206 Chapter 6. Cold plasma waves in a magnetized plasma
Suppose now that waves are being excited by a line sourceqδ(x)δ(z)exp(−iωt), i.e., a
wire antenna lying along theyaxis oscillating at the frequencyω.In this case Eq.(6.109)
becomes
∂^2 φ
∂ξ^2+
∂^2 φ
∂η^2=
q
|SP|^3 /^2 ε 0δ(ξ)δ(η) (6.110)so that the equipotential contours excited by the line source are just static concentric circles
inξ,ηor equivalently, static concentric ellipses inx,z.
However, ifSandPhaveopposite signs, the situation is entirely different, because
now the equation is hyperbolic and has the form
∂^2 φ
∂x^2=
∣
∣
∣
∣
P
S
∣
∣
∣
∣
∂^2 φ
∂z^2. (6.111)
Equation (6.111) is formally analogous to the standard hyperbolic wave equation
∂^2 ψ
∂t^2=c^2∂^2 ψ
∂z^2(6.112)
which has solutions propagating along the characteristicsψ=ψ(z±ct).Thus, the solu-
tions of Eq.(6.111) also propagate along characteristics, i.e.,
φ=φ(z±√
−P/Sx) (6.113)which are characteristics in thex−zplane rather than thex−tplane. For a line source,
the potential is infinite at the line, and this infinite potential propagates from the source
following the characteristics
z=±√
−
P
S
x. (6.114)If the source is a point source, then the potential has the form
φ(r,z)∼q4 πε 0(
r^2
S+
z^2
P) 1 / 2 (6.115)
which diverges on the conical surface having cone angletanθcone=r/z=±
√
−S/P
as shown in Fig.6.4. These singular surfaces are called resonance cones and were first
observed by Fisher and Gould (1969). The singularity results because the cold plasma ap-
proximation allowskto be arbitrarily large (i.e., allows infinitesimally short wavelengths).
However, whenkis made larger thanω/vT,the cold plasma assumptionω/k>>vTbe-
comes violated and warm plasma effects need to be taken into account. Thus, instead of
becoming infinite on the resonance cone, the potential is large and finite and has afine
structure determined by thermal effects (Fisher and Gould 1971).