Wave Propagation in Linear Media | re-examined

3Wave propagation in electromagnetic transmission lines
and had been of purely academic interest at that time, as Brillouin himself admitted [2].
The work on wave propagation in plasma, however, was stimulated by the research on the
properties of the ionosphere, which plays an important role in the transmission of radio
signals. A survey of this work and the dispersive e ects in the ionosphere was given by
McIntosh and Malaga [98].
In the previous chapter we mentioned microwave experiments that were thought to prove
superluminal wave propagation in the evanescent region of a wave guide or plasma that
exhibits the same dispersion properties. As a follow-up to this discussion we choose the
frequency of our signal source to be below the cuto frequency of the plasma. Of course the
spectrum of the pulse will extend into the pass band, however the spectral peak about the
carrier frequency can be con ned to the stop band.
At rst, we seek the steady-state solution of the wave. With a strictly monochromatic source
I(0;t)=I0cos !0t, wewould obtain a current
where 0 is the imaginary part of the wave number at !0,
Is
I0
0 = !p
c
= e , 0x cos !0t ; (3.69)
s
1 , !0
!p
2
: (3.70)
Bearing in mind that the characteristic impedance is imaginary, we readily obtain the voltage
from U = Z0I with the relation Re je j!t = , sin !t,
Us p
I0 L0 =C0 = ,
e , 0x
r
2
!p
!0
, 1
sin !0t : (3.71)
If the signal source is never turned o , then the complete expressions of voltage and current
must converge to this steady-state solution for t !1. To get the e ect of turn-on, we need
additional transient solutions such that the wave complies with the initial conditions for t =0
I(x; 0) = 0 ; U(x; 0) = 0 ; x>0: (3.72)
The voltage in (3.71) already satis es this requirement, the current does not. Hence we resort
to a little trick: we add a function that cancels the current everywhere except for the origin
x = 0, where the wave front I0 emerges at t = 0. We construct this desired function by
expanding the exponential function ,e , 0x antisymmetrically into the negative half space
x