6.5 High frequency waves: Altar-Appleton-Hartree dispersion relation 199Equation (6.71) can be Taylor-expanded in the vicinity of the principle anglesθ= 0
andθ=π/ 2 to give dispersion relations for quasi-parallel or quasi-perpendicular propaga-
tion. The terms quasi-longitudinal and quasi-transverse are commonly used to denote these
situations. The nomenclature is somewhat unfortunate because of the possible confusion
with the traditional convention that longitudinal and transverse refer tothe orientation of
krelative to the wave electric field. Here, longitudinal meanskis nearly parallel to the sta-
tic magnetic field while transverse meanskis nearly perpendicular to the static magnetic
field.
6.5.1 Quasi-transverse modes (θ≃π/2)
For quasi-transverse propagation, the first term inΓdominates, that is
ω^4 ce
ω^4sin^4 θ>> 4ω^2 ce
ω^2(
1 −
ω^2 pe
ω^2) 2
cos^2 θ. (6.73)In this case a binomial expansion ofΓgives
Γ =
ω^2 ce
ω^2sin^2 θ
1+4ω2
ω^2 ce(
1 −
ω^2 pe
ω^2) 2
cos^2 θ
sin^4 θ
1 / 2≃
ω^2 ce
ω^2sin^2 θ+2(
1 −
ω^2 pe
ω^2) 2
cot^2 θ. (6.74)Substitution ofΓinto Eq.(6.71) shows that the generalization of the ordinary mode disper-
sion to angles in the vicinity ofπ/ 2 is
n^2 +=1 −
ω^2 pe
ω^2
1 −ω^2 pe
ω^2cos^2 θ. (6.75)
The subscript+here means that the positive sign has been used in Eq.(6.71). This mode
is called the QTO mode as an acronym for ‘quasi-transverse-ordinary’.
Choosing the−sign in Eq.(6.71) gives the quasi-transverse-extraordinary mode or
QTX mode. After a modest amount of algebra (cf. assignments) the QTX dispersion is
found to be
n^2 −=(
1 −
ω^2 pe
ω^2) 2
−
ω^2 ce
ω^2sin^2 θ1 −
ω^2 pe
ω^2−
ω^2 ce
ω^2sin^2 θ. (6.76)
Note that the QTX mode has a resonance near the upper hybrid frequency.
6.5.2 Quasi-longitudinal dispersion (θ≃0)
Here, the term containingcos^2 θdominates in Eq.(6.72). Because there are no cancellations
of the leading terms inΓwith any remaining terms in the denominator of Eq.(6.71), it