The problem of a helix-type traveling-wave amplifier tube, under certain simplifying assumptions, is solved as a boundary-value problem. The results indicate that the presence of the beam in the helix causes the normal mode to break up into three modes with different propagation characteristics. Over a finite range of electron velocities one of the three waves has a negative attenuation, and is thus amplified as it travels along the helix. If the electron velocity is too high or too low for net energy interaction, all three waves have purely imaginary propagation constants; no amplification occurs. Consideration of the beam admittance functions shows that, during amplification, the electron beam behaves like a generator with negative conductance, supplying power to the fields through a net loss of kinetic energy by the electrons. Curves are shown for a typical tube, and the effects of beam current and beam radius are indicated. The initial conditions are investigated, as are the conditions of signal level and limiting efficiency. In the Appendix a simple procedure for computing the attenuation constant is given.