Stochastic Green's functions and random walks

Abstract

In previous work [D. P. Vasholz, J. Acoust. Soc. Am. 60, S33(A) (1976)] a general black-box approach to wave propagation through a randomly fluctuating medium based upon the stochastic Green's function G was presented. Motivated by parallels with quantum field theory the quantities 𝒢 and M, related to the first and second moments of G, were defined. It was then shown how these quantities may be conveniently used to describe such effects as frequency broadening and coherence losses induced by the medium. In the present work it is pointed out that stochastic Green's functions readily lend themselves to interpretations of recent experimental work which have been made in terms of the random walk taken by the time-varying transfer function in the complex plane [W. H. Munk and G. O. Williams, Nature 267, 775 (1977)]. In particular it is shown how these random walks arise in a stochastic Green's function formulation and how their principal parameters may be very simply expressed in terms of 𝒢 and M.