Wave Propagation in Linear Media | re-examined

3Wave propagation in electromagnetic transmission lines
the wave front are dominated by the plasma frequency, which again is not surprising. This
becomes most clear by inspection of the voltage at the source at low signal frequencies,
where the in nitely high source impedance permits U to oscillate freely. It shows therefore
a superposition of the high resonance frequency and a component determined by the signal
( gs. 3.13 and 3.14).
Remark (Dispersion e ects) That the plasma frequency ultimately dominates the
free oscillations is not at all unexpected. The resonance frequency has the smallest group
velocity vg = d!=d (note that we may safely use this de nition here since the wave
number k = + j is purely real in the non-evanescent mode). The fastest components,
on the other hand, are the high-frequent ones that constitute the sharp wave front. They
rush through, and the slow components around !p remain. That the local frequency
equals exactly the plasma frequency holds of course only in the limit t !1.
The local wavelength exhibits a complementary e ect. If we take a snapshot of the wave
along the line at some given time, we nd that a long way behind the wave front, the
oscillation gradually fades away. Here, it is the wavenumber that tends to zero as ! ! !p,
and thus the local wavelength grows in nitely.
The explanations given so far have been of a rather mathematical nature. There is, however,
also a physical interpretation why the wave front travels undistorted through the medium.
The electrons in the plasma have a nite mass and consequently a certain amount of inertia.
Hence they cannot react to the arriving wave front instantaneously, but only with a short
delay, and the wave front at the very instant t = x=c remains una ected. This explanation
was already given by Sommerfeld, who in turn owed it to Voigt [2, 12].
Remark (Confession of a graphical retouch) In the gures, the values of the waves
at X = T have been set to unity. This is, admittedly, not the correct numerical result
and has been introduced deliberatly to illustrate the actual height of the wave front. The
true value, as dictated by Fourier, naturally is 0:5 , and 1 is only the left-sided limit, as
will be demonstrated below.
The space-time evolution diagrams have a sampling grid that has been chosen su ciently
coarse to keep the computational e ort at a reasonable value. This impaired particularly the
area about the wave front, which we examine more closely now. A re ned computation of the
wave front shows that it remains indeed unaltered even for large values of X and T , as depicted
in g. 3.16 for the current and g. 3.17 for the voltage. The graphs show the spatial evolution
of the wave for di erent instances in time and reveal two remarkable features. First, voltage
and current are in phase for some time after X = T . This is plausible if we consider that
the high frequency components at the wave front see a real-valued characteristic impedance.
Thus any phase shift is impossible.
Second, the trailing edge of the wave front becomes steeper as it travels down the line. The
left point of the time interval in each of the gures indicates to a good approximation a
constant value of the wave function. It can be calculated in dependence on the X-value as
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