Long Time Evolution of Wavefronts in Random Media

Abstract

This work studies the evolution of the front of an impulsive plane wave in a layered random medium. A general second-order hyperbolic equation with an inhomogeneous plane wave solution in a frame moving with random propagation speed is considered. In this frame it takes the form of a coupled system of ordinary differential equations along characteristics for the propagation modes. The author shows that in the asymptotic limit in which the amplitude of random inhomogeneities decreases but the distance at which the pulse is observed increases the leading part of the pulse, the wavefront, stabilizes to a deterministic waveform. This waveform depends on the statistical properties of the medium and the initial pulse only, and it is explicitly calculable. The theory is applied to the elastic wave equations in a two-dimensional medium.