Wave Propagation in Linear Media | re-examined

3Wave propagation in electromagnetic transmission lines
What we want to study is the evolution of the elds if we switch on the transverse component
of the magnetic eld at t = 0 in such a way that only the TE01 mode is excited. This
assumption is quite reasonable since this mode has the lowest cuto frequency and is thus
the dominating mode in the wave guide [15, 93]. So the boundary condition reads
Hy(0;t)= (t)H0cos !0t sin y
b
: (3.91)
With the substitution H0 = j 0
! E and taking the real parts of the complex functions, we
nally obtain the steady-state elds
H0
Ez;s
Hx;s
H0
Hy;s
H0
=
p =" =
r
1
1 , !0
!p
= sin y
b e, 0x cos !0t
r !p
!0
1
2
2
, 1
cos y
b e, 0x cos !0t
sin y
b e, 0x sin !0t ;
(3.92)
where 0 = 1p
!p c
2 , !0 2 is the attenuation constant at the signal frequency. Comparing
these solutions with the ones we obtained for the current (3.69) and voltage (3.71) in the
plasma transmission line, we notice a striking similarity: the transverse component of the
magnetic eld, Hx, corresponds to the current in the transmission line, and the electric eld
to the negative voltage. As the initial conditions are also the same, we can apply the same
method we used successfully in the last section to obtain the transient parts of the
With the familiar scaled variables
elds.
= c
!p
; T = !pt; X= !px
c
; = !0
!p
; Y = y
b
we can immediately write down the complete solutions for the transverse elds,
H0
Hy
H0
= sin Y
Ez p = sin Y
="
h e ,X p 1, 2
, 2 Z 1
0
cos T ,
sin ( X) cos T p 1+ 2
1+ 2 , 2
h
,X
e p 1, 2 1
p sin T ,
1= 2 , 1
, 2 Z 1
0
p 1+ 2 cos ( X) sin T p 1+ 2
62
1+ 2 , 2
d
i
; (3.93)
d
i :
(3.94)
(3.95)