The failure of "classic" perturbation theory at a rough Neumann boundary near grazing

Rice's "classic" perturbation theory predicts an erroneous limit at grazing for vertically polarized plane wave scatter from an infinite perfectly conducting rough surface; likewise, the attendant result for the specularly reflected mode also fails at grazing. We show where and why in the system of perturbational equations this difficulty occurs. We then reformulate the perturbational approach to handle the low-incidence angle region for a one-dimensionally (1-D) rough Neumann boundary (vertical polarization from a perfectly conducting surface). The result for scattered fields vanishes in direct proportion to incidence angle above grazing and the result for the normalized roughness-modified surface impedance becomes constant with angle near grazing. For-completeness and comparison, we give results for the horizontal polarization at a Dirichlet boundary, where perturbation results encounter no difficulties, Scatter dependence on grazing angle is explained in terms of the "classic" perturbation result multiplied by a propagation factor to the cell. The latter includes the sum of the direct and specularly reflected waves at the surface. This quantity can be replaced by the appropriate surface-wave propagation factor for radiation from dipole antennas, thereby explaining the strong observed vertically polarized sea scatter at high frequency (HF) on and below the horizon.