Forward Scattering by Long Thin Bodies

The parabolic approximation of diffraction theory is extended to problems involving thin obstacles at grazing incidence to a short plane wave. Although the most direct application is to shallow-water problems in hydrodynamics, the same analysis applies in more general contexts, such as for acoustic and electromagnetic waves. The parabolic approximation reduces the task to that of solving an Abel-type integral equation of the second kind, for the scattered amplitude. Closed-form solutions are obtainable for opaque obstacles, i.e., islands. Results are also obtained for submerged islands, i.e., partially-transparent obstacles, where the wave speed is reduced, and for submarine canyons, where the wave speed is increased compared to the value outside the obstacle. For submerged islands, there is a possibility of energy trapping, and numericalsolutions of the Abel equation are used to illustrate the resulting resonance phenomenon.