Propagation in Electron-Ion Streams

A mathematical theory is given for the propagation of electromagnetic waves in electron-ion streams composed of $N$ discrete beams. The solution, which is fully relativistic, is obtained in vector form by an extension of Hansen's theory and takes explicit account of the initial and boundary conditions. When certain restructions are placed upon the transverse boundary conditions the general solution satisfying arbitrary initial conditions can be expanded in terms of a complete orthogonal set of elementary vector solutions. For this case the necessary and sufficient conditions are found for amplification and instability both in the terminated and the unterminated stream. The correct physical interpretation of the conventional "Ansatz" solutions is found together with the conditions under which they are valid. One is then able to distinguish amplified growing waves from reverse waves attenuated in the reverse direction. Finally the analysis is extended to the continuous velocity distribution. It is shown that the present treatment differs significantly from Landau's theory for the thermal plasma and the consequences of this are discussed.