A variational technique for determining the propagating modes of a finite plasma

Abstract

The use of a variational method to determine the propagating characteristics of a slow wave mode along a finite plasma column is described. The density of the column is assumed to be non-uniform, and to have a variation of the form n = n₀(l - αr²/a²), where a is the radius of the column. Rayleigh-Ritz procedures are used to determine the best approximation for the r.f. field variation over the cross section. For this purpose, a set of Jacobi orthogonal polynomials is constructed which are orthogonal over the cross section of the plasma. Both the d.c. density variation of a plasma and the r.f. field variations within the plasma are expressed in terms of this set of polynomials. This procedure yields results in good agreement with an exact field theory in the range βa < 2, where β is the propagation constant. For small βa, the propagating characteristics depend on the average value of the density within the column.