Nonlinear equilibrium and stability analysis of rippled, partially neutralized, magnetically focused electron beams

After giving a higher-order solution of the equation describing the nonlinear equilibrium of rippled electron beams, in the second part of the paper we derived mathematical expressions for the growth rates of electrostatic perturbations for the resonant coupling of (i) fast and slow and (ii) fast—fast or slow—slow waves. Actually, viewed from a physical point of view, only type (i) of wave coupling may lead to the kind of instability considered in the paper.