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Let us consider the transmission of electromagnetic waves along the
axis of a wave guide, which is simply a long, axially symmetric, hollow
conductor with open ends. In order to represent a wave propagating along
the direction,
we can write the dependence of the fields on the coordinate
variables and the time as

(1100) 
The guide propagation constant, , is just the
of previous sections, except that it is no longer restricted
by the boundary conditions to take discrete values. The general considerations
of Section 6.5 still apply, so that we can treat TM and TE modes separately.
The solutions for are identical to those for axially symmetric cavities
already discussed. Although is not restricted in magnitude,
we note that for every eigenvalue of the twodimensional equation,
, there is
a lowest value of , namely
(often designated for wave guides), for which is real.
This corresponds to the cutoff frequency below which waves are
not transmitted by that mode, and the fields fall off exponentially
with increasing . In fact, the wave guide dispersion relation for
a particular mode can easily be shown to take the form

(1101) 
where

(1102) 
is the cutoff frequency. There is an absolute cutoff frequency associated
with the mode of lowest frequency; i.e., the mode with the
lowest value
of .
For real (i.e.,
) it is clear from Eq. (6.50) that the wave is propagated
along the guide with a phase velocity

(1103) 
It is evident that the phase velocity is greater than that of electromagnetic
waves in free space.
This velocity is not constant, however, but depends on the frequency.
The wave guide thus behaves as a dispersive medium. The group
velocity of a wave pulse propagated along the guide is given by

(1104) 
It can be seen that is always smaller than , and also that

(1105) 
For a TM mode () Eqs. (6.34) yield
where use has been made of
.
For TE modes () Eqs. (6.35) give
The timeaverage component of the Poynting vector is
given by

(1110) 
It follows that

(1111) 
for TE modes, and

(1112) 
for TM modes. The subscript 0 denotes the peak value of a wave quantity.
Wave guide losses can be estimated by integrating Eq. (6.14) over the
wall of the guide for any given mode. The energy flow of
a propagating wave attenuates as , where

(1113) 
Thus,

(1114) 
where the numerator is integrated over unit length of the wall and the denominator
is integrated over the transverse cross section of the guide. It is
customary to define a guide impedance by writing

(1115) 
It follows from Eqs. (6.58) and (6.59) that

(1116) 
for TE modes, and

(1117) 
for TM modes. For both types of mode
.
Next: Dielectric wave guides
Up: Resonant cavities and wave
Previous: Cylindrical cavities
Richard Fitzpatrick
20020518