Dispersion Relation for Relativistic Streams of Finite Radius Page: 6 of 14
This report is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided to Digital Library by the UNT Libraries Government Documents Department.
The following text was automatically extracted from the image on this page using optical character recognition software:
relativistic factor). It is otherwise arbitrary. Each beam is characterized
by a particle density, particle mass, charge sign, velocity distribution, and
average velocity. These five quantities may differ for each beam but the
radius of each beam is the same. The cylinder is isolated in space. (See
Fig. 1. )
This system is in neutral equilibrium. At time t = 0 the system is
disturbed. The subsequent time evolution of the fields and other variables is
obtained by the Laplace transform and the assumption that all variables are
first order small. The result may be expressed in the following form. The
system behaves in time like ane , where the discrete wn are solutions
of a dispersion relation. There is also a term which decays with time, and
for given k becomes negligible as t >> 1/kc - w/k. This term apparently
establishes causality, i. e., that no signal can travel faster than c, so that at
time t = r, for example, there is no disturbance further than c'r away from.the
edge of the plasma cylinder. This term arises from a branch cut integral and
is apparently related to the branch cut integrals discussed by Brillouin8 and
Stratton9 in their analyses of signal velocities in dispersive media. All
quantities vary as eikz
The dispersion relation obtained may be simplified in two limits. One
where Ka >> 1, la [1 - ( ) ka) and the other where Ka << 1. Thus
the form of the equations depends not only on ka, but also on the phase velocity
of the wave. The 'thin beam' equation (Ka << 1) will then be obtained even for
ka >> 1, if w/kc = 1.
The physical significance of K when w/kc < 1 is that r is the wave vector
measured in a system moving with the wave velocity w/kc. Only in this sys-
tem can the longitudinal electric field be obtained from Poisson's equation, as
the charge distribution is stationary. (The particles move,. but the charge
Here’s what’s next.
This report can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Report.
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/m1/6/: accessed March 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.