Model of electron beam transformation in a narrow tube

The problem of transformation of an electron beam in a long narrow conducting tube is considered. General wave solutions for the propagation of an arbitrary (including aperiodic) modulation of a homogeneous beam are obtained in the case of weak modulation. The solution corresponding to harmonic modulation is compared to the solutions obtained earlier and described in the literature. The problem of beam transformation corresponding to the bunching regime in high-power klystrons is also considered (including overtaking). A discrete analytic solution to the problem is obtained in the frozen beam approximation. This solution is tested for the problem of spilling of a monovelocity bunch and for the problem of the efficiency of a two-cavity klystron with infinitely narrow gaps. In both cases, physically correct results are obtained.