An exact spectral representation of the wave equation for propagation over a terrain

An exact spectral representation of the wave equation above a dielectric ground is proposed. The formulation is based on the diagonalisation of the vertical operator, takes into account the angle-dependance of the reflexion coefficient, and does not include any paraxial approximation. The expressions of the spectrum comprise two parts: a continuous part and a discrete part. The latter corresponds to a possible surface wave. The use of this result in split-step algorithms to simulate wave propagation requires a discretization of the spectrum. To render the discretization consistent, an alternative discrete spectral representation is proposed that intrinsicaly includes the truncation of the computation domain at a finite high.