Energy and momentum densities of a Landau-damped wave packet

Abstract

The energy of a Landau-damped electrostatic wave is a long-standing problem. Calculations based on a spatially infinite wave are deficient. In this paper, a wave packet is analysed. The energy density depends on second-order initial conditions that are independent of the first-order wave. Pictures of energy and momentum transfer to resonant electrons are presented. On physical grounds, suitable second-order initial conditions are proposed. The resulting wave energy agrees with fluid theory. The ratio between energy and momentum is not the phase velocity up, as predicted by fluid theory, but ½up, in agreement with Landau-damping physics.