Path-integral analysis of scalar wave propagation in multiple-scattering random media

In this work we consider the general problem of scalar wave propagation in a continuously inhomogeneous random medium, applying the approach originally proposed by Fock for the integration of quantum-mechanical equations. The principal idea of the method is based on the introduction of an additional pseudotime variable and the transfer to a higher-dimensional space, in which the propagation process is described by the generalized parabolic equation similar to the nonstationary Schrödinger equation in quantum mechanics. We present its solution in a form of Feynman path integral, the asymptotic evaluation of which in the far field allows us to estimate the so-called wave correction terms. These corrections are related to coherent backscattering and repeated multiple-scattering events on the same inhomogeneities, i.e., to the phenomena that are not described in the framework of the conventional theories of radiative transfer or small-angle scattering. As an example of the approach we consider the first statistical moments of the field for a point source located in a statistically homogeneous Gaussian random medium. The correction term obtained for the mean field coincides exactly with the classical result of the Bourret approximation for the Dyson equation, but with a much weaker restriction on the value of wave number k, which allows us to analyze the correction as a function of k. The main feature of the result obtained for the second moment of the field is that the normalized mean intensity is not equal to unity. We relate such behavior to the localization phenomenon. The dependence of the correction term does not differ significantly from that obtained in works concerning the localization of classical waves in discrete random media.