Courant-Hilbert waves in the troposphere

The Courant-Hilbert solutions of the wave equation in free space are form invariant along their propagation, an important property for those interested in transmitting undistorted signals. Their amplitude does not remain constant but the fading may be compensated by convenient amplifiers. This property is not expected to persist along propagation in nonhomogeneous media but it is proved that in the troposphere with refractive index n(z)=1+βf(z) and small β, Gaussian-like beams of the Courant-Hilbert type are form invariant to the 0(β³) order of approximation. Courant-Hilbert solutions of Maxwell´s equations in the troposphere are discussed and the 0(β³) explicit expressions for TE and TM Gaussian beams are given. Calculations are performed for a troposphere above a flat Earth in the vertical direction.