Wave Propagation in Linear Media | re-examined

3Wave propagation in electromagnetic transmission lines
which enables one to see that the modulated pulse penetrates deeper into the medium than
the base-band signal. This is in accordance with the dispersion relation and the respective
attenuation coe cient.
The bulk of the signal energy is concentrated within the parabola-shaped region in the timespace
graph. But we also recognise a small fraction of the pulse running away down the
line when the major part of the signal is already diminishing. The high oscillation frequency
of this low but very broad pulse (twice the carrier frequency) shows that it consists of the
spectral components just above cuto . In g. 3.24, which shows exactly the spectrum of this
particular case, the spectrum seems to be su ciently below cuto , but evidently it is not. We
can roughly calculate the frequency that dominates this part of the signal by measuring the
slope X= T parallel to the peaks and troughs visible in the graph. They give an estimate
for the phase velocity, while the lines connecting the outer ends of the ripples are related to
the group velocity. Using the de nition of the dispersion relation (3.97), we readily nd the
normalised expressions for the phase velocity,
and the group velocity,
vp
c =
r 2
vg
c =
r
1 , 1
2
2 , 1 ; (3.108)
; (3.109)
respectively. With these two equations, which directly correspond to the appropriately measured
slope X= T , we nally obtain an interval for the carrier frequency, 2 [1; 1:02].
The left bound of the interval is obtained from the left end of the propagating wave close
to X = 0 where the tangents to the ripples are horizontal and the ends of the valleys lie
on a vertical line. The same consideration applied to the right end of the wave gives the
upper limit of the interval. It is clear, though, that the data we can gather from the graph
to nd this limit is directly dependent on the resolution we choose for the voltage amplitude.
In reality, the interval is as unlimited as the spectrum of the pulse, and only the numerical
accuracy imposes constraints on the reliability of the results (which indeed prevented us from
discovering the same e ect in the base-band pulse). One must of course not overlook the fact
that the amplitude of the propagating part is nine orders of magnitude smaller than the main
pulse. Therefore it is likely to be missed in an experiment.
The third example ( g. 3.27) shows a pulse where the carrier frequency equals the cuto frequency
( = 1). Here, half the spectrum lies in the pass band. Thus the pulse is propagated,
but it is severely distorted. The reason for this behaviour is evidently the dispersion relation.
A moderate pulse distortion requires k(!) to be approximately linear in the frequency range
that makes up the dominating part of the spectrum. About cuto , this condition is de -
nitely not met, and we end up with a mixture of spectral components having grossly di erent
propagation velocities.
In g. 3.28, = 1:5 was chosen for the carrier. As we could have expected, this gives a
pulse propagating straight through the medium. This result can now be used to verify the
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