Fundamentals of Plasma Physics

214 Chapter 7. Waves in inhomogeneous plasmas and wave energy relations

IfDis construed to be the Hamiltonian, then the first term in Eq.(7.17) can be identified as
the ‘kinetic energy’ while the second term can be identified as the ‘potential energy’. As an
example, consider the simple case of an electromagnetic mode in an unmagnetized plasma
which has nonuniform density, so that

D(k,x)=

c^2 k^2

ω^2

−1+

ω^2 pe(x)

ω^2

=0. (7.18)

The wave propagation can be analyzed in analogy to the problem of a particle in a poten-
tial well. Here the kinetic energy isc^2 k^2 /ω^2 ,the potential energy is−1+ω^2 pe(x)/ω^2 ,and
the total energy is zero. This is a ‘wave-particle’ duality formallylike that of quantum me-
chanics since there is a correspondence between wavenumber and momentum and between
energy and frequency. Cutoffs give wave reflection in analogy to the reflection of a particle
in a potential well at points where the potential energy equals the total energy. As shown
in Fig.7.2(a), when the potential energy has a local minimum, waves willbe trapped in
the potential well associated with this minimum. Electrostatic plasma waves can also ex-
hibit wave trapping between two reflection points;these trapped waves are called cavitons
(the analysis is essentially the same;one simply replacesc^2 by 3 ω^2 peλ^2 De).The bouncing
of short wave radio waves from the ionospheric plasma can be analyzed using Eq.(7.18) to-
gether with Eqs.(7.15) and (7.16). As shown in Fig.7.2(b) a wave resonance (i.e.,k^2 →∞)
would correspond to a deep crevice in the potential energy. One must be careful to use geo-
metric optics only when the plasma is weakly inhomogeneous, so that the waves change
sufficiently gradually as to satisfy the WKB criterion.

(a) (b)

totalenergy totalenergy

‘potentialenergy’

‘potentialenergy’

‘kineticenergy’ ‘kineticenergy’

wave

resonance

Figure 7.2: (a) Effective potential energy for trapped wave;(b) for wave resonance.

7.3 Surface waves - the plasma-filled waveguide

An extreme form of plasma inhomogeneity occurs when there is an abrupt transition from