Space charge and field waves in an electron beam

W. C. Hahn has shown that the basic characteristics of a new type of vacuum tube using a velocity modulated electron beam may be explained by means of waves propagating along the beam. For an "ideal" tube in which the beam was assumed to be of uniform density throughout its length he described the small amplitude, slow "space charge" waves which have axial symmetry. In the following paper a study is made first of the more general slow space charge waves which do not necessarily possess symmetry about the axis. Two cases are considered. First, a very high magnetic focusing field is assumed, so that the motion of electrons in any but the axial direction may be neglected. Then the magnetic focusing field is assumed to be completely absent, and waves having components of velocity of electrons in all directions are treated. Also in the following, attention is given to the fast "field waves" which may exist in the idealized tube under certain conditions. The waves have been termed "space charge" waves and "field waves" because, for the former type, the phase velocities are close to beam velocity and the wave energy is mainly in the electrons. In the case of the field waves, the phase velocities are large compared to beam velocity and the energy is mainly in the electromagnetic field