The diffraction of an H-polarized plane wave is investigated when the wave is obliquely incident on a flanged parallel-plate waveguide or on a slit in a thick screen. Using discrete spectra of periodic functions inside the waveguide and of Mathieu functions in a half-space, the solution can be obtained by matching the different field representations in the aperture plane. The technique has the advantage that it operates in a wide frequency range, ka > 0 through ka ≈ 20, and yields rapid convergent results of excellent accuracy (k is the free-space wavenumber, 2a is the aperture width). The calculated fields satisfy the edge condition. Investigations into a proper ratio between series truncation numbers to avoid relative convergence appear to be unnecessary. A number of numerical results are presented. The transmission coefficient is critically dependent on the cutoff frequencies of the waveguide modes and on the thickness of the screen, which causes strong resonances, whereas the near- and far-field patterns are influenced by the thickness, especially for oblique incidence.