Alternative calculations of initial value problem for electromagnetic field in dielectric waveguide

Two ways of solving an initial value problem in a dielectric waveguide is presented. Each way describes one of two complementary approaches: a progressive or Brillouin waves approach and an oscillatory or eigenfrequency approach. The evolution of a transient electromagnetic field in the waveguide after an instant change of its permittivity is considered and calculated. The results of the approaches coincide completely, however, the Brillouin approach is more intuitive, visual, and less demanding computationally.