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