Metal pipe waveguides are often used to guide electromagnetic waves.
The most common waveguides have rectangular cross-sections and so are
well suited for the exploration of electrodynamic fields that depend
on three dimensions. Although we confine ourselves to a rectangular
cross-section and hence Cartesian coordinates, the classification of
waveguide modes and the general approach used here are equally
applicable to other geometries, for example to waveguides
of circular cross-section.

The parallel plate system considered in the previous three
sections illustrates much of what can be expected in pipe waveguides.
However, unlike the parallel plates, which can support TEM modes as
well as higher-order TE modes and TM modes, the pipe cannot transmit a
TEM mode. From the parallel plate system, we expect that a waveguide
will support * propagating* modes only if the frequency is high
enough to make the greater interior cross-sectional dimension of the
pipe greater than a free space half-wavelength. Thus, we will find
that a guide having a larger dimension greater than 5 cm would
typically be used to guide energy having a frequency of 3 GHz.

** Figure 13.4.1 **Rectangular waveguide.
We found it convenient to classify two-dimensional fields as
transverse magnetic (TM) or transverse electric (TE) according
to whether **E** or **H** was transverse to the direction of
propagation (or decay). Here, where we deal with three-dimensional
fields, it will be convenient to classify fields according to whether
they have **E** or **H** * transverse to the axial direction of
the guide*. This classification is used regardless of the
cross-sectional geometry of the pipe. We choose again the *y*
coordinate as the axis of the guide, as shown in Fig. 13.4.1. If we
focus on solutions to Maxwell's equations taking the form

then all of the other complex amplitude field components can be
written in terms of the complex amplitudes of these axial fields,
*H*_{y} and *E*_{y}. This can be seen from substituting fields having the
form of (1) and (2) into the transverse components of Ampère's
law, (12.0.8),

and into the transverse components of Faraday's law, (12.0.9),

If we take _{y} and _{y} as specified, (3) and
(6) constitute two algebraic equations in the unknowns _{x}
and _{z}. Thus, they can be solved for these components.
Similarly, _{x} and _{z} follow from (4) and (5).

We have found that the three-dimensional fields are a
superposition of those associated with *E*_{y} (so that the magnetic
field is transverse to the guide axis ), the TM fields, and those due
to *H*_{y}, the TE modes. The axial field components now play the role
of "potentials" from which the other field components can be derived.

We can use the *y* components of the laws of Ampère and
Faraday together with Gauss' law and the divergence law for **H** to
show that the axial complex amplitudes _{y} and _{y}
satisfy the two-dimensional Helmholtz equations.

** TM Modes** *(H*_{y} = 0):

where

and

** TE Modes** *(E*_{y} = 0):

where

These relations also follow from substitution of (1) and (2) into the
*y* components of (13.0.2) and (13.0.1).

The solutions to (11) and (12) must satisfy boundary conditions on
the perfectly conducting walls. Because *E*_{y} is parallel to the
perfectly conducting walls, it must be zero there.

** TM Modes:**

The boundary condition on *H*_{y} follows from (9) and (10), which
express _{x} and _{z} in terms of _{y}.
On the walls at *x = 0* and *x = a, *_{z} = 0. On the walls
at *z = 0, z = w, *_{x} = 0. Therefore, from (9) and
(10) we obtain

** TE Modes:**

The derivative of _{y} with respect to a coordinate
perpendicular to the boundary must be zero.

The solution to the Helmholtz equation, (11) or (12), follows a
pattern that is familiar from that used for Laplace's equation in Sec.
5.4. Either of the complex amplitudes representing the axial fields is
represented by a product solution.

Substitution into (11) or (12) and separation of variables then gives

where

Solutions that satisfy the TM boundary conditions, (13), are then

** TM Modes:**

so that

When either *m* or *n* is zero, the field is zero, and thus *m* and *n*
must be equal to an integer equal to or greater than one. For a given
frequency and mode number *(m, n)*, the wave number *k*_{y} is
found by using (19) in the definition of *p* associated with (11)

with

Thus, the TM solutions are

For the TE modes, (14) provides the boundary conditions, and we are
led to the solutions

** TE Modes:**

Substitution of _{m} and _{n} into (17) therefore gives

The wave number *k*_{y} is obtained using this eigenvalue in the
definition of *q* associated with (12). With the understanding that
either *m* or *n* can now be zero, the expression is the same as that
for the TM modes, (20). However, both *m* and *n* cannot be zero. If
they were, it follows from (22) that the axial **H** would be uniform
over any given cross-section of the guide. The integral of Faraday's
law over the cross-section of the guide, with the enclosing contour *C*
adjacent to the perfectly conducting boundaries as shown in Fig.
13.4.2, requires that

where *A* is the cross-sectional area of the guide. Because the contour
on the left is adjacent to the perfectly conducting boundaries, the
line integral of **E** must be zero. It follows that for the *m =
0, n = 0* mode, *H*_{y} = 0. If there were such a mode, it would
have both **E** and **H** transverse to the guide axis. We will
show in Sec. 14.2, where TEM modes are considered in general, that
TEM modes cannot exist within a perfectly conducting pipe.

