In this paper, the linearized relativistic Vlasov equation is solved and a dispersion relation is calculated for the interaction between a relativistic electron beam and the electromagnetic fields of a rectangular waveguide supporting TE
mnmodes. The dispersion relation is simplified to the special case of a frame of reference moving with the electrons and the resulting coupling coefficient

is calculated for various TE
mnmodes. The dependence of

on the parity of the harmonic number l and various mn modes of the waveguide are discussed.