Diffusion equation derived from the space–time transport equation in anisotropic random media

Abstract

The diffusion equation is derived from the ordinary space–time transport equation with an arbitrary scattering cross section σ (Ω‖Ω′) ≠σ (Ω′‖Ω). After various equations are Fourier transformed with respect to the space–time coordinates, the transport equation is shown to be reduced to an integral equation of the Fredholm type with respect to the angle of scattering, with the variables of Fourier transformation kept constant. The solution is then expanded in the eigenfunction series and, under the diffusion condition, its convergence of series is generally good enough to keep only the first term of the series except for the case where the waves are scattered mostly in forward direction, as in turbulent air. The boundary equations on the surface of medium discontinuity are then obtained, also based on the eigenfunction expansion of the quantities continuous across the boundary surface. A simple application is made to the pulse wave propagation in a space of semi‐infinite scattering medium.