In this section, we delve into the relationship between the
two-dimensional higher-order modes derived in Sec. 13.2 and their
sources. The examples are chosen to relate directly to case studies
treated in quasistatic terms in Chaps. 5 and 8.

The matching of a longitudinal boundary condition by a superposition
of modes may at first seem to be a purely mathematical process. However,
even qualitatively it is helpful to * think* of the influence of
an excitation in terms of the resulting
modes. For quasistatic systems, this has already been our experience.
For the purpose of estimating the dependence of the output signal on
the spacing *b* between excitation and detection electrodes, the EQS
response of the capacitive attenuator of Sec. 5.5 could be pictured in
terms of the lowest-order mode. In the electrodynamic situations of
interest here, it is even more common that one mode dominates. Above
its cutoff frequency, a given mode can propagate through a waveguide
to regions far removed from the excitation.

Modes obey orthogonality relations that are mathematically useful
for the evaluation of the mode amplitudes. Formally, the mode
orthogonality is implied by the differential equations governing the
transverse dependence of the fundamental field components and the
associated boundary conditions. For the TM modes, these are
(13.2.5) and (13.2.11).

** TM Modes:**

where

and for the TE modes, these are (13.2.8) and (13.2.12).

** TE Modes:**

where

The word "orthogonal" is used here to mean that

These properties of the modes can be seen simply by carrying out the
integrals, using the modes as given by (13.2.13) and (13.2.16). More
fundamentally, they can be deduced from the differential equations and
boundary conditions themselves, (1) and (2). This was illustrated in
Sec. 5.5 using arguments that are directly applicable here
[(5.5.20)-(5.5.26)].

The following two examples illustrate how TE and TM
modes can be excited in waveguides. In the quasistatic limit, the
configurations respectively become identical to EQS and MQS situations
treated in Chaps. 5 and 8.

#### Example 13.3.1. Excitation of TM Modes and the EQS Limit

In the configuration shown in Fig. 13.3.1, the parallel plates
lying in the planes *x = 0* and *x = a* are shorted at *y = 0* by a
perfectly conducting plate. The excitation is provided by
distributed voltage sources driving a perfectly conducting plate in
the plane *y = b*. These sources constrain the integral of **E**
across narrow insulating gaps of length between the respective
edges of the upper plate and the adjacent plates. All the conductors
are modeled as perfect. The distributed voltage sources maintain the
two-dimensional character of the fields even as the width in the *z*
direction becomes long compared to a wavelength. Note that the
configuration is identical to that treated in Sec. 5.5. Therefore, we
already know the field behavior in the quasistatic (low frequency)
limit.

** Figure 13.3.1 **Configuration for excitation of TM
waves.
In general, the two-dimensional fields are the sum of the TM and
TE fields. However, here the boundary conditions can be met by the TM
fields alone. Thus, we begin with *H*_{z}, (13.2.19), expressed as a
single sum.

This field and the associated **E** satisfy the boundary conditions
on the parallel plates at *x = 0* and *x = a*. Boundary conditions
are imposed on the tangential **E** at the longitudinal boundaries,
where *y = 0*

and at the driving electrode, where *y = b*. We assume here that the
gap lengths are small compared to other dimensions of interest.
Then, the electric field within each gap is conservative and the line
integral of *E*_{x} across the gaps is equal to the gap voltages *
v*. Over the region between *x = * and *x = a - *, the perfectly
conducting electrode makes *E*_{x} = 0.

Because the longitudinal boundary conditions are on *E*_{x}, we
substitute *H*_{z} as given by (5) into the *x* component of Faraday's
law [(12.6.6) of Table 12.8.3] to obtain

To satisfy the condition at the short, (6), *A*_{n}^{+} = A_{n}^{-} and (8)
becomes

This set of solutions satisfies the boundary conditions on three
of the four boundaries. What we now do to satisfy the "last" boundary
condition differs little from what was done in Sec. 5.5. The
*A*_{n}^{+}'s are adjusted so that the summation of product solutions
in (9) matches the boundary condition at *y = b* summarized by
(7). Thus, we write (9) with *y = b* on the right and with the
function representing (7) on the left. This expression is
multiplied by the *m*'th eigenfunction, *cos (m x/a)*, and
integrated from *x = 0* to *x = a*.

Because the intervals where _{x}(x, b) is finite are so
small, the cosine function can be
approximated by a constant, namely * 1* as appropriate. On the
right-hand side of (10), we exploit the orthogonality condition so
as to pick out only one term in the infinite series.

Of the infinite number of terms in the integral on the right in (10),
only the term where *n = m* has contributed. The coefficients follow
from solving (11) and replacing *m n*.

With the coefficients *A*_{n}^{+} = A_{n}^{-} now determined, we
can evaluate all of the fields. Substitution into (5), and (8) and
into the result using (12.6.7) from Table 12.8.3 gives

Note the following aspects of these fields (which we can expect
to see in Demonstration 13.3.1). First, the magnetic field is
directed perpendicular to the *x-y* plane. Second, by making the
excitation symmetric, we have eliminated the TEM mode. As a result,
the only modes are of order *n = 1* and higher. Third, at frequencies
below the cutoff for the TM_{1} mode, _{y} is imaginary and the
fields decay in the *y* direction.

