Acoustic Waves in Long Range Random Media

Abstract

Abstract. 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.Key words. wave propagation, random media, long range processes, fractional Brownian motion AMS subject classifications. 34F05, 34E10, 37H10, 60H20 DOI. 10.1137/07068610X1. Introduction. Modeling in terms of a multiscale medium is important for propagation problems in, for instance, the earth's crust, the turbulent atmosphere, turbulent boundary layers, sea ice, and outer space [9,15,19,20,34,42,44]. Communication, remote sensing, and laser beam propagation schemes are affected by bad weather and multiscale medium variations. Large scale research projects (the ABLE ACE program Kirtland AFB, for instance [43]) have focused on gathering atmospheric turbulence data and numerically simulating propagation of wave fields through synthetic turbulence models that derive from these. Above a boundary layer atmospheric turbulence may occur within a stratified environment, and the turbulent temperature variations may be highly anisotropic; see [16,36]. Rough and long range medium fluctuations associated with multiscale modeling are also important in medical imaging, device modeling, and nuclear technology, to name a few. The zone in between different tissue types (or in between different dielectrica) may in particular be strongly heterogeneous with variation on a continuum of scales. In general, the detailed pointwise variation of a multiscale medium, the refractive index, say, cannot be identified. However, the statistics of this variation can be characterized. Optimal design of, for instance, imaging and communication algorithms requires insight about how the wave is affected by the rough medium fluctuations, that is, the nonlinear coupling between medium and wave field statistics. This is particularly the case with modern high resolution sampling and imaging technology. Insight about the wave medium interaction is also important in a range of other applications like design of sound-absorbing materials and non...

Asymptotics of the Solutions of the Random Schrödinger Equation

“…Thus, the slow decay of correlations leads to time-separation of the energy and phase evolutions, a phenomenon we plan to address in detail elsewhere. Let us mention that to the best of our knowledge the first study of wave propagation in random media with slowly decaying correlations was done in the onedimensional case [12,20], where it was shown that a pulse going through a random medium with long range correlations performs a fractional Brownian motion around its mean position, as opposed to the regular Brownian motion in the rapidly decorrelating case [11]. On the other hand, motion of particles in such random media leading to fractional Brownian limits was considered in [10,17,18].…”

Pulse Propagation in Random Media with Long-Range Correlation

“…Wave propagation in multiscale and rough media, with longrange fluctuations, has recently attracted a lot of attention, as more and more data collected in real environments confirm that this situation can be encountered in many different contexts, such as in geophysics [11] or in laser beam propagation through the atmosphere [13,16,25]. Recently it has been shown that the main effect of such fluctuations of the medium parameters is a random time shift for the wave front, that obeys a Gaussian statistics described in terms of a fractional Brownian motion [22]. Here we observe the wave front along its random characteristics and we show that the wave front also experiences a deterministic shape modification, that can be described in terms of a pseudo-differential operator that depends on the power decay rate of the autocorrelation function of the fluctuations of the medium parameters.…”

Fractional precursors in random media

“…These results are derived in [14]. Related results for particular situations, respectively paraxial propagation, strongly heterogeneous media with long-range correlations and Goupillaud media can be found in [10,21,29]. In [28] the transmitted pulse in so called locally layered media was considered and it was shown that central results on the pulse deformation are robust with respect to such medium perturbations, while in [13] solitons in random media with long range perturbations was considered and it was shown that aspects of the pulse deformation theory that predicts power law rather than exponential decay generalizes to nonlinear situations.…”