The survival threshold for plasma oscillations

The classical relativistic Vlasov-Maxwell system is used to model the dynamics of charged particles, and plasma oscillations or Langmuir’s waves are shown to exist and obey a Klein-Gordon’s type dispersion relation. Amusingly, a threshold of the wave number is provided for the survival of pure electrical oscillations at which waves are damped by the classical Landau damping (the faster electrons decay, the weaker Landau damping is), while magnetic fields disperse like a Klein-Gordon wave for the full range of wave numbers.