  Results of our investigations of the properties of the electromagnetic
oscillations in the resonators based on the periodic waveguides are
represented. These results point out on the possibility of existing in resonant
cavities the new type of eigen oscillations based on the evanescent waves. Such
oscillations do not exist in the smooth waveguides. They have a longitudinal
distribution of electromagnetic field with decreasing (increasing) dependence
on the coordinate. We discuss the using of such oscillations for bunching and
accelerating of electron beams.
