Effective fractional acoustic wave equations in random multiscale media

Wave propagation in a one-dimensional random medium with shortor long-range correlations is analyzed. Multiple scattering is studied in the regime where the fluctuations of the medium parameters are small and the propagation distance is large. In this regime pulse propagation is characterized by a random time shift described in terms of a standard or fractional Brownian motion and a deterministic deformation described by a pseudo-differential operator. This operator is characterized by a frequency-dependent attenuation that obeys a power law with an exponent ranging from 0 to 2. The exponent is between 1 and 2 for a long-wavelength pulse and it is determined by the power decay rate at infinity of the autocorrelation function of the random medium parameters. The exponent is between 0 and 1 for a short-wavelength pulse and it is determined by the power decay rate at zero of the autocorrelation function of the random medium parameters. This frequency-dependent attenuation is associated with a frequency-dependent phase responsible for dispersion, which ensures causality and that the Kramers-Kronig relation is satisfied. In the time domain the effective wave equation has the form of a linear integro-differential equation with a fractional derivative.