About permittivity profiles having joint analytic solutions for horizontally and vertically polarized waves

We consider two model permittivity profiles for which we obtain joint analytic solutions for horizontally polarized electromagnetic waves (H-waves) and vertically polarized waves (V-waves): hyperbolic and square hyperbolic profiles. For the case of hyperbolic profile solutions for H- and V-waves are confluent hypergeometric functions with the integral second indices. For the case of the square hyperbolic profile we have confluent hypergeometric functions with non-integral second indices. The first profile can model the transition layer, and the second-region with the permittivity maximum. Solutions obtained can be used for testing of computer programs, for modelling of soil moisture distribution influence on the polarization properties of scattered microwave signals