Resonant interaction of an electrostatic wave with electrons in a current sheet

A one-dimensional model of a current-sheet with a static electric potential is considered. Free electrons and protons move in opposite directions under the influence of the potential and we seek a solution for the potential in a bounded region in space (over the sheet width). The general solution is found to be plasma oscillations. Then introducing a short wavelength, monochromatic wave as a perturbation, it is shown by the method of stationary phase that electrons in resonance give a growth- (damping-) rate similar to the Landau formula. There is the possibility, however, that the wave can both grow and damp in different regions when traversing the sheet as it will sample electrons from various parts of the zero-order distribution function.