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Cutoffs
We have seen that electromagnetic wave propagation (in one dimension) through an
inhomogeneous plasma, in the physically relevant limit in which the variation
lengthscale of the plasma is much greater than the wavelength of the wave,
is well described by the WKB solutions, (569)(570). However, these
solutions break down in the immediate vicinity of a cutoff, where
, or a resonance, where
. Let us
now examine what happens to electromagnetic waves propagating through
a plasma when they encounter a cutoff or a resonance.
Suppose that a cutoff is located at , so that

(571) 
in the immediate vicinity of this point, where . It is evident, from the
WKB solutions, (569)(570), that
the cutoff point lies at the boundary between a region () in which
electromagnetic
waves propagate, and a region () in which the waves are evanescent.
In a physically realistic solution, we would expect the wave amplitude to
decay (as decreases) in the evanescent region . Let us search for
such a wave solution.
In the immediate vicinity of the cutoff point,
, Eqs. (555) and (571) yield

(572) 
where

(573) 
Equation (572) is a standard
equation, known as Airy's equation, and possesses two
independent solutions, denoted
and
.^{} The second solution,
,
is unphysical, since it blows up as
.
The physical solution,
, has the asymptotic
behaviour

(574) 
in the limit
, and

(575) 
in the limit
.
Suppose that a unit amplitude plane electromagnetic wave, polarized in the
direction, is launched
from an antenna, located at large positive , towards the cutoff point at .
It is assumed that at the launch point.
In the nonevanescent region, , the wave can be
represented as a linear combination
of propagating WKB solutions:

(576) 
The first term on the righthand side of the above equation represents the
incident wave, whereas the second term represents the reflected wave.
The complex constant is the coefficient of reflection.
In the vicinity of the cutoff point (i.e., small and positive,
or large and positive)
the above expression reduces to

(577) 
However, we have another expression for the wave in this region. Namely,

(578) 
where is an arbitrary constant.
The above equation can be written

(579) 
A comparison of Eqs. (577) and (579) yields

(580) 
In other words, at a cutoff point there is total reflection, since
, with a phaseshift.
Next: Resonances
Up: Waves in Cold Plasmas
Previous: Wave Propagation Through Inhomogeneous
Richard Fitzpatrick
20110331