^{6} *sin (ju) = j sinh
(u)* and *cos (ju) = cosh (u)*

Indeed, in the quasistatic limit
where ^{2} ( /a)^{2}, the electric field is the same
as that given by taking the gradient of (5.5.9). In this same
quasistatic limit, the magnetic field would be obtained by using this
quasistatic **E** to evaluate the displacement current and then
solving for the resulting magnetic field subject to the boundary
condition that there be no normal flux density on the surfaces of the
perfect conductors. Fourth, above the cutoff frequency for the *n =
1* mode but below the cutoff for the *n = 2* mode, we should find
standing waves having a wavelength *2 /*_{1}.

Finally, note that each of the expressions for the field
components has *sin (*_{n} b) in its denominator. With the
frequency adjusted such that _{n} = n /b, this function goes
to zero and the fields become infinite. This resonance condition
results in an infinite response, because we have pictured all of the
conductors as perfect. It occurs when the frequency is adjusted so
that a wave reflected from one boundary arrives at the other with
just the right phase to reinforce, upon a second reflection, the wave
currently being initiated by the drive.

The following experiment gives the opportunity to probe
the fields that have been found in the previous example. In practical
terms, the structure considered might be a parallel plate waveguide.

#### Demonstration 13.3.1. Evanescent and Standing TM Waves

The experiment shown in Fig. 13.3.2 is designed so that the field
distributions can be probed as the excitation is varied from below to
above the cutoff frequency of the TM_{1} mode. The excitation
structures are designed to give fields approximating those found in
Example 13.3.1. For convenience, *a = 4.8* cm so that the excitation
frequency ranges above and below a cut-off frequency of 3.1 GHz. The
generator is modulated at an audible frequency so that the amplitude
of the detected signal is converted to "loudness" of the tone from
the loudspeaker.

** Figure 13.3.2 ** Demonstration of TM evanescent and
standing waves.
In this TM case, the driving electrode is broken into segments,
each insulated from the parallel plates forming the waveguide and each
attached at its center to a coaxial line from the generator. The
segments insure that the fields applied to each part of the
electrode are essentially in phase. (The cables feeding each segment
are of the same length so that signals arrive at each segment in
phase.) The width of the structure in the *z* direction is of the
order of a wavelength or more to make the fields
two dimensional. (Remember, in the vicinity of the lowest cutoff
frequency, *a* is about one-half wavelength.) Thus, if the feeder were
attached to a contiguous electrode at one point, there would be a
tendency for standing waves to appear on the excitation electrode,
much as they did on the wire antennae in Sec. 12.4. In the
experiment, the segments are about a quarter-wavelength in the *z*
direction but, of course, about a half-wavelength in the *x* direction.

In the experiment, **H** is detected by means of a one-turn coil.
The voltage induced at the terminals of this loop is proportional to
the magnetic flux perpendicular to the loop. Thus, for the TM fields,
the loop detects its greatest signal when it is placed in an *x - y*
plane. To avoid interference with **E**, the coaxial line connected
to the probe as well as the loop itself are kept adjacent to the
conducting walls (where *H*_{z} peaks anyway).

The spatial features of the field, implied by the normalized
versus *k*_{y} plot of Fig. 13.3.2, can be seen by moving the probe
about. With the frequency below cutoff, the field decays in the *-y*
direction. This exponential decay or evanescence decreases to a linear
dependence at cutoff and is replaced above cutoff by standing waves.
The value of *k*_{y} at a given frequency can be deduced from the
experiment by measuring the quarter-wave distance from the short to
the first null in the magnetic field. Note that if there are
asymmetries in the excitation that result in excitation of the TEM
mode, the standing waves produced by this mode will tend to obscure
the TM_{1} mode when it is evanescent. The TEM waves do not have a
cutoff!

As we have seen once again, the TM fields are the electrodynamic
generalization of two-dimensional EQS fields. That is, in the
quasistatic limit, the previous example becomes the capacitive
attenuator of Sec. 5.5.^{7}

^{7}The example which was the
theme of Sec. 5.5 might equally well have been called the "microwave
attenuator," for a section of waveguide operated below cutoff is
used in microwave circuits to attenuate signals.

We have more than one reason to expect that the two-dimensional
TE fields are the generalization of MQS systems. First, this was seen
to be the case in Sec. 12.6, where the TE fields associated with a
given surface current density were found to approach the MQS limit as
^{2} k_{y}^{2}. Second, from Sec. 8.6 we know that
for every two-dimensional EQS configuration involving perfectly
conducting boundaries, there is an MQS one as
well.

^{8} The **H** satisfying the condition that **n**
**B** = 0 on the perfectly conducting boundaries was obtained
by replacing * A*_{z} in the solution to the analogous
EQS problem.

In particular, the MQS analog of the capacitor
attenuator is the configuration shown in Fig. 13.3.3. The MQS **H**
field was found in Example 8.6.3.

