1. AN EFFECTIVE FRACTIONAL PARAXIAL WAVE EQUATION FOR WAVE-FRONTS IN RANDOMLY LAYERED MEDIA WITH LONG-RANGE CORRELATIONS.
- Author
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GOMEZ, CHRISTOPHE
- Subjects
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ORDINARY differential equations , *PSEUDODIFFERENTIAL operators , *TRAVEL time (Traffic engineering) , *MATHEMATICAL analysis , *THEORY of wave motion , *REACTION-diffusion equations , *WAVE equation , *POWER law (Mathematics) - Abstract
This work concerns the asymptotic analysis of high-frequency wave propagation in randomly layered media with fast variations and long-range correlations. The analysis takes place in the three-dimensional physical space and weak-coupling regime. The role played by the slow decay of the correlations on a propagating pulse is twofold. First we observe a random travel time characterized by a fractional Brownian motion that appears to have a standard deviation larger than the pulse width, which is in contrast with the standard O'Doherty-Anstey theory for random propagation media with mixing properties. Second, a deterministic pulse deformation is described as the solution of a paraxial wave equation involving a pseudodifferential operator. This operator is characterized by the autocorrelation function of the medium fluctuations. In case of fluctuations with long-range correlations this operator is close to a fractional Weyl derivative whose order, between 2 and 3, depends on the power decay of the autocorrelation function. In the frequency domain, the pseudodifferential operator exhibits a frequency-dependent power-law attenuation with exponent corresponding to the order of the fractional derivative, and a frequency-dependent phase modulation, both ensuring the causality of the limiting paraxial wave equation as well as the Kramers-Kronig relations. The mathematical analysis is based on an approximation-diffusion theorem for random ordinary differential equations with long-range correlations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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