Saturation of plasma beat waves and de-tuning of relativistic Langmuir waves

An analysis is made of the saturation mechanism for the electrostatic plasma wave excited by two electromagnetic waves, and it is shown that the level is determined not only by the relativistic frequency shift but also by other nonlinear effects which are of the same order. It is shown that the difference between this and previous results, which suggest that there is only a relativistic frequency shift, arises from the boundary conditions imposed on the plasma. It is argued that the correct condition to be imposed is that the average Eulerian velocity be zero. The analytic results are directly checked by numerical simulations carried out using a 1-D particle in cell code showing excellent agreement between the two approaches. (author)