Theory of waves in a homogeneous random medium

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Abstract
A novel theory is developed to cope with the difficulty of the multiple-scattering problem in a random medium (RM). The theory is given for a one-dimensional homogeneous RM which is represented by a strictly stationary random process. Some possible forms of the stochastic solution are determined by a group-theoretic consideration based on the shift-invariance property of the homogeneous RM. The form of the solution has some analogy with Floquet's solution for a periodic medium. It is shown that there are two kinds of solutions in the one-dimensional RM: a travelling-wave mode and a cutoff mode. The former exists only when the power spectrum of the medium becomes zero at nearly double the wave number. Otherwise the wave is in the cutoff mode which is almost a standing wave whose envelope increases or decreases exponentially with distance. For a Gaussian RM with small fluctuations, an approximate stochastic solution given in the possible form is obtained in terms of multiple Wiener integrals with respect to the Brownian-motion process. The average and variance are calculated for the phase and amplitude of the wave in terms of the power spectrum of the RM. The law of large numbers is shown to hold concerning the fluctuations of the phase and amplitude. The average value of the wave and the transmission coefficient of a medium with finite thickness are also studied using the stochastic solution.

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