Azimuthal surface waves are eigen electromagnetic waves of cylindrical metallic waveguides with plasma filling. Azimuthal surface waves of extraordinary polarization are known to be effectively excited by a beam of electrons gyrating in an axial external static magnetic field along large Larmor radii around the plasma column. Beam excitation of azimuthal surface waves was well studied theoretically in both linear and nonlinear approaches. An important limitation of the earlier studies was the approximation of zero axial wavenumber. In the present paper, perturbation theory is applied to account for finite magnitude of the axial wavenumber of electromagnetic surface waves in studying their excitation by an electron beam. The dependence of the wave growth rate on the axial wavenumber and parameters of the plasma waveguide is studied. The results obtained in this paper are of interest for application in plasma electronics.