Acoustic Waves in Long Range

We consider waves propagating through multiscale media. Much is known about waves propagating through a medium that satisfies a scale separation assumption with random fluctuations on a microscale. Here we go beyond this situation and consider waves propagating through a medium defined in terms of a long range process. Such a medium can, for instance, be modeled in terms of a one-dimensional fractional Brownian motion with variations on a continuum of scales. Fractal medium models are used to model, for example, the heterogeneous earth and the turbulent atmosphere. We set forth a framework using the theory of rough paths in which propagation problems of this nature can be analyzed in the case with anticipative medium fluctuations with a Hurst exponent H > 1/2. We show how the wave interacts with the medium fluctuations in this case and that the interaction is qualitatively different from the situation where the medium satisfies a separation of scales assumption. In the long range case considered here the travel time depends strongly on the particular medium realization, but in fact the pulse shape does not.