Diffusion equation derived from space-time transport equation

The diffusion equation is derived from the ordinary space-time transport equation having a scattering cross section of the form σ(Ω·Ω′), Ω and Ω′ being the azimuth vectors of incident and scattered waves. The condition of applicability is examined in detail. It is shown that the diffusion equation necessarily becomes first order in time in contrast to a recent paper on a similar subject in which it is given as second order in time. This is further confirmed for the case of isotropic scattering where an exact analytical solution is obtainable. The boundary conditions on a surface of medium discontinuity are also derived in the diffusion approximation and, in this connection, the pulse-width broadening in random media is investigated in terms of the pulse moments, which are obtained by solving boundary-value problems of a simple equation in space.