Theory of beam-plasma instability in a periodic plasma-filled waveguide

The beam-plasma wave interaction in a periodic plasma-filled waveguide is treated in a mathematically correct manner on the basis of the integral equation (IE) method. It has been shown that the relevant boundary-value problem could be reduced to an IE with a singular kernel for the longitudinal component of the electric field on the waveguide axis. The regularization of the IE was performed by extracting the static part of the kernel. The resulting IE of the second kind with a regular kernel, being rather convenient for a numerical analysis, is treated in a quasistatic approximation as a spectral problem. First-order expressions for eigenfunctions, and an infinite set of dispersion relations linking a wave number and frequency of plasma oscillations which separate radial branches of plasma oscillations from axial ones, have been obtained in the closed analytical form, thus enabling us to avoid the problem with the so-called "dense" spectrum. The solutions of the relevant "cold" dispersion relations establish a periodical dependence of the frequency on the wave number over several periods within the accuracy of order of the neglected terms. In the presence of an electron beam they turn out to be unstable near frequencies providing the resonances of the beam with spatial plasma harmonics. Evaluations of the instability saturation level predict a more efficient beam-plasma wave energy transfer compared with those following from a conventional theoretical analysis based on the formulation of a dispersion relation in terms of an infinite determinant, with following truncation of the latter to the finite sized relation.