On the applicability limits of the small-parameter theory in a resonance long -time-interaction O-type oscillator

Under the resonance oscillator the author considers a comb-line resonator of the length L and the quality factor Q, in which an infinitely thin electron beam flows above the comb-line. The electric field on the comb-line has the appearance of the standing wave. Its amplitude is supposed to be constant along the comb-line. This wave is spatially modulated by the comb periodical structure and, hence, presents a series of spatial harmonics with a decreasing phase velocity. The electron beam is non-modulated when entering the interaction space. Due to the present longitudinal magnetic field the electrons in the beam can move only along the x axis oriented along the comb-line. The author considers only the steady-state generation mode, when the wave amplitude does not depend on the time. Due to the energy conservation law, resulting from the Maxwell equations, the power rejected from the wave field (as output and loss) is equal to the power transferred by the beam to the wave field