Coupled-mode theory is employed to obtain the threshold gains for oscillation on hybrid modes of a passive-core corrugated-waveguide laser pumped in a central portion of its length. Methods are developed for calculating intermode coupling coefficients and loss coefficients resulting from diffraction into any order of waveguide corrugations of arbitrary shape. Experimental observations that have been reported in the literature are compared with the hybrid-mode threshold gains computed from coupled-mode theory. It is shown that the oscillation frequencies for such a laser are determined by the length of the corrugated structure, the proportion that is pumped, and the effects of coupling into leaky modes.