Pulse Propagation in Random Media with Long-Range Correlation

Abstract

This paper analyzes wave propagation in a one-dimensional random medium with long-range correlations. The asymptotic regime where the fluctuations of the medium parameters are small and the propagation distance is large is studied. In this regime pulse propagation is characterized by a random time shift described in terms of a fractional Brownian motion and a deterministic spreading described by a pseudodifferential operator. This operator is characterized by a frequency-dependent attenuation that obeys a power law with an exponent ranging from 1 to 2 that is related to the power decay rate of the autocorrelation function of the fluctuations of the medium parameters. This frequency-dependent attenuation is associated with a frequency-dependent phase, which ensures causality of the filter that realizes the approximation. A discussion is provided showing that the mean-field theory cannot capture the correct attenuation rate; this is because it also averages the random time delay. Numerical results are given...