[en] A mode analysis is presented for the case of a planar electron beam in a ridged waveguide slow-wave structure. By matching boundary conditions between a Fourier expansion of the mode between the ridges with a space-harmonic expansion of the mode in the region below the ridges, a dispersion relation for a traveling-wave interaction is found. The dispersion relation is numerically solved for both the case with and without beam. For the nominal geometry, gains as high as 30 dB/cm at 300 GHz are found for a 15 A, 155 keV beam