Non-Linear and Dispersive Effects in the Propagation and Growth of Longitudinal Waves on a Coasting Beam

Experimental Observations of Soliton Wave Trains in Electron Beams

“…Meanwhile, when the steepened wavefront width is comparable to the pipe radius, the wave becomes dispersive [5] and could balance the steepening. These two opposing phenomenasteepening and dispersion -are the mechanisms behind the generation of a stable, dispersion-less KdV type soliton pulse.…”

“…However, the beam system, a bounded nonneutral plasma, can behave in ways that differ fundamentally from an unbounded plasma. Since the 1980s, solitons have been predicted in charged particle beams, both theoretically and in simulations [5][6][7][8][9][10][11]. Experiments on proton beams exhibited longitudinal single-soliton hole structures [12][13].…”

“…Larger perturbations are theoretically predicted to evolve into solitons, if the space charge in the beam is sufficiently strong [5,8]. The beam evolution can be shown to approximately evolve according to the Korteweg-deVries (KdV) equation [6,8] known to describe soliton evolution:…”

Scaled electron studies at the University of Maryland

“…Since these waves propagate, at most, approximately a beam diameter per focusing lens period [9], measurements of longitudinal wave phenomena generally require long beam propagation lengths. Longitudinal wave propagation is therefore an active area of experimental study on UMER [10].…”

Threshold Behavior for Longitudinal Stability of Induction LINAC Bunches

“…This expression is obtained from a Bessel function expansion of the longitudinal field in a finite pipe, with the value 2.4 the first zero of J 0 • For a purely rectangular, cold bunch of length L, space charge wave modes can be obtai ned from a fluid mode 1. The eigenfrequencies of these modes are given by {2) (3) where n is any integer and L is the bunch length. The plasma wave velocity, V 0 , is given by (4) for charge q, g = 1 + 2Ln (B/A), line density (A) beam radius A, and mass m. Note that at high frequencies the modes coalesce, and the group velocity tends to zero.…”