Abstract
Expressions for the kinetic energy and the kinetic momentum of longitudinal waves are derived including the nonlinear motion up to second-order in the perturbation. The Landau dispersion equation is then derived from the energy-momentum conservation laws. It is shown that the effect of the electric field of a wave on the motion of particles results both in quasi-harmonic oscillations (damped or growing) and in the changed value of uniform motion. Landau damping is due to the changed value of uniform motion, the field energy being transformed into the kinetic energy of this change in uniform motion. The Landau contour of integration includes the contribution from this change in uniform motion by means of integration around a pole. In the case of an instability the change in uniform motion, being negligible in comparison with the growing oscillatory motion, does not contribute to the energy transfer.