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This paper concerns wave phenomena in electron beams and plasmas with internal motions, in the presence of a static magnetic field. It is known that, axially-rotating beams show wave propagation when the Doppler-shifted frequency seen by the beam electrons is approximately a harmonic of the electron cyclotron frequency. Similar propagation in an infinite series of coupled modes can occur for beams and plasmas in which the electrons gyrate about the magnetic field lines, rather than the axis. Associated with each Doppler-shifted cyclotron harmonic, a slow and a fast space-charge wave are propagated along an electron beam. These sets of waves can interact with each other, with slow or fast wave circuits, or with a plasma, to yield growing waves. The nature of these interactions is discussed in the paper. It is suggested that this mechanism of wave amplification along an unneutralized electron beam may provide an explanation of the noise growth observed in electron beams from shielded guns, and in crossed-field devices. [ABSTRACT FROM AUTHOR]