Parametric instability in a bounded plasma-filled system

The parametric instabilities for modulated beams in a one-component system arise when the condition 2L=nλ, is satisfied, where L is the wavelength of the density or velocity modulation of the beam, λ is the plasma wavelength, and n = 1, 2, hor-ellipsis. The problem of the excitation of a parametric instability in a plasma-filled system was first treated by Fainberg and Shapiro in the linear approximation. They considered a beam in slab geometry, consisting of a periodic sequence of charged-neutralized infinitely wide uniform clumps moving with constant velocity in an unbounded plasma. It was found that the instability in this case also develops according to the condition. In the present work we describe the discovery of a parametric induced by the boundedness of a plasma-filled system. The excitation takes place under a condition analogous to (1), but instead of L in Eq. (1) the length of the system enters as a parameter. One of its boundaries is a conducting surface with an oscillating charge density and the other is virtual; about this more will be said in what follows. 13 refs., 1 fig