3. Rectangular Waveguides

3.1 Geometry and Maxwell's equations inside

Consider a hollow rectangular pipe with internal width $a$ along $x$ and height $b$ along $y$, infinite extent along $z$, walls assumed perfectly conducting. The cross-section convention is $a > b$ (wide direction is $a$), and standard sizes have $a/b \approx 2$.

                  y
                  │
              b   ├──────────────┐
                  │              │   ← perfectly conducting walls
                  │              │
                  └──────────────┼──── x
                                 a
                  z (out of page): propagation direction

Inside the pipe, no charges and no currents (free space). Maxwell's equations reduce to the wave equation for each Cartesian component of $\vec{E}$ and $\vec{H}$:

$\nabla^2 \psi + k^2 \psi = 0, \quad k^2 = \omega^2 \mu_0 \varepsilon_0$

with boundary conditions imposed by the perfectly-conducting walls: tangential $\vec{E}$ vanishes at the walls, and normal $\vec{H}$ vanishes at the walls. We assume traveling-wave dependence in the propagation direction: $\psi(x, y, z, t) = \psi(x, y) e^{j(\omega t - \beta z)}$. Substituting,

$\nabla_t^2 \psi(x, y) + (k^2 - \beta^2)\psi(x, y) = 0$

where $\nabla_t^2 = \partial^2/\partial x^2 + \partial^2/\partial y^2$ is the transverse Laplacian and $\beta$ is the (longitudinal) phase constant. Define $k_c^2 \equiv k^2 - \beta^2$. The cross-section equation is

$\nabla_t^2 \psi + k_c^2 \psi = 0$

This is the 2D Helmholtz equation, and it splits cleanly under separation of variables. For TE modes, work with $H_z(x, y) = H_0 \cos(m\pi x / a)\cos(n\pi y / b)$. For TM modes, work with $E_z(x, y) = E_0 \sin(m\pi x / a)\sin(n\pi y / b)$. The boundary conditions select these specific sinusoidal patterns: cosines for TE so that $\partial H_z/\partial n = 0$ at the walls (which makes tangential $E$ zero), sines for TM so that $E_z$ itself vanishes at the walls.

Each solution is labeled by two non-negative integers $(m, n)$. We write TE$_{mn}$ and TM$_{mn}$ modes. The integers count the number of half-wave variations of the field across the $a$ and $b$ dimensions respectively.

3.2 Cutoff frequency derivation

Substituting the assumed form back into the Helmholtz equation gives

$k_c^2 = \left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2$

But we also have $k_c^2 = k^2 - \beta^2$, so

$\beta^2 = k^2 - k_c^2 = \omega^2 \mu_0 \varepsilon_0 - \left(\frac{m\pi}{a}\right)^2 - \left(\frac{n\pi}{b}\right)^2$

For a propagating wave we need $\beta^2 > 0$, which means

$\omega > \omega_c \equiv \frac{1}{\sqrt{\mu_0 \varepsilon_0}}\sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2}$

Converting to frequency,

$\boxed{f_c^{(mn)} = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}}$

This is the cutoff-frequency formula. Each mode has its own cutoff. Below $f_c$, $\beta^2 < 0$ and $\beta$ is imaginary; the wave is evanescent and decays as $e^{-|\beta|z}$. Above $f_c$, the mode propagates with phase constant $\beta = \sqrt{k^2 - k_c^2}$.

For TM modes we need $m \geq 1$ and $n \geq 1$ (else the field collapses). For TE modes we need at least one of $m, n$ nonzero. The absolute lowest cutoff in a rectangular waveguide therefore comes from TE$_{10}$ (since $a > b$), with

$f_c^{(10)} = \frac{c}{2a}$

For X-band waveguide with $a = 22.86$ mm, $f_c^{(10)} = 6.56$ GHz. Below 6.56 GHz this guide is opaque to any signal you push into it; above 6.56 GHz the TE$_{10}$ mode propagates. The next mode to come in is TE$_{20}$ at $f_c^{(20)} = c/a = 13.12$ GHz, and below that TE$_{01}$ at $f_c^{(01)} = c/(2b) = 14.76$ GHz (since $b = 10.16$ mm in WR-90).

