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A TE/TM modal solution for a longitudinally corrugated rectangular waveguide is developed. These longitudinal corrugations can be used to excite a quasi-TEM wave and form a hard waveguide by correctly choosing the impedance at the guide wall. The correctly chosen impedance is referred to as the hard boundary condition. The modal solution developed here solves the problem of longitudinal corrugations filled with a dielectric material by first finding and solving the characteristic equation for a complete TE/TM modal set. It is shown that this TE/TM mode solution can be used to achieve the hard boundary condition resulting in the quasi-TEM wave in a hard waveguide for discrete values of corrugation depth. Beyond each of these depths, a mode becomes a surface wave. The theoretical mode set is amenable to the solution of problems using the mode-matching method. A combination of the mode-matching method and the TE/TM modal solution will allow the solution of larger problems.