A small‐signal analysis of smooth electron beams with periodic variations in their dc parameters reveals the existence of infinite sets of space harmonics of the fast and slow space‐charge waves. For a finite beam coupling between the space‐charge waves and field waves exists at an infinite number of frequencies. The periodicity of a beam has a very small effect on the space‐charge wave propagation constants. Velocity‐jump, rippled‐stream, and rippled‐wall amplification are shown to result from coupling between fast and slow space‐charge waves of adjacent harmonics. The periodicity of an electron beam will have negligible effect on most conventional traveling‐wave devices. Although the frequency dependence of the amplitudes of the beam harmonics may introduce difficulties in some devices, this dependency may also be made use of in a variety of ways. Beam harmonics make millimeter‐wave interactions possible in smooth circuits, but practical devices at these frequencies will probably be limited to those employing positional periodicity of the beam.