A Compact Derivation of Formulas for Propagation over a Plane Earth in a Homogeneous Atmosphere

The method of Fourier transforms is used to separate variables in the two-dimensional wave equation. The resultant ordinary differential equation in the transformed height function is solved, subject to boundary conditions at the surface of the earth, the source, and very great height. Use of the inverse transform then leads to the complete solution, in the form of an integral, as a function of height and radial distance. This integral cannot be evaluated exactly except by numerical methods. However, for all except the shortest distances, a practical solution can be obtained by the saddlepoint method. This procedure demonstrates the existence of a diffracted wave in addition to the components obtained from geometrical optics.