multiple scattering by a collection of randomly located obstacles Part III: Theory - slab geometry

In this paper, scattering of electromagnetic waves by discrete, randomly distributed objects inside a (finite thickness or semi-infinite) slab is addressed.
In general, the non-intersecting scattering objects can be of arbitrary form, material and shape with a number density of $n_0$ (number of scatterers per volume).
The main aim of this paper is to calculate the coherent reflection and transmission characteristics for this configuration.
Applications of the results are found at a wide range of frequencies (radar up to optics), such as attenuation of electromagnetic propagation in rain, fog, and clouds etc. The integral representation of the solution of the deterministic problem constitutes the underlying framework of the stochastic problem.
Conditional averaging and the employment of the Quasi Crystalline Approximation lead to a system of integral equations in the unknown expansion coefficients. With a uniform distribution of scatterers the analysis simplifies to a system of integral equations in the depth variable. Explicit solutions for tenuous media and low frequency approximations can be obtained for spherical obstacles.