Propagation of radiation in time-dependent three-dimensional random media

In Ref. [1] (Appendix A) we derived equations governing the frequency and spatial spectrum of radiation propagating in three-dimensional time-dependent random media with randomly varying sound speed c (x, t). From the spectral equations we determine equations for the energy flux in both the forward and backward directions. We consider media that are spatially homogeneous and isotropic and stationary in time. In order to allow an independence assumption the analysis is restricted to fluctuations that satisfy the conditions τ_μ ≫ L_z/c₀ and τ_μ ≪ L_FS/ c₀ where τ_μ is the characteristic time of the fluctuations, k₀ is the mean radiation wavenumber, L_z is the characteristic correlation length of the random fluctuations in the mean propagation direction and LFS is a mean scattering length. We consider various values of γ = (k₀ L_z)²/2. When γ ≪ 1 we find the usual radiation transfer equations. When γ ≫ 1, but back-scattering can be neglected, we find the forward-scattering equations. We also consider γ ≫ 1, when back-scattering cannot be neglected. We consider as initial boundary conditions a plane wave and an infinite incoherent source. We present numerical solutions for γ ≪ 1, γ = O (1) and γ ≫ 1 using a simple Gaussian form for the fluctuation correlation function.