Wave Propagation in Linear Media | re-examined

3Wave propagation in electromagnetic transmission lines
classical understanding of the phase and group velocities. The slope of the stripes yields the
phase velocity, and the contour of the entire pulse (most easily traced at the edge where they
fade into the grey surroundings) de nes the group velocity. Obviously, these two velocities
are di erent, which empirically a rms the theoretical ndings of section 1.1: in a dispersive
medium, the envelope of a pulse propagates at a distinct velocity with the train of carrier
oscillations moving indepently underneath its overall shape. As for the determination of
from the graph, both (3.108) and (3.109) give the correct answers.
Remark (Dispersion e ects) In a dispersive medium, we should of course be able
to observe a broadening of the pulse. We are, however, at a loss to do so in this case
because the initial pulse is too wide and the distance too short for this e ect to become
visible. But if we su ciently reduced the plotted voltage range (by some nine orders of
magnitude), we would encounter an e ect similar to that in g. 3.26: namely, a small
portion of the wave travelling, if at all, at very low speed. Again, this is caused by the
small frequency range immediately above cuto where the group velocity is hardly larger
than zero. In contrast, the e ect of the evanescent components (which are present also in
this case) is not noticeable at all.
In accordance with the theory, the examples show that if the spectrum of a pulse is bundled
below cuto , there is no phase shift, and the pulse decays exponentially without allowing for
wave propagation. The amount of attenuation is determined by the centre frequency of the
pulse, !0, so that we can estimate the voltage amplitude by
U(X) / e ,Xp 1, 2
: (3.110)
If in a data transmission system we decided, for the sake of proper signal detection, to
tolerate an attenuation of 10 ,4 (80 dB), we could calculate the admissible length of the
line. For = 0:5 this would give X = 10:64, whereas with = 0:8, we could cover a
normalised distance of X =15:35. The actual length of the transmission line can be obtained
from the de nition of the scaled space coordinate, x = Xc=!p. In a plasma with a cuto
frequency of 10 GHz, X = 10 corresponds to an actual distance of x = 4:77 cm. So if we
were to use signals with superluminal group velocities (as they are sometimes called) for data
transmission, we could cover but a few centimetres. This, however, makes their usefulness for
practical applications rather questionable.
The maximum of a band-limited pulse can be transmitted at a velocity greater than that of
light, there is no doubt about it. But does this mean that information can also be transmitted
superluminally? The response of a linear system to an input signal can be written as a
convolution integral,
u(x; t) =
Z 1
,1
of the input u(0;t) and the transfer function
g(x; t) = 1
2
u(0; ) g(x; t , ) d ; (3.111)
Z 1
e
,1
,jk(!)x e j!t d! : (3.112)
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