Focusing of waves in turbulent inhomogeneous media

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Abstract
Growth of large fluctuations in the amplitude of high‐frequency waves or shocks propagating through turbulent, inhomogeneous media is investigated stochastically with geometrical acoustics. Previous efforts use linearization which is not uniformly valid at large distances and large variations in the ray tube area. The present analysis retains the nonlinear terms crucially important to focusing and growth of singular fluctuations in the amplitude. It is shown, in two dimensions, that the fluctuations of the ray‐tube area grow exponentially and caustics occur on every ray, repeatedly, on a distance scale of a σ−2/3, where the fluctuations in ray directions are still small, O(σ2/3). Here, σ≪1 is the standard deviation and a is the correlation length of the wave velocity distribution in the medium. Probability densities for the occurrence of caustics are given. Also, the moments of the ray‐tube area distribution and amplitude related statistics are developed for distances well into the region of caustic formation. Finally, a method is presented for relating the theoretical predictions to measurements on an image plane.

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