Wave Propagation in Linear Media | re-examined

3.8 Turn-on e ects in a wave guide
b
y
z
x
Figure 3.21: Rectangular wave guide.
constant along one axis and sinusoidal along the other, the sine spanning only a half period.
The cuto frequency then is
!p = c
;
b
(3.86)
b being the width of the wave guide, which is then half the transverse wavelength of the eld.
Remark (Plasma and cuto frequencies) To emphasise the similarity to the plasma
example, we retain the symbol !p for the cuto frequency, although it is normally denoted
by !c in the literature.
The steady-state components of this mode are well known (Piefke [16], Harrington [93]). For
frequencies above cuto , !>!p,wehave
Ez=Esin y x
e,j
b
Hx = j b! E cos y
b
e,j x
Hy = , ! E sin y
b e,j x ;
(3.87)
with the dispersion relation
= 1q
! 2 , !p
c
2 : (3.88)
Note that we haveomitted the explicit time dependence of the elds. In the evanescent region,
the propagation constant becomes imaginary,
j = = 1q
!p
c
2 , ! 2 ; (3.89)
and the eld components change to
Ez = E sin y x
e,
b
Hx = j b! E cos y
b
e, x
Hy = j ! E sin y
b e, x :
61
(3.90)