A variational theory for wave propagation in a one-dimensional inhomogeneous medium

A variational theory is developed to study wave behavior in an isotropic medium with one-dimensional inhomogeneities in both permittivity and permeability. The formulation is compact so that one variational equation is adequate for handling permittivity and permeability profiles of various kinds, as well as polarizations of both types (perpendicular and parallel). The variational equation is then solved numerically, using the finite element method. Numerical results for cosinusoidal and logarithmic profiles are presented, and their errors are estimated by the criterion of complex power conservation. Various propagation characteristics for the cosinusoidal profile, including complete transmission, total reflection, and effects of losses, are investigated in detail. The possibility of having a complete transmission window for the lossless ionized cosinusoidal profile is pointed out and interpreted physically by the concept of leaky resonance. An analysis of the logarithmic profile is also included for an approximate description of a nonuniform coaxial line.