Page 399 - Electromagnetics

2.5
v /v
p
2.0
v / v or v / v g 1.5

p
1.0
v /v
0.5 g

0.0
0.0 0.5 1.0 1.5 2.0 2.5
ω/ω c

Figure 5.5: Phase and group velocity for a hollow-pipe waveguide.

dω !
2
2
v g = = v 1 − ω /ω , (5.147)
c
dβ
√
2
where v = 1/ µ . Note that v g v p = v . We showlater that v g is the velocity of energy
transport within a lossless guide. We also see that as ω →∞ we have v p → v and
v g → v. More interestingly, as ω → ω c we find that v p → ∞ and v g → 0. This is shown
graphically in Figure 5.5.
We may also speak of the guided wavelength of a monochromatic wave propagating
with frequency ˇω in a waveguide. We define this wavelength as
2π λ λ
λ g = = = .
2
2
β 1 − ω / ˇω 2 1 − λ /λ 2
c c
Here
2π 2π
λ = √ , λ c = .
ˇ ω µ k c

Orthogonality of waveguide modes. The modal fields in a closed-pipe waveguide
ˇ
ˇ
obey several orthogonality relations. Let (E n , H n ) be the time-harmonic electric and
ˇ
ˇ
magnetic fields of one particular waveguide mode (TE or TM), and let (E m , H m ) be
the fields of a different mode (TE or TM). One very useful relation states that for a
waveguide containing lossless materials

ˇ ∗
ˆ z · ˇ e n × h dS = 0, m
= n, (5.148)
m
CS
where CS is the guide cross-section. This is used to establish that the total power carried
by a wave is the sum of the powers carried by individual modes (see below).

© 2001 by CRC Press LLC