Wave Propagation in Linear Media | re-examined

3.7 Turn-on e ects in a lossless plasma
1
0.8
0.6
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0.2
9.9 9.95 10.05 10.1
1
0.8
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0.2
99.99 99.995 100.005 100.01
1
0.8
0.6
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0.2
999.999 1000 1000.
Figure 3.16: Wave front of the current for = 0:5 depending on the spatial coordinate.
1
0.8
0.6
0.4
0.2
9.9 9.95 10.05 10.1
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99.99 99.995 100.005 100.01
1
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999.999 1000 1000.
Figure 3.17: Wave front of the voltage for = 0:5 depending on the spatial coordinate.
T = X ,1=X, so that an arbitrary initial point (X0;T0) follows a phenomenological trajectory
given by
X , T
X0 , T0
= T0
T
: (3.84)
This again is due to dispersion. The high frequencies move ahead, the lower ones more and
more lag behind, such that the high-pass lter e ect becomes more marked. The behaviour
of the wave is therefore locally similar to the output of a high-pass lter whose characteristic
frequency is tuned higher and higher, i. e. the needle of the output pulse becomes sharper.
As last item of the examination, we discuss the propagation of a rectangular pulse through
the plasma. Owing to the linearity of the medium, we already stated in (3.68) that such a
pulse can be constructed from two delayed step functions. If the duration of the pulse is an
integer multiple n of the signal period, we can use identical step responses for switch-on and
switch-o and need not take care of the additional phase shift otherwise introduced by an
arbitrary pulse length. In our scaled variables, the pulse duration then becomes
T t= = 2n : (3.85)
We expect that both the leading and the trailing edge of the pulse cause a sharp wave front
travelling at the velocity of light. The contour plots for short pulses with low ( g. 3.18)
and high ( g. 3.19) input frequencies con rm this suspicion. They also show how the phase
trajectories of the peaks and troughs of the wave asymptotically approach the wave front.
Note that if we trace the evolution of such a peak, we nd that it travels always at a velocity
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