Wave Propagation in Linear Media | re-examined

3Wave propagation in electromagnetic transmission lines
After some simple rearrangements, we nally obtain the complete expression for the current
along the line
I
I0
Likewise, we nd the voltage
I0
= e ,Xp1, 2
cos T ,
, 2 Z 1
0
sin X cos T p 1+ 2
1+ 2 , 2
U
p
L0 =C0 = ,e,Xp 1, 2 1
p sin T +
1= 2 , 1
+ 2 Z 1
0
d :
p 1+ 2 cos X sin T p 1+ 2
1+ 2 , 2
d :
(3.82)
(3.83)
Before giving numerical evaluations of the above expressions, we take a look at the their
range of validity. The lower end of the evanescence region is !0 = 0, i. e. a DC excitation. It
can be readily seen by setting =0that this raises no particular problem, and the special
case of a non-oscillating step source is covered by our model as we could have expected.
Things are di erent for a source oscillating with the plasma frequency !p, which is exactly
the boundary between evanescent and propagation mode. The steady-state part of the current
(3.69) reduces to the unattenuated oscillation cos !pt, and for t =0,we nd with the identity
sign x = 2 R 1 sin x
d that the current still complies with the boundary conditions. The
0
voltage, however, does not, because the steady-state solution (3.71) becomes unbounded.
The deeper reason for this is our assumption of a lossless plasma, which entails a pole of the
characteristic impedance at ! = !p. We could have circumvented this particular problem
by choosing a voltage instead of a current source, thereby forcing the current to vanish at
resonance.
Numerical evaluations of (3.82) and (3.83) are shown in g. 3.13 for the DC case = 0, g. 3.14
for = 0:2 , and g. 3.15 for = 0:8. The most striking property of the result is that the wave
front indeed goes straight through the medium with a velocity ofX=T = 1 or, expressed with
the unscaled variables, x=t = c. Thus the plasma is completely at rest before the disturbance
arrives with the speed of light. This nding is, in principle, nothing spectacular in that it
only con rms the more general results obtained by Sommerfeld. It emphasises, however, that
even in the evanescent region not the faintest evidence of superluminal wave propagation can
be found.
Apart from the wave front, there are other similarities between the three cases as well. The
in uence of the forced oscillation applied by the signal source diminishes with the attenuation
distance 1= 0 = c p 1
, so that further down the line, the wave is determined exclusively
!p 1, 2
by the transient solutions (3.74) and (3.76). Note that the transients consist solely of frequency
components above the plasma frequency. The evanescent region does not come into play here,
which is of course merely a mathematical e ect and bears no physical signi cance. The
free oscillations of the plasma far away from both the entrance (as for the current) and
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