Electrostatic wave generation above and below the plasma frequency by electron beams

The growth of electrostatic waves near the plasma frequency fp resulting from an unstable electron beam is investigated by solving the unmagnetized electrostatic dispersion equation numerically. Particular attention is given to finding the dispersion relations, frequencies of maximum growth, and resonant or nonresonant character of the waves for particular beam parameters. These numerical solutions are compared with analytic theories for reactive (or fluid-type) and kinetic versions of the beam instability, and for the O’Neil and Malmberg [Phys. Fluids 11, 1754 (1968)] connection of the beam and Langmuir modes. Conditions for these theoretical descriptions to be relevant, and conditions for growth significantly above or below fp are given. Three general results are found: (i) the unstable waves do not have the Langmuir dispersion relation except in the limit of a very dilute beam with growth on the connected mode of O’Neil and Malmberg; (ii) the properties of the unstable mode depend strongly on beam parameters such as beam density, speed, and temperature; (iii) the frequency of maximum growth frequently lies significantly above or below fp, and differs significantly from that predicted by the Langmuir dispersion relation. These results imply important consequences for strong turbulence theory, theories for nonlinear wave–wave processes such as the Langmuir wave decay and electromagnetic emission at multiples of fp, and observational identifications of fp from observed wave frequencies.