Two-Term Ritz–Galerkin Solution for the Low Frequency Scattering by a Rectangular Trough in a Soft Ground Plane

An analytic form is derived for the diffracted field that originates when a transverse-magnetic plane-wave excites a rectangular trough cut in a ground plane. The low-frequency analysis is valid for trough apertures that are less than half a free-space wavelength in width. Trough depth is arbitrary. The quasi-static nature of the long wave problem is exploited by approximating the aperture field as a complex superposition of decoupled symmetric and antisymmetric solutions to Laplace´s equation in the vicinity of the aperture, subject to Dirichlet boundary conditions. Geometrical contributions to the final algebraic expressions for the two aperture coefficients arise from two sources: A rapidly convergent series governs the aperture interaction with the trough modes and the effect of the half-space is embodied in a small number of terms of low powers in kappaalpha and factors of In kappaalpha.