The propagation of a plasma wave with trapped particles in a weakly inhomogeneous plasma

The propagation of a Langmuir wave with a small group of trapped particles in a weakly non-uniform plasma along a density gradient is considered. The dynamics of trapped electrons accelerated by the wave is investigated in the adiabatic approximation, including the relativistic case. The set of equations describing the self-consistent spatial evolution of the wave—trapped-particles system is obtained and analysed. It is shown that the possibility exists of wave transmission through a classically forbidden barrier as a consequence of the reversibility of the wave—particle interaction. The conditions of validity of the theory developed are also discussed.