Transient electromagnetic fields in unbounded space with inhomogeneous medium

To solve propagation problems in time domain directly the modal representation of transient electromagnetic fields is considered. Using evolutionary approach the initial nonstationary three-dimensional electrodynamic problem is transformed into the problem for the system of connected one-dimensional evolutionary equations. The system describes the interconversions of all modes on inhomogeneities in cross-section. After elimination of the longitudinal components of electrical and magnetic field from Maxwell equation system the four-dimensional differential operators are formed. It is proved that the operators are self-adjoint ones. The completeness of the basis is proved by means of Weyl Theorem about orthogonal splitting of Hilbert space. The expansion coefficients of transient electromagnetic field can be found from the set of evolutionary equations.