The Multiple Scattering of Waves. I. General Theory of Isotropic Scattering by Randomly Distributed Scatterers

TLDR

Multiple scattering of particles has been studied with the Boltzmann equation, but wave scattering has largely been ignored, with prior work limited to the geometrical optics limit that neglects interference. This paper investigates multiple scattering of scalar waves by a random distribution of isotropic scatterers using a fully consistent wave‑theoretic approach. By averaging over a statistical ensemble of scatterer configurations, the authors derive equations for the mean wave function, its mean intensity, and the mean flux. The mean intensity satisfies an integral equation analogous to that for particle scattering, and the authors discuss its physical interpretation and possible extensions.

Abstract

While the problem of the multiple scattering of particles by a random distribution of scatterers has been treated classically through the use of the Boltzmann integro-differential equation, the corresponding problem of the multiple scattering of waves seems to have received scant attention. All previous treatments have considered the problem in the "geometrical optics" limit, where the rays are regarded as trajectories of particles and the treatment for particles is then applied, so that the interference phenomena in wave scattering are neglected. In this paper the problem of the multiple scattering of scalar waves by a random distribution of isotropic scatterers is considered in detail on the basis of a consistent wave treatment. The introduction of the concept of "randomness" requires averages to be taken over a statistical ensemble of scatterer configurations. Equations are derived for the average value of the wave function, the average value of the square of its absolute value, and the average flux carried by the wave. The second of these quantities satisfies an integral equation which has some similarities to the corresponding equation for particle scattering. The physical interpretation of the results is discussed in some detail and possible generalizations of the theory are outlined.