The single-mode operating range of WR-90 is therefore roughly 6.56 GHz to 13.12 GHz; in practice, the IEEE-defined operating band is 8.2 to 12.4 GHz, leaving safety margins on both ends. Below 8.2 GHz the attenuation rises too fast as you approach cutoff; above 12.4 GHz you risk exciting TE$_{20}$ and getting unpredictable mode-conversion losses at imperfections.

3.3 The dominant TE$_{10}$ mode

TE$_{10}$ is the dominant rectangular-waveguide mode and the one almost all practical microwave systems use. Its field distribution comes from $H_z = H_0 \cos(\pi x / a)$ (no $y$-dependence since $n = 0$), and after computing the curl relations,

$E_y = -\frac{j\omega\mu_0 a}{\pi}H_0 \sin(\pi x / a)$ $H_x = \frac{j\beta a}{\pi}H_0 \sin(\pi x / a)$ $E_x = E_z = H_y = 0$

In words: only $E_y$ exists in the transverse-electric direction, peaked at the center of the broad wall ($x = a/2$), zero at the side walls. $H$ has both $H_x$ and $H_z$ components and forms closed loops in the $x$-$z$ plane. $E$ is uniform across the narrow dimension $b$.

   E-field arrows in cross-section (E_y, into/out of page in 3D side view):
 
   ┌───────────────────────────┐
   │ ↑  ↑↑↑  ↑↑↑↑↑  ↑↑↑  ↑    │   E peaks at center
   │ ↑  ↑↑↑  ↑↑↑↑↑  ↑↑↑  ↑    │   E zero at side walls
   │ ↑  ↑↑↑  ↑↑↑↑↑  ↑↑↑  ↑    │
   │ ↑  ↑↑↑  ↑↑↑↑↑  ↑↑↑  ↑    │
   └───────────────────────────┘
   x=0          x=a/2           x=a

The fact that $E_y$ peaks at the center and is uniform across $b$ has a practical consequence: probes inserted into the broad wall, perpendicular to the broad wall, at the centerline, couple maximally to the mode. This is the standard way to feed a rectangular waveguide from a coaxial line. Insert a coax with the inner conductor sticking into the wide dimension at the center, and you launch a TE$_{10}$ mode efficiently. Off-center probes couple poorly. Probes through the narrow wall do not couple at all.

3.4 Phase velocity, group velocity, and the "$v_p > c$" paradox

From $\beta = \sqrt{k^2 - k_c^2}$ with $k = \omega/c$ in air, $\beta = k\sqrt{1 - (f_c/f)^2}$. The phase velocity inside the guide is

$v_p = \omega/\beta = \frac{c}{\sqrt{1 - (f_c/f)^2}}$

This is greater than $c$. Just above $f_c$, $v_p$ tends to infinity. At $f \gg f_c$, $v_p$ approaches $c$ from above. We were promised nothing travels faster than light.

The resolution is the classical phase-versus-group-velocity distinction. Phase velocity is the speed of a single sinusoidal phase pattern: track a particular crest of a steady-state sine wave. But a steady sine carries no information. To send information you modulate the wave, creating a wavepacket made of multiple frequencies, and the envelope travels at the group velocity $v_g = d\omega/d\beta$. Differentiating $\beta^2 = (\omega/c)^2 - k_c^2$, $2\beta d\beta = (2\omega/c^2) d\omega$, so

$v_g = c^2 \frac{\beta}{\omega} = c\sqrt{1 - (f_c/f)^2}$

This is less than $c$. Combining,

$\boxed{v_p \cdot v_g = c^2}$

phase fast, group slow, product fixed. Information travels at $v_g < c$ and causality is preserved. Phase travels at $v_p > c$ but carries no information, so relativity is unbothered.

Beach analogy. Stand on the shore and watch ocean waves rolling in obliquely. The point where each crest meets the shoreline can sweep along the beach faster than any water molecule moves, but that sweeping point is not a "thing" you could send a message with. A bottle floating on the water moves at the actual wave speed, slower. Phase velocity is the sweeping point; group velocity is the bottle.

