Space-charge wave propagation on rippled electron beams

A measurement of the phase characteristics of space-charge waves propagating on a rippled magnetically confined electron beam reveals a shift in the average value of the wave number from that expected for an unrippled beam. An analysis of space-charge wave propagation for such a beam can be done by assuming a small-amplitude sinusoidal variation of the beam radius about its average. This analysis shows that there is a periodic variation in the space-charge wave number as a function of distance along the beam. The average value of the wave number taken over one ripple period is shifted by an amount that agrees with experimental observations. This periodic variation of wave numbers results in sidebands similar to those associated with the frequency modulation of radio waves.