Scalar waves in a one-dimensional randomly inhomogeneous medium: Allowance for spatial dispersion in pair-interaction approximation

Abstract

The Helmholtz equation for the average field in an ubounded one-dimensional randomly inhomogenous medium described by the normalized binary correlation function ϕ(z1, z2)=exp(−/z//a), where a is the correlation scale and z≡z1−z2, is solved in a pair-interaction approximation (Bourret approximation). Allowance for the macroscopic spatial dispersion (caused by the inhomogeneity of the medium) expands the applicability range of the Bourret approximation by including ultrashort waves. The scattering indices y± and phase v± and group c± propagation velocities of monochromatic waves corresponding to two roots x of the dimensionless wave numbers x± of the dispersion equation are calculated. It is shown that, unlike in the three-dimensional case [1], the scattering index has the same asymptotic dependence on frequency for long and short waves: V+ ∝ ω2a. The wavelength-dependent applicability conditions of the pair-interaction approximation (M criterion) and negligibility of the macroscopic spatial dispersion (N criterion) are investigated for each of the roots. The field of a point source is represented as a superposition of divergent waves defined by roots x+ and x−. The absolute values of the amplitudes of these waves reach maximum values at the boundary of the short-wave and ultrashort-wave ranges. Anomalous properties of y±, v±, and c± are observed in this region.