An idealized problem is investigated which illustrates the role of wave-particle interactions in the evolution of runaway beams. The model considers the interaction between a weak cold beam, driven by an external static electric field E, and waves quantized by the geometry. The waves may correspond to Gould-Trivelpiece modes fixed by the length of the experimental device, or to the finite length of the experimental device, or to the finite Fourier modes encountered in computer simulations. The physics consists of the sweeping of the accelerated beam through the resonance provided by each cavity mode. This process is formulated in analogy with the O'Neil-Winfrey-Malmberg problem but uses a spatially averaged description based on the exact energy and momentum conservation laws with the dynamics simplified through a WKB representation of the dispersion relation. This model shows that the beam can be clamped in velocity with the momentum push being transferred to the waves. The model has been extended to the relativistic and multi-mode cases. For E = 0 the nonrelativistic results are in good agreement with the work of O'Neil, et al., while in the relativistic case the model reproduces the nontrivial features of the computer study by Lampe and Sprangle. © 1979 American Institute of Physics.
