A model is presented that describes the interaction of electrons with both longitudinal and transverse waves in a cold plasma. Starting from the Lagrangian for the system of fields, background plasma, and particles, a finite dimensional self-consistent model is derived using the envelope approximation for the waves. The (squared) wave amplitudes and phases form action-angle variables in the closed system of waves and particles. The system conserves energy and momentum, and thus is natural for solving the beam-loading problem. Numerical simulations are performed to compare with earlier electrostatic problems.
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Evstatiev et al. (2003) studied this question.
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