Invariant Imbedding and Wave Propagation in Stochastic Media

by Richard Ernest Bellman, Robert E. Kalaba

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An application of the theory of invariant imbedding to problems of propagation in stochastic media. The particular example is discussed in which a plane wave is incident on a stratified slab which is characterized by stochastic wave numbers in each stratum. The distribution functions for the amplitude of the random reflected and transmitted waves can then be determined by Monte Carlo, or analytically under very special assumptions. The study illustrates the applicability of functional equations and principles of invariance to the study of various types of wave propagation; indicates the importance of functional equations in stochastic variables prior to the appearance of any expected values; and emphasizes that to determine the nonlinear aspects of the recurrence relations, the actual probability distribution of the random variables appearing must be examined.

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