ABSTRACT
One of the main results of chapter 3 can now be summarized as follows. Any ‘hydrodynamic’ instability of a charged particle beam occurs because the slow wave of the beam space charge is synchronous with the wave excited in the medium, pumps this medium wave and thereby is enhanced itself; obviously, the wave energy is drawn from the energy of the beam. For instance, the synchronism condition in the case of the Budker–Buneman instability signifies (roughly) that u = ω 1/k ≃ ω +/k, and in the case of the pumping of Langmuir waves, that u − ω 1/k ≃ ω p/k, which agrees with the theory outlined above. This picture has one spectacular feature which I will now discuss in detail: it is found that the slow beam wave grows precisely because it loses energy by exciting a wave in the medium. The explanation of this apparent paradox lies in realizing, as will be shown below, that the slow wave of the beam space charge carries, in contrast to the fast wave, a negative energy. By the definition of this concept, this means that the total energy (kinetic and electric) of the beam in which the wave has been excited is less than the kinetic energy of the same beam without the wave, and that this energy deficiency is the result of energy transfer from the slow beam wave to the wave carrying positive energy.
