Apparent Pulse Diffusion Due to Disordered Microstructure

Wave propagation in disordered (random) media is the underlying theme. Many types of wave propagation problems can most conveniently be analyzed in this framework. Acoustic waves in the earth's crust and shallow water (or surface gravity) waves are two examples. The interaction of sound waves with the heterogeneities in the earth's crust is important in seismology. The interaction of surface gravity waves with rough submerged obstructions has been analyzed extensively because of its importance in oceanography. It is also important in the development of new techniques for analyzing scattering problems in uid mechanics. The theory to be outlined in this paper had its rst application in seismic exploration. The earth is strongly heterogeneous, also on small scales, and it is important to describe when and how ne scale heterogeneities interact with a traveling seismic pulse. We are interested in pulse shaped waves that interact with the rapidly varying features (i.e. microstruture) of the medium. The multiple scattering associated with the disordered microstruture, leads to the apparent diiusion of the propagating pulse. That is, the pulse diiuses about its moving center due to the multiple scattering of the wave energy. We present a summary of the asymptotic theory that describes the interaction of a propagating pulse with the medium's microstructure. Also we describe the connection between acoustic and shallow water pulse shaping or apparent diiusion. We consider media having general smooth background variations, with a locally laminated disordered microstructure. Theoretical results are validated through numerical experiments. Acoustic wave experiments are performed in a Goupillaud medium. In the context of water waves the numerical model consists of a semi{implicit, semi{Lagrangian discretization of the hydrostatic Navier{Stokes equation in three dimensions. We show that the numerical experiments capture very well diierent types of behavior predicted by the theory.