Space-charge waves in a randomly inhomogeneous medium

The propagation and amplification of space-charge waves generated in the transmission of charged-particle beams through randomly inhomogeneous and time-varying media and also in beams with fluctuating parameters are investigated. We consider potential waves generated in the motion of a cold nonrelativistic beam of charged particles in a medium with a fluctuating dielectric constant. The nature of the propagation of electromagnetic waves in a statistical medium is determined both by the space scale and by the intensity of the dielectric constant fluctuations. Here we consider the cases of one-dimensional dielectric constant fluctuations of arbitrary intensity and small space scale, smooth three-dimensional large-scale fluctuations, and three-dimensional fluctuations of arbitrary scale and small intensity