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Propagation of any type of wave in a random medium is analyzed on the assumption that the medium differs slightly from a homogeneous medium. An equation satisfied by the average wave is deduced which is correct through terms of order ε2, where ε measures the deviation of the medium from homogeneity. From this equation, the propagation constant of the medium is determined. The general formulation applies to any type of linear differential or integral equation with random coefficients. It is applied to time-harmonic waves satisfying the reduced wave equation, to the equations of elasticity and to Maxwell's equations. The propagation constant for the average or coherent wave is complex even for a nondissipative medium, because the coherent wave is continually scattered by the inhomogeneities and converted into the incoherent wave. The propagation velocity of the average wave is also diminished by the inhomogeneities. This propagation constant depends upon certain trigonometric integrals of the auto- and cross-correlation functions of the coefficients in the original equations, i.e., of the various coefficients characterizing the medium. To illustrate the results, media with particular random variations are considered and the propagation constants are determined for them.
Karal et al. (1964) studied this question.