The guide wavelength is

$\lambda_g = \frac{2\pi}{\beta} = \frac{\lambda_0}{\sqrt{1 - (f_c/f)^2}}$

longer than the free-space wavelength. A 10 GHz signal in WR-90 has $\lambda_0 = 30$ mm but $\lambda_g \approx 40$ mm. Quarter-wave transformers in waveguide are sized in $\lambda_g$, not $\lambda_0$.

3.5 Wave impedance

For TE modes,

$\eta_{TE} = \frac{E_y}{H_x} = \frac{\omega \mu_0}{\beta} = \frac{\eta_0}{\sqrt{1 - (f_c/f)^2}}$

Greater than free-space $\eta_0 = 377$ Ω. For TM modes,

$\eta_{TM} = \frac{\beta}{\omega \varepsilon_0} = \eta_0 \sqrt{1 - (f_c/f)^2}$

Less than 377 Ω. These wave impedances are not the same as a "characteristic impedance" of the guide (which is ill-defined for waveguides because there is no unique voltage; the relationship between line voltage and the field integral depends on which two points you choose). For TE$_{10}$ specifically, common conventions normalize to $Z_{TE10} = (b/a) \cdot \eta_{TE}$ to get something with the right impedance scaling for matching to coax.

3.6 Power flow: integrating the Poynting vector

The time-averaged Poynting vector $\langle\vec{S}\rangle = \frac{1}{2}\text{Re}(\vec{E}\times\vec{H}^*)$ tells you power per unit area. To get the total power transported by the mode, integrate across the cross-section. For TE$_{10}$,

$P_{10} = \frac{1}{2}\int_0^a \int_0^b E_y H_x^* \, dy \, dx = \frac{|E_0|^2 ab}{4 \eta_{TE10}}$

This formula is what radar engineers use to figure out maximum power handling. The breakdown field of dry air at 1 atm is about 3 MV/m, so for $E_0 = 3 \times 10^6$ V/m in WR-90, the maximum power before arc-over is around 1 megawatt at 10 GHz. Real waveguides derate that for safety margins, humidity, dust, and connector imperfections, but megawatt-class radar transmitters routinely run pressurized waveguide (sulfur-hexafluoride or just dry nitrogen at a few atmospheres) to push that breakdown limit higher.

3.7 Losses and mode conversion

Real walls have finite conductivity. Copper has skin-depth resistance that turns a small fraction of wall current into heat each pass. The attenuation constant has the rough form

$\alpha_c \propto \frac{R_s}{\eta_0 b}\cdot\frac{1 + (b/a)(2/(f/f_c)^2 - 1)}{\sqrt{1 - (f_c/f)^2}}$

with $R_s = \sqrt{\omega \mu_0 / (2\sigma)}$. Two takeaways: loss diverges as you approach $f_c$ from above, and loss grows slowly with frequency above $f_c$. Pick operating bands with margin on both sides of cutoff. For X-band copper, 0.1 dB/m is typical; silver-plated guides shave that to 0.08 dB/m.

A separate loss mechanism is mode conversion at imperfections. A bend, flange joint, or internal scratch couples a fraction of TE$_{10}$ into TE$_{20}$, TM$_{11}$, or higher modes. If those higher modes are above their cutoffs, energy walks down the guide as the wrong mode and converts again at the next discontinuity, causing unpredictable amplitude and phase variation. Single-mode operation defends against this entirely: anything converted into TE$_{20}$ etc. is below cutoff and dies within a few cm.

3.8 Standard sizes (WR designations)

The North American convention uses **WR-**xx labels, where xx is the broad-wall dimension in hundredths of an inch.

WR-90 is the workhorse of X-band radar. Find any 10-GHz radar transmitter and you will find WR-90 piping the megawatts from the magnetron or klystron toward the antenna. WR-28 shows up in the back-end of millimeter-wave 5G radios. The dimensions tell you the cutoff and therefore the band.