Propagation of initially plane waves in the region of random caustics (Q802356)

It is assumed that a high frequency plane wave is described by the equations of ray theory with a random, smoothly-varying, homogeneous and isotropic propagation speed, which is a small, O(\(\sigma)\) for \(0<\sigma \ll 1\), mean zero perturbation of a constant. In previous work by \textit{V. A. Kulkarny} and the second author [Phys. Fluids 25, 1770-1784 (1982; Zbl 0531.76059)], it has been shown that caustics form along every ray, on a large propagation distance scale of order \(O(\sigma^{-2/3})\), when angular deviations of the ray are still small but deviations in ray position and raytube area are O(1). Furthermore, the probability of caustic formation is, up to a single scale factor, as \(\sigma\) \(\downarrow 0\) a universal curve which defines the region of random caustics. In this paper, we derive a forward Kolmogorov (Fokker-Planck) equation for the joint statistics of an arbitrary number of ray positions and associated quantities in the random caustic region. Equations are derived for the two-point correlation function of intensity or energy in a plane orthogonal to the initial propagation direction of the wave. It is shown that for 'short' distances on the \(O(\sigma^{-2/3})\) scale the solution agrees with that of weak fluctuation theory. The two-dimensional version of the theory is tested by Monte-Carlo simulations.