In treating MQS fields in the presence of perfect conductors, we
recognized that the condition of zero tangential **E** implied
that there be no time-varying normal **B**. This made it
possible to determine **H** without regard for **E**. We could then
delay taking detailed account of **E** until Sec. 10.1. Thus, in the
MQS limit, a system involving essentially a two-dimensional
distribution of **H** can (and usually does) have an **E** that
depends on the third dimension. For example, in the configuration of
Fig. 13.3.3, a voltage source might be used to drive the current in
the *z* direction through the upper electrode. This current is
returned in the perfectly conducting -shaped walls. The
electric fields in the vicinities of the gaps must therefore increase
in the *z* direction from zero at the shorts to values consistent
with the voltage sources at the near end. Over most of the length of
the system, **E** is across the gap and therefore in planes
* perpendicular* to the *z* axis. This MQS configuration does not
excite pure TE fields. In order to produce (approximately)
two-dimensional TE fields, provision must be made to make **E** as
well as **H** two dimensional. The following example and
demonstration give the opportunity to further develop an
appreciation for TE fields.

** Figure 13.3.3 ** Two-dimensional MQS configuration
that does not have TE fields.

#### Example 13.3.2. Excitation of TE Modes and the MQS Limit

An idealized configuration for exciting standing TE modes is
shown in Fig. 13.3.4. As in Example 13.3.1, the perfectly conducting
plates are shorted in the plane *y = 0*. In the plane *y = b* is a
perfectly conducting plate that is segmented in the *z* direction.
Each segment is driven by a voltage source that is itself distributed
in the *x* direction. In the limit where there are many of these
voltage sources and perfectly conducting segments, the driving
electrode becomes one that both imposes a *z*-directed **E** and has
no *z* component of **B**. That is, just below the surface of
this electrode, *w E*_{z} is equal to the sum of the source voltages.
One way of approximately realizing this idealization is used in the
next demonstration.

** Figure 13.3.4 **Idealized configuration for
excitation of TE standing waves.
Let be defined as the flux per unit length (length taken
along the *z* direction) into and out of the enclosed
region through the gaps of width between the driving electrode
and the adjacent edges of the plane parallel electrodes. The
magnetic field normal to the driving electrode between the gaps is
zero. Thus, at the upper surface, *H*_{y} has the
distribution shown in Fig. 13.3.5a.

** Figure 13.3.5 **Equivalent boundary conditions on
normal **H** and tangential **E** at *y = b*.
Faraday's integral law applied to the contour *C* of Fig. 13.3.4
and to a similar contour around the other gap shows that

Thus, either the normal **B** or the tangential **E** on the surface
at *y = b* is specified. The two must be consistent with each other,
i.e., they must obey Faraday's law. It is perhaps easiest in this
case to deal directly with *E*_{z} in finding the coefficients
appearing in (13.2.20). Once they have been determined (much as in
Example 13.3.1), **H** follows from Faraday's law, (12.6.29) and
(12.6.30) of Table 12.8.3.

In the quasistatic limit, ^{2} (m /a)^{2}, this
magnetic field reduces to that found in Example 8.6.3.

A few observations may help one to gain some
insights from these expressions. First, if the
magnetic field is sensed, then the detection loop must have its
axis in the *x - y* plane. For these TE modes, there should be no signal
sensed with the axis of the detection loop in the *z* direction. This
probe can also be used to verify that **H** normal to the perfectly
conducting surfaces is indeed zero, while its tangential value peaks at
the short. Second, the same decay of the fields
below cutoff and appearance of standing waves above cutoff is predicted
here, as in the TM case. Third, because **E** is perpendicular to
planes of constant *z*, the boundary conditions on **E**, and hence
**H**, are met, even if perfectly conducting plates are placed over
the open ends of the guide, say in the planes *z = 0* and *z = w*.
In this case, the guide becomes a closed pipe of rectangular
cross-section. What we have found are then a subset of the
three-dimensional modes of propagation in a rectangular waveguide.

#### Demonstration 13.3.2. Evanescent and Standing TE Waves

The apparatus of Demonstration 13.3.1 is altered to give TE
rather than TM waves by using an array of "one-turn inductors" rather
than the array of "capacitor plates." These are shown in Fig. 13.3.6.

** Figure 13.3.6 **Demonstration of evanescent and
standing TE waves.
Each member of the array consists of an electrode of width *a - 2*,
driven at one edge by a common source and shorted to
the perfectly conducting backing at its other edge. Thus, the magnetic
flux through the closed loop passes into and out of the guide through
the gaps of width between the ends of the one-turn coil and the
parallel plate (vertical) walls of the guide. Effectively, the
integral of *E*_{z} created by the voltage sources in the idealized
model of Fig. 13.3.4 is produced by the integral of *E*_{z} between the
left edge of one current loop and the right edge of the next.

The current loop can be held in the *x - z* plane to sense *H*_{y} or
in the *y - z* plane to sense *H*_{x} to verify the field distributions
derived in the previous example. It can also be observed that placing
conducting sheets against the open ends of the parallel plate guide,
making it a rectangular pipe guide, leaves the characteristics of
these two-dimensional TE modes unchanged.