An O-mode wave of frequency travels through a slowly varying

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Plasma Physics · Waves in plasmas

Plasma Physics, Ch Waves in a non-uniform plasma: rays, reflection, mode conversion and radiofrequency heating

Problem

An O-mode wave of frequency

f = 90\,\mathrm{GHz}

travels through a slowly varying, unmagnetised plasma layer whose local dispersion relation is

\omega^2 = \omega_{pe}^2(x) + c^2k^2(x)

. ((a)) Starting from the constancy of the WKB energy flux

|E|^2v_g = \mathrm{const}

along the ray, and using the local group velocity

v_g = c^2k/\omega

for this dispersion relation, show that the WKB field amplitude scales as

|E| \propto k^{-1/2}

. ((b)) The wave passes from a point where

n_e = 1.0\times10^{19}\,\mathrm{m^{-3}}

to a point where

n_e = 4.0\times10^{19}\,\mathrm{m^{-3}}

. Using

|E| \propto k^{-1/2}

, find the ratio of the field amplitude at the second point to that at the first.