Rectangular Waveguides

The previous article proved that a hollow pipe carries only TE and TM modes, each with a cutoff, all slaved to two longitudinal field profiles that satisfy a drumhead eigenvalue problem. Proofs of existence are not field maps. This article solves the problem — completely — for the geometry that owns the microwave industry: the rectangular box, width $a$ , height $b$ . It is the chapter's flagship boundary-value solve, and it runs on machinery this track already built: separation of variables from the E2 articles, the PEC boundary package from E6, assembled here into the full mode catalog TE $_{mn}$ and TM $_{mn}$ with the cutoff ladder $\omega_{mn} = c\pi\sqrt{(m/a)^2 + (n/b)^2}$ falling out as the eigenvalue spectrum. One mode dominates the catalog and the industry. TE $_{10}$ — half a sine of electric field arching across the wide dimension, uniform along the short one — has the lowest cutoff of any mode, and beneath the next rung of the ladder lies an octave-wide window where it propagates alone. Single-mode operation is why waveguide plumbing works: one mode means one arrival time, one field map, one impedance, and a wave you can steer through bends and junctions without it scrambling into siblings. The article maps TE $_{10}$ 's fields and wall currents in full (explaining, among other things, why a slot down the center of the broad wall is invisible to the mode while a transverse slot is an antenna — the principle behind both the slotted measuring line and the slot-array radar antennas on ships), integrates its Poynting flux to get the megawatt-scale power ratings of real guides, and audits the standard WR-90 X-band guide number for number against its datasheet. What the eigenvalues mean dynamically — the hyperbolic dispersion, the strange velocities, evanescence below cutoff — is deliberately left to the chapter's closing article, by name. Here: the box, its modes, and the engineering the mode map buys.

Key takeaways

The framework article's transverse Helmholtz problem, solved in a rectangle by separation of variables (E2 machinery, recalled): TE modes $B_z \propto \cos(m\pi x/a)\cos(n\pi y/b)$ (Neumann walls), TM modes $E_z \propto \sin(m\pi x/a)\sin(n\pi y/b)$ (Dirichlet walls).
One eigenvalue ladder serves both families: $\gamma_{mn}^2 = (m\pi/a)^2 + (n\pi/b)^2$ , hence cutoffs $f_{mn} = \tfrac{c}{2}\sqrt{(m/a)^2 + (n/b)^2}$ . TE allows one index to vanish (TE $_{10}$ , TE $_{01}$ , ...); TE $_{00}$ does not exist (constant $B_z$ slaves to zero field); TM needs both indices (TM $_{11}$ lowest).
With $a > b$ , TE $_{10}$ ( $f_{10} = c/2a$ ) is the dominant mode; the single-mode window runs to the next cutoff ( $f_{20} = c/a$ for $a \geq 2b$ ) — a full octave.
TE $_{10}$ anatomy: $E_y = E_0\sin(\pi x/a)$ — no field variation along $y$ , E arching wall to wall; B loops in the $x$ – $z$ plane; wall currents axial down the broad-wall center (slots there are legal) and transverse elsewhere (slots there radiate).
WR-90 audited: $a = 22.86$ mm, $b = 10.16$ mm → $f_{10} = 6.56$ GHz, next cutoffs 13.1 (TE₂₀) and 14.8 GHz (TE₀₁); recommended band 8.2–12.4 GHz sits inside with guard margins; at 10 GHz the guide wavelength stretches to 3.97 cm; breakdown-limited power ~0.26 MW (derated air).
KEY IDEA: separation of variables in the rectangle turns the guided-wave eigenproblem into two quantized sine/cosine ladders; the resulting cutoff spectrum $f_{mn} = \tfrac{c}{2}\sqrt{(m/a)^2+(n/b)^2}$ hands TE₁₀ the lowest rung and an octave of solitude — the single-mode band on which microwave engineering is built.

The core insight

Solving the transverse eigenvalue problem in a rectangular cross-section quantizes the mode spectrum — TE $_{mn}$ with cosine profiles and Neumann walls, TM $_{mn}$ with sine profiles and Dirichlet walls, sharing the cutoff ladder $\omega_{mn} = c\pi\sqrt{(m/a)^2 + (n/b)^2}$ — and the lowest rung, TE $_{10}$ , owns an octave-wide single-mode band that makes practical waveguide transmission possible.

