Consider an electron pulse (or “bunch”) of average density and average bunch velocity
in a surrounding plasma
of average electron density
. One is interested in deriving the propagation equations for plasma waves with relativistic phase velocities. A simplifying assumption is the presence of relatively slow moving ions at a very small fraction of the speed
of light c which is realistic for plasma ion temperatures of less than 10,000 K. One may also neglect in a first approximation the influence of the excited wake-field that affects the time-evolution of the electron pulse shape. Furthermore, one can consider the configuration of a cylindrical plasma in the absence of external magnetic fields; along the plasma containing tube
- axis one has a one-dimensional system
for which Maxwell's equations
can be written in the following simplified form for the electrical field
, average electron velocity in plasma
, charge
density
, current density
The equation of motion of a plasma electron with momentum
in the wake of a relativistic electron bunch of average velocity
can be then written as:
Because the driving electron pulse has a relativistic average velocity one can expect solutions of the equations of motion to be of the form of travelling waves:
Molecular dynamics experiments or computer simulations that include these equations provide results in the form of numerical data that are consistent with such travelling wave solutions.
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