Longitudinal Dynamics of a Charged-Particle Bunch in a Traveling Wave Field

In this work, which is devoted to the longitudinal dynamics of charged particles in the field of traveling wave, on which the theory of the resonance method of acceleration is based, a transition is made from the dynamics of individual charges to the dynamics of charge bunches as a whole in order to increase the effectiveness of mathematical modeling. A similar problem has already been solved for nonrelativistic bunches. Using the same method, the solution is extended to relativistic bunches. The method is based on the Lagrange equation for the longitudinal motion of a charged-particle bunch in the field of traveling wave; the solutions are the longitudinal size of a bunch and the width of the energy spectrum. The self-field of the bunch is taken into account in the equation and the limitations of linear approximations are eliminated.