The wave equation governing the propagation of waves in a medium, often stochastic, that scatters radiation predominantly at small angles to the forward direction θ = 0 is usually approximated by a parabolic form. The parabolic equation neglects all scattering contributions except those in a narrow cone around θ = 0. However, in many cases of interest, e.g., optics in turbulent air, there are also small contributions at larger angles. Although they are small, these contributions can accumulate over larger distances and thus render the parabolic equation invalid. Here, the wave equation is not approximated but is recast into two coupled quasi-parabolic equations in a forward field and a backward field. Solutions for mean fields and coherence functions are obtained with some well-established stochastic approximations that decouple wave–medium ensemble averages.
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