Instabilities and amplification in bounded beam-plasma systems

The linear differential equation governing the perturbed potential and the corresponding boundary conditions are obtained for homogeneous cold beam-plasma systems bounded by metallic cylindrical walls in an external magnetic field. The expression for amplification rate of the premodulated beam perturbations and the dispersion equation for eigenfrequencies are then derived and discussed in detail in the one-dimensional limit. We demonstrate that the usually accepted exponential growth of spatial amplification inevitably changes to only a linear one with increasing the system length and present also analytical description of the rather complicated pattern of evolution of the unstable eigenmode spectra.