Multimode time-dependent gyrotron equations for different time scales

Development of gyrotrons requires careful understanding of different regimes of gyrotron oscillations. For a long time, the gyrotron theory was developed assuming that the transit time of electrons through the interaction space is much shorter than the cavity fill time. Correspondingly, it was assumed that during this transit time, the amplitude of microwave oscillations remains constant. However, there are situations when this assumption is not fulfilled, or is marginally fulfilled. In such cases, a different mathematical formalism has to be used. The present paper generalizes the new formalism to the multi mode case. The particular example considered indicates that in some cases the results obtained by means of the old and the new formalism differ significantly.