Plasma Waves in a Different Frame: a Tutorial for Plasma-based Electron Accelerators
We present an analysis of a nonlinear, relativistic longitudinal wave with sub-luminal phase velocity vp in a cold plasma. By a Lorentz transformation to a frame co-moving at velocity vp we reduce the problem to a simple second-order ordinary differential equation. Thus, the nonlinear waveform and important properties such as the maximum possible amplitude (wave-breaking limit) and the energy attainable by trapped electrons are easily obtained by exploiting the analogy with the mechanical motion of a particle in a potential well. This approach seems particularly suitable for a compact, tutorial introduction to plasma-based electron accelerators with little mathematical complexity.