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

The use of an intense beam of heavy ions to ignite a thermonuclear pellet at a distance from the final lens system requires a long thin beam with a very small longitudinal velocity spread. For perturbations long compared with the beam diameter, the longitudinal electric field due to the beam space charge can be approximated by the first derivative of the line charge density. This system of equations exhibits steepening of a finite amplitude wave. When other effects are included, such as the dispersion which occurs as the perturbation wavelength approaches the beam pipe diameter, a variation in density along the beam, or the exponential growth due to wall resistivity, the wave dynamics increase in complexity. Computer simulations are presented which illustrate some of the effects which can occur.