Dispersion Relation for Relativistic Streams of Finite Radius Page: 7 of 14
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distribution in this moving frame is stationary. ) Since the longitudinal elec-
tric field is Lorentz invariant, this field is the same in the lab frame and
hence is the physically significant quantity.
These ideas immediately suggest the existence of two branches of the
dispersion relation. One where (Re w)/kc < 1 and the other where Re w/kc > 1.
For the former case, we have the situation just discussed, where a Lorentz
transformation may be made to a system where the charges are independent
of time though varying in z. For the latter case a Lorentz transformation
may be made to a system where the currents are independent of z, though
varying in time. These ideas are covered in more detail by Landau and
. For the case where the phase velocity of the wave is larger than c, in
the lab frame, the waves will radiate by a Cerenkov-type process.11 The
two-stream instability mechanism may provide an energy source for this
radiation. Unfortunately, our dispersion relation has not, as yet, been solved
to see if the above qualitative arguments are true.
From our dispersion relation the works of other authors may be obtained.
The results of Jackson,2 Bludman et al.,4 and others for the infinite beam case
may be recovered when xa >> 1. As mentioned, these results are now seen to
be invalid when w/kc = 1. It is moreover clear that a discontinuity in the
equations exists when w/kc z 1 and that only solutions with complex w are
found if w/kc-> 1. This is not found in the strictly one-dimensional case
considered by Buneman,12 where the waves are undamped.
The thin beam results of Sturrock6 and Finkelstein and Sturrock13 are
also recovered when Ka << 1. It is seen that the results obtained by these
authors are valid only if w/kc << 1.
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Landau, R. W. Dispersion Relation for Relativistic Streams of Finite Radius, report, December 21, 1964; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc1030342/m1/7/: accessed March 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.