Exact wide-angle formulation of the Beilis-Tappert method

The Beilis-Tappert method was originally developed for narrow-angle acoustic propagation under a rough sea surface [A. Beilis and F. D. Tappert, J. Acoust. Soc. Am. 66, 811–826 (1979) ]. The method has also been applied to narrow-angle propagation over irregular terrain for acoustic waves and radar. It is shown here that an exact wide-angle formulation of the Beilis-Tappert method can be derived simply by replacing ∂/∂z with ∂/∂z + ik0 tanφ, where k0 = 2π/λ, λ is the physical wavelength, φ is the slope angle, and i= √-1. The exact formulation makes clear that for large slope angles, much of the acoustic field does not propagate, but decays exponentially with range. Existing finite difference methods and all narrow-angle methods fail to properly account for the exponential decay with a range of the non-propagating components of the acoustic field. The exponential decay with this range is qualitatively explained in terms of rays and quantitatively explained in terms of waves. Properly accounting for the pro...