The finitedifference theory of excitation of periodic waveguides is used to derive the characteris tic equation for electron waves formed during interaction of an electron beam with forward and counter propagating electromagnetic waves of periodic waveguides (slowwave structures). The derived characteristic equation describes interaction of electrons and waves in passbands and stopbands of periodic waveguides and contains known solutions for “smooth” slowwave structures and resonator slowwave structures near cutoff frequencies as particular cases. Several analytical solutions allowing comparison of amplification and propa gation properties of electron waves inside and at the edges of passbands and stopbands of periodic waveguides are found.
In the linear approximation is carried out numerical simulation and analysis of the features of the interaction of colliding electron beams in a drift tube with the reduction ratio of plasma oscillations and electronic interaction between threads. In the hydrodynamic approach in the weak signal wave technique developed theories related to space-charge waves colliding electron beams. Write an equation relating the plasma waves and electron beams obtained dispersion equation continuous four-two-beam interaction, which is valid over a wide bandwidth for different values of the parameters of decreasing Coulomb forces with distance, and the coefficients of the electronic interaction between threads. In the single-mode approximation in each of the interacting flows spread fast and slow space-charge wave (electronic or plasma wave beams). It was obtained solution of the dispersion equation in three modes and fourwave communications space-charge waves. The features of the modes of interaction such as BWO-amplifier, directional coupler with a periodic or aperiodic wave coupling in the case of interaction of colliding wide and narrow electron beams in the drift tube.