Propagation of initially plane waves in the region of random caustics
type: Item
instance of: scholarly article
title: Propagation of initially plane waves in the region of random caustics
zbMATH Open document ID: 0558.73019
DOI: 10.1016/0165-2125(85)90007-1
author: Daniel I. Zwillinger
author: Benjamin S. White
published in: Wave Motion
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 74J99
Mathematics Subject Classification ID: 74A40
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 3890821
zbMATH Keywords: short distance expansion
zbMATH Keywords: high frequency plane wave
zbMATH Keywords: equations of ray theory
zbMATH Keywords: random, smoothly-varying, homogeneous and isotropic propagation speed
zbMATH Keywords: mean zero perturbation of a constant
zbMATH Keywords: forward Kolmogorov (Fokker- Planck) equation
zbMATH Keywords: joint statistics
zbMATH Keywords: arbitrary number of ray positions
zbMATH Keywords: random caustic region
zbMATH Keywords: two-point correlation function of intensity or energy
zbMATH Keywords: two-dimensional version
zbMATH Keywords: tested by Monte-Carlo simulations
MaRDI profile type: Publication
full work available at URL: 0165-2125(85)90007-1
OpenAlex ID: W1982869501
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description: scientific article; zbMATH DE number 3890821