** Figure 13.4.2 ** Cross-section of guide with contour
adjacent to perfectly conducting walls.
Even though the dispersion equations for the TM and TE modes only
differ in the allowed lowest values of *(m, n)*, the field distributions
of these modes are very different.

^{9} In other geometries,
such as a circular waveguide, this coincidence of *p*_{mn} and
*q*_{mn} is not found.

The superposition of TE modes gives

where *m n 0*. The frequency at which a given mode
switches from evanescence to propagation is an important parameter.
This * cutoff frequency* follows from (20) as

** TM Modes:**

** TE Modes:**

Rearranging this expression gives the normalized cutoff frequency as
functions of the aspect ratio *a/w* of the guide.

These normalized cutoff frequencies are shown as functions of *w/a*
in Fig. 13.4.3.

** Figure 13.4.3 **Normalized cutoff frequencies for
lowest rectangular waveguide modes as a function of aspect ratio.
The numbering of the modes is standardized. The dimension *w* is
chosen as *w a*, and the first index *m* gives the variation of
the field along *a*. The TE_{10} mode then has the lowest cutoff
frequency and is called the * dominant* mode. All other modes
have higher cutoff frequencies (except, of course, in the case of the
square cross-section for which TE_{01} has the same cutoff
frequency). Guides are usually designed so that at the frequency of
operation only the dominant mode is propagating, while all
higher-order modes are "cutoff."

In general, an excitation of the guide at a cross-section *y =*
constant excites all waveguide
modes. The modes with cutoff frequencies higher than the frequency
of excitation decay away from the source. Only the dominant mode has
a sinusoidal dependence upon *y* and thus possesses fields that
are periodic in *y* and "dominate" the field pattern far away from
the source, at distances larger than the transverse dimensions of the
waveguide.

#### Example 13.4.1. TE_{10} Standing Wave Fields

The section of rectangular guide shown in Fig. 13.4.4 is excited
somewhere to the right of *y = 0* and shorted by a conducting plate in the
plane *y = 0*. We presume that the frequency is above the cutoff
frequency for the TE_{10} mode and that *a > w* as shown. The
frequency of excitation is chosen to be below the cutoff frequency
for all higher order modes and the source is far away from *y = 0*
(i.e., at *y a*). The field in the guide is then that of the
TE_{10} mode. Thus, *H*_{y} is given by (25) with *m = 1* and
*n = 0*. What is the space-time dependence of the standing waves
that result from having shorted the guide?

** Figure 13.4.4 ** Fields and surface sources for
**TE**_{10} mode.
Because of the short, *E*_{z} (x, y = 0, z) = 0. In order to relate the
coefficients *C*^{+}_{10} and *C*^{-}_{10}, we must determine _{z}
from _{y} as given by (25) using (10)

and because _{z} = 0 at the short, it follows that

so that

and this is the only component of the electric field in this mode.
We can now use (29) to evaluate (25).

In using (7) to evaluate the other component of **H**, remember that
in the *C*^{+}_{mn} term of (25), *k*_{y} = _{mn}, while in the
*C*^{-}_{mn} term, *k*_{y} = -_{mn}.

To sketch these fields in the neighborhood of the short and
deduce the associated surface charge and current densities, consider
*C*^{+}_{10} to be real. The *j* in (31) and (32) shows that *H*_{x} and
*H*_{y} are 90 degrees out of phase with the electric
field. Thus, in the field sketches of Fig. 13.4.4, **E** and
**H** are shown at different instants of time, say **E** when
* t = * and **H** when * t = /2*. The surface
charge density
is where *E*_{z} terminates and originates on the upper and lower
walls. The surface current density can be inferred from Ampère's
continuity condition. The temporal oscillations of these fields
should be pictured with **H** equal to zero when **E** peaks, and
with **E** equal to zero when **H** peaks. At planes spaced by
multiples of a half-wavelength along the *y* axis, **E** is always
zero.

The following demonstration illustrates how a movable probe designed
to couple to the electric field is introduced into a waveguide with
minimal disturbance of the wall currents.

#### Demonstration 13.4.1. Probing the TE_{10}Mode.

A waveguide slotted line is shown in Fig. 13.4.5. Here the
line is shorted at *y = 0* and excited at the right. The probe
used to excite the guide is of the capacitive type, positioned so
that charges induced on its tip couple to the lines of electric field
shown in Fig. 13.4.4. This electrical coupling is an alternative to
the magnetic coupling used for the TE mode in Demonstration 13.3.2.

** Figure 13.4.5 **Slotted line for measuring axial
distribution of **TE**_{10} fields.
The *y* dependence of the field pattern is detected in the
apparatus shown in Fig. 13.4.5 by means of a second capacitive
electrode introduced through a slot so that it can be moved in the
*y* direction and not perturb the field, i.e., the wall is cut
* along* the lines of the surface current **K**. From the
sketch of **K** given in Fig. 13.4.4, it can be seen that **K** is in
the *y* direction along the center line of the guide.

The probe can be used to measure the wavelength *2 /k*_{y} of
the standing waves by measuring the distance between nulls in the
output signal (between nulls in *E*_{z}). With the frequency somewhat
below the cutoff of the TE_{10} mode, the spatial decay away from
the source of the evanescent wave also can be detected.