On the method of solution of the wave equation with periodic coefficients

The analytic solution method of the nonuniform wave equation of the problem of radiation of the magnetic moment, moving uniformly in a waveguide of arbitrary cross section with time-space periodic dielectric filling is given. The fields and the analytic expression for the Cherenkov energy losses of the magnetic moment in the region of strong (resonance) interaction between the radiation wave and the modulation wave are found.