Transverse components of the electromagnetic field in a waveguide with modulated in space and in time magnetodielectric filling

The propagation of transverse magnetic (TM) and transverse electric (TE) electromagnetic waves in a regular ideal waveguide of arbitrary cross section is considered. It is assumed that the permittivity and permeability of the magnetodielectric filling of the waveguide are functions that depend on the coordinate and time. Analytical expressions for the transverse components of the magnetic and electric vectors of the TM- and TE-fields in the waveguide are obtained from the system of Maxwell equations. They are expressed in terms of the longitudinal components of the electric and magnetic vectors, which describe the transverse magnetic and transverse electric fields in the waveguide. For the above longitudinal components of the electric and magnetic vectors, the wave equations are given, which are also obtained from the system of Maxwell's equations.