Instability of oscillations in the flow moving through a random medium as the counterpart of the localization of waves in a passive random medium

This paper deals with waves propagating in a one-dimensionial flow moving through a randomly layered medium: The flow velocity is assumed to be greater than the group velocity of the waves in the reference system of the flow. As a result, in the laboratory reference system, all the waves propagate in a single direction. Amplitudes of these waves moving through a randomly inhomogeneous medium are growing exponentially. This effect is analogous to the wave localization phenomenon in a randomly inhomogeneous passive medium. The only difference is that the wave propagation in a passive medium is described by the boundary value problem, while all the oscillations in a medium with flow propagate in a single direction and hence the corresponding problem is formulated in the form of the initial value Cauchy problem. In the former case exponentially decreasing solutions are realized in the direction of the wave incidence, while in the latter case (as in the case of parametric resonances) the exponentially increasing solutions are realized.