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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.