An exact vectorial spectral representation of the wave equation for propagation over a Terrain in 3D

A spectral representation for the electromagntic fields is developed in 3D in cylindrical coordinates for propagation over a Terrain. The result is obtained for a planar terrain and for an homogeneous atmosphere and ground. The spectral representation is related with both the vertical and azimuthal variables. For the vertical variable, this spectral representation contains a discrete component associated with a possible surface wave, and a continuous component. For the azimuthal variable, the spectral representation amounts to a Fourier series.