We investigate TE-wave propagation in a hollow waveguide with a graded dielectric barrier, using an equivalent model of the waveguide filled with a stratified medium. General formulae for the electric field components of the TE-waves, applicable to hollow waveguides with arbitrary cross sectional shapes, are presented. As an illustration, we obtain the exact analytical results for the electric field components in a rectangular waveguide, as well as the exact analytical results for reflection and transmission coefficients which are valid for waveguides of arbitrary cross sectional shapes.