One Dimensional Random Wave Propagation

Abstract

We study wave propagation in a one-dimensional random medium whose index of refraction is characterized randomly and is assumed to have small fluctuations about the mean. The appropriate stochastic boundary value problem for the scattering region is transformed into a Cauchy type initial value problem for the boundary values of the random Green’s function. The stochastic differential equation derived is a first order, nonlinear equation of the Riccati type. The initial value problem is solved in two ways: (i) by conventional power series perturbation expansion, and (ii) by quasi-linearization. In both cases we consider the refracting medium to be characterized by a general stationary process in the broad sense, and for such a process, general expressions for the statistical properties of the reflected and transmitted amplitude waves are derived. In the former case the solutions obtained are uniformly valid for $\lambda < O( 1/\varepsilon )$, $\lambda $ being the size of the scattering region and $\varepsilon \ll 1$, while in the second case the solutions are uniformly valid for all $\lambda $. As an example, we consider an Ornstein–Uhlenbeck process and its limiting white-Gaussian process, for which expressions for the mean power reflected and transmitted are obtained.