Wave Propagation in Certain One-Dimensional Random Media

Abstract

In this paper we investigate the stochastic ordinary differential equation u″+k02[1+εy(t)]u=0 with y(t) a random process. Two specific types of process y(t) are considered. Both of these arise from a bounded mapping y(t) = f(x(t)) of a countable state space Markov process x(t). Exact equations are derived for the statistical moments of u(t), and the behavior of the first two moments is discussed in the limit of small ε. A description of the layered media to which our results apply is given and a comparison of our exact results with certain perturbation methods is made.