The multiple scattering of waves by weak random irregularities in the medium

Abstract
Papers by Lighthill (1953) and Fejer (1953) have treated multiple scattering by supposing a wave to be scattered any number of times in accordance with the cross-section for single scattering. This paper extends this idea, and uses the equation of energy transfer for radiation in a uniform scattering atmosphere to describe the variation of average intensity in a randomly inhomogeneous medium. In part I, the results of the single-scattering theory are reviewed, and an estimate is made of the conditions under which they should be correct. The justification for the treatment of multiple scattering by an equation of energy transfer is then discussed, and conditions under which it may be expected to be valid are obtained. In part II, the general solution of the equation of transfer for a spatially homogeneous radiation field, varying with time, is given first, and compared with Lighthill’s result for the angular distribution of radiation in terms of the length of path travelled. The much more difficult problem of a steady-state field with spatial variation has been treated by Chandrasekhar (1950), who gives many exact solutions for special types of scattering (such as isotropic and Rayleigh scattering). But his methods are not well suited to some other types, especially small-angle forward scattering. Most of part II is devoted to finding approximate solutions for this case, first generalizing Fejer’s solution for a slab of scattering medium which produces a small total angular deviation of the radiation, and then deriving an approximate partial differential equation of transfer to treat problems where the total angular deviation is not small. Methods of solving this equation by eigenfunction expansions are explained, and some numerical results are given, especially angular distributions of emergent and reflected radiation for a semi-infinite scattering region.

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