Using the theory of waveguide excited, the full three-dimensional self-consistent working equations of interaction between slow electromagnetic traveling wave and electronic beam have are obtained in traveling-wave tubes, considering multi-signal inputs and relativistic effect. The equations include excited equation, motion equations, energy conversion equation, phase evolving equation etc., which can be applied to simulate most of nonlinear interactions between slow electromagnetic traveling wave and electronic beam in traveling wave tubes (TWT). The crossmodulation in a wide-band helix TWT is simulated using this theory, and is compared with the experimentation results to confirm the theory. The nonlinear beam-wave interaction processes in a relativistic disk-loaded waveguide TWT are also been simulated using the theory.