Propagation of space-charge waves, similarly to field wave propagation on periodic circuits, may be described by modes, each mode containing an infinite number of space harmonics. Expressions for harmonic amplitudes and their propagation constants are derived for infinite as well as for laterally finite beams.nKinetic power flow in the periodic beam modes is analyzed. It is determined that kinetic power decreases near the unstable-beam region (beam period equals half the plasma wavelength) and is iden¬tically zero within it.nThe coupled-mode analysis of microwave tubes is extended to interactions of periodic beam modes with periodic structures or other periodic beams.nAn analysis of a folded-line circuit interacting with a periodic beam is carried out to illustrate the application of the theoretical results. Some conclusions about the significance of beam harmonics in this and similar devices are drawn from this analysis.nAn experimental tube using a folded-line slow-wave structure and a periodic electron beam has been built and tested. Beam space-harmonic amplitudes and phase velocities have been measured and found to be in good agreement with the calculated values. The periodic nature of the beam has been found to have little effect upon the beam fundamental space-harmonic interactions with the circuit.n"A fast-wave" device using interaction between the TM.. mode in a circular waveguide and a higher space harmonic of a periodic beam is analyzed. The device is found to be suitable for high-power millimeter-wave generation and amplification.
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