The 3-D motion of the relativistic electrons is analysed under the combination of a cylindrical waveguide TE mode (or TM mode) and an axial guide field. Both linear and nonlinear calculations are carried out. Generally speaking, the projection of the electron orbits onto the transverse plane is an annulus, the thickness of which is dependent on the amplitude of the electromagnetic wiggler. However, the electron orbits may diverge when ω+kzvzo+ ( m±1 )ωc≈0 Discussion is also given to the special case that the electromagnetic wiggler is a slow wave propagating parallel to the electron motion.