Where this lives today

The rectangular guide is the power spine of the microwave world: radar transmitters, satellite ground stations, industrial heating, and the RF distribution of every large particle accelerator (kilometer-scale runs of pressurized guide delivering tens of megawatts in pulses to accelerating cavities — where the resonant cavity, a waveguide with ends, is this article's mode catalog turned into standing waves and quality factors). The WR standard series — WR-2300 down to WR-3 — is the ladder formula (2) industrialized: each size an octave, each octave a catalog number, from below 1 GHz to above 300 GHz. The modern frontier miniaturizes and integrates. Substrate-integrated waveguides (SIW) fabricate the rectangular geometry inside a printed circuit board — via-fence walls, dielectric filling shrinking all cutoffs by $\sqrt{\varepsilon_r}$ — bringing single-mode guide discipline to 5G/6G millimeter-wave front ends at PCB cost. At the other extreme of scale and stakes: fusion experiments pipe megawatt-class millimeter waves (electron-cyclotron heating, 100–200 GHz) through corrugated and oversized guides where the mode catalog and its coupling rules are the difference between heating plasma and cooking the transmission line; and the same TE-mode physics in superconducting resonant cavities hosts the qubits of circuit QED — the drumhead spectrum of a box, read out one microwave photon at a time.

Origins · 1 minLord Rayleigh and how this was discovered

Lord Rayleigh

1842–1919 · Terling Place and Cambridge

John William Strutt, third Baron Rayleigh, solved this article's problem in 1897 — "On the passage of electric waves through tubes" — deriving the TE/TM mode structure, the cutoff frequencies, and the impossibility of the TEM mode in a hollow pipe, four decades before anyone could use it (his motivation was partly to explain why Lodge's attempts to send Hertzian waves down pipes behaved so oddly). It was a characteristic Rayleigh production: a complete, quietly definitive analysis of a wave system, of which he wrote dozens — the theory of sound (his two-volume classic, begun on a Nile houseboat honeymoon), Rayleigh scattering and the blue of the sky, Rayleigh waves in solids (the destructive surface waves of earthquakes), the Rayleigh criterion for optical resolution, the Rayleigh–Jeans law whose "ultraviolet catastrophe" helped force Planck's hand. He ran his physics from a laboratory in his country house, won the 1904 Nobel Prize for the meticulous density measurements (with Ramsay) that uncovered argon — a discovery born from a discrepancy in the third decimal place — and served as Cavendish professor, Royal Society president, and the model of the complete classical physicist. When Southworth's Bell Labs team rediscovered waveguide propagation experimentally in the 1930s, the theory they needed was waiting, fully formed, in a paper older than their careers.

History

1893

— J.J. Thomson analyzes electric oscillations in conducting cylinders; the cavity idea stirs

1897

— Rayleigh's tube paper: TE/TM modes, cutoffs, and the no-TEM verdict, complete

1920

s — Waveguides languish as curiosities; coax and open lines carry the radio age

1933

— Southworth (Bell Labs) and Barrow (MIT) independently propagate microwaves in hollow pipes

1940

— The cavity magnetron makes centimeter waves practical; waveguide plumbing becomes war materiel

1942

— The MIT Radiation Laboratory codifies waveguide engineering; the WR standard series and slotted-line practice emerge

1953

— Slot-array antennas turn Derivation B's current map into steerable radar apertures

2000

s — Substrate-integrated waveguides put the rectangular mode catalog inside printed circuit boards

The physical picture

Before the algebra, know what you are looking for. A waveguide mode is a self-reproducing transverse pattern: a field shape that, after the wall reflections have their say, reconstructs itself as it slides down the axis. In the rectangle the candidate shapes are products of sines and cosines — the drumhead functions of a rectangular membrane, and the same separated solutions the E2 Cartesian article built for the Laplace equation, now with an eigenvalue attached. The walls do the quantizing: an integer number of half-oscillations must fit across each dimension, giving each mode its index pair $(m, n)$ — $m$ half-periods across the width $a$ , $n$ across the height $b$ — and its price, the transverse budget $\gamma_{mn}^2 = (m\pi/a)^2 + (n\pi/b)^2$ that (via the framework article's wavenumber budget) sets its cutoff. The two families differ only in which longitudinal field carries the pattern, and the wall conditions do the sorting. TM modes carry $E_z$ , which is tangential to every wall and must therefore vanish there (Dirichlet): sine profiles, and both indices at least 1 — a sine needs room to rise and fall. TE modes carry $B_z$ , whose wall condition works out to a vanishing normal derivative (Neumann — Derivation A audits this carefully, because it is the step textbooks wave at): cosine profiles, which are perfectly happy to be flat across one dimension, so a single index may vanish. That asymmetry is not a technicality; it is the entire reason hollow-pipe transmission works. It hands TE the $\,_{10}$ mode — flat along $y$ , one half-sine along $x$ — whose cutoff $c/2a$ undercuts every other mode's by enough to open an octave of single-mode operation. (And it kills TE $_{00}$ : a constant $B_z$ satisfies Neumann conditions but slaves to identically zero transverse fields — the framework's $1/\gamma^2$ machinery with nothing to differentiate. No mode without structure.) Two pictures to carry out of this article. First, the field map: TE $_{10}$ is an electric field arching from broad wall to broad wall, strongest down the center line, zero at the side walls, with magnetic field looping around it in the $x$ – $z$ plane — a pattern you can draw from memory once you have seen it, and should. Second, the current map its walls carry: where the currents run along a slot, the slot is invisible; where they must cross it, the slot interrupts them and radiates. That single idea is a measuring instrument (the slotted line), a family of antennas (slot arrays), and a microwave-oven door, all at once.

Figure

The TE10 mode mapped — electric field arching across the guide with one half-sine of structure, magnetic loops around it, and the wall currents that decide where a slot may be cut

The mode catalog

The framework article's eigenproblem, in the rectangle $0 \leq x \leq a$ , $0 \leq y \leq b$ (convention: $a > b$ ): solve $(\nabla_\perp^2 + \gamma^2)\psi = 0$ for $\psi = B_z$ (TE, Neumann walls) or $\psi = E_z$ (TM, Dirichlet walls). Separation of variables — the method owned by the E2 Cartesian article and recalled, not re-taught — delivers both catalogs at once (Derivation A):

\text{TE}_{mn}:\ B_z = B_0\cos\frac{m\pi x}{a}\cos\frac{n\pi y}{b}, \qquad \text{TM}_{mn}:\ E_z = E_0\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{b}, \qquad (1)

where: the cosines satisfy Neumann conditions at all four walls, the sines Dirichlet; $m$ counts half-oscillations across $a$ , $n$ across $b$ . Index rules: TM needs $m, n \geq 1$ (a sine with a zero index is identically zero); TE allows one zero index but not both — TE $_{00}$ 's constant $B_z$ slaves to zero transverse field. Both families draw cutoffs from one ladder:

\gamma_{mn}^2 = \left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2, \qquad f_{mn} = \frac{c\,\gamma_{mn}}{2\pi} = \frac{c}{2}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2}, \qquad (2)

where: the ladder is the Pythagorean spectrum of the box — each mode's cutoff is set by how much transverse structure it must maintain. Lowest rung, since $a > b$ : TE $_{10}$ at $f_{10} = c/2a$ — the dominant mode, cut off when the width is half a free-space wavelength (the guide is "too narrow for the wave to fit sideways" below that, a slogan the closing article makes exact with the bouncing-wave picture). The single-mode window runs from $f_{10}$ up to the next rung — $f_{20} = c/a$ if $a \geq 2b$ (the standard proportion, chosen exactly so TE₂₀ arrives no later than TE₀₁) — a guaranteed octave. The dominant mode's full anatomy (Derivation B), with propagation factor $e^{i(kz - \omega t)}$ understood:

E_y = E_0\sin\frac{\pi x}{a}, \qquad B_x = -\frac{k}{\omega}E_0\sin\frac{\pi x}{a}, \qquad B_z = \frac{i\pi}{\omega a}E_0\cos\frac{\pi x}{a}, \qquad (3)

where: the single electric component arches wall to wall (zero at the sides, as tangential E must be), the transverse B partners it Fresnel-style through $k/\omega$ , and the longitudinal $B_z$ — a quarter-cycle out of phase, note the $i$ — closes the magnetic loops in the $x$ – $z$ plane. Everything measurable follows: the wall-current map (axial along the broad-wall centerline — the slot rule), the time-averaged power (Derivation C)

\langle P\rangle = \frac{E_0^2\,ab}{4Z_0}\sqrt{1 - \frac{f_{10}^2}{f^2}}, \qquad (4)

where: $Z_0 = 377\ \Omega$ , and the square root — the mode's obliquity factor — collapses the power to zero at cutoff (nothing travels there; the closing article's story) and saturates toward the plane-wave value far above. Numbers for the workhorse WR-90 X-band guide ( $a = 22.86$ mm, $b = 10.16$ mm): ladder rungs 6.56, 13.1 (TE₂₀), 14.8 (TE₀₁), 16.1 (TE₁₁/TM₁₁) GHz; recommended band 8.2–12.4 GHz — tucked inside the octave with guard margins (worked example 1 explains both margins); breakdown-limited power about a quarter megawatt derated (example 1); guide wavelength at 10 GHz stretched to 3.97 cm from the free-space 3.0 cm (the dispersion article cashes that stretch).

Figure

The mode gallery — half-wave counting across each dimension, with the index rules that forbid TE00 and any zero-index TM mode

Derivation AThe complete mode catalog by separation of variables

The goal is (1)–(2), both families, with the boundary-condition step — why TE means Neumann — done honestly rather than asserted. The strategy: separate the Helmholtz equation (the E2 Cartesian method, recalled by name: assume a product, split the constant), then impose the PEC wall package through the framework article's slaving relations, which is where the Neumann condition actually comes from.

Step 1. Separate. Put $\psi(x,y) = X(x)Y(y)$ into $(\nabla_\perp^2 + \gamma^2)\psi = 0$ :

\frac{X''}{X} + \frac{Y''}{Y} + \gamma^2 = 0 \quad\Longrightarrow\quad X'' = -k_x^2 X,\ \ Y'' = -k_y^2 Y,\ \ k_x^2 + k_y^2 = \gamma^2,

with separation constants forced negative (oscillatory) by the boundary conditions to come — a decaying exponential cannot vanish, or have vanishing slope, at both ends of an interval. Each direction contributes sines and cosines; the walls choose between them.

Step 2. TM walls are Dirichlet by inspection. $E_z$ is tangential to every wall of the guide (the walls are parallel to $z$ ), and the PEC package (electrodynamic-boundary-conditions article, recalled) demands tangential E vanish: $E_z = 0$ on all four walls. Sines it is: $E_z = E_0\sin(m\pi x/a)\sin(n\pi y/b)$ , $m, n \geq 1$ .

Step 3. TE walls are Neumann — via the slaving relations, not by analogy. The PEC package says nothing about $B_z$ directly (it is tangential B, which may be anything). The real conditions are on the slaved transverse fields: tangential E must vanish at the walls. Take the wall $x = 0$ : the tangential electric components there are $E_y$ and $E_z$ ; $E_z = 0$ already (TE), and the framework's slaving relation with $E_z = 0$ reads $E_y = -\tfrac{i\omega}{\gamma^2}\,\partial_x B_z$ . Demanding $E_y = 0$ at $x = 0, a$ forces $\partial_x B_z = 0$ there; the walls $y = 0, b$ likewise force $\partial_y B_z = 0$ . (The normal-B condition, $B_x = \tfrac{ik}{\gamma^2}\partial_x B_z = 0$ at $x = 0, a$ , is then satisfied automatically — one audit, two conditions honored.) Cosines: $B_z = B_0\cos(m\pi x/a)\cos(n\pi y/b)$ . Why this step deserves its paragraph: "B_z has Neumann conditions" is a slogan whose justification routes through the slaving algebra; guides with dielectric walls or losses modify exactly this step, and knowing where the condition comes from is knowing when it fails.

Step 4. The ladder and its exclusions. Both families inherit $\gamma_{mn}^2 = (m\pi/a)^2 + (n\pi/b)^2$ and cutoff $f_{mn}$ per (2). TM excludes any zero index (sine dies). TE excludes only the double zero: for TE $_{00}$ , $B_z$ = const has $\gamma = 0$ and every slaved component carries $\partial B_z = 0$ — no fields. Equivalently and deeper: a nontrivial $\gamma = 0$ mode would be TEM, and the previous article's theorem forbids TEM in a hollow pipe. The counting conspires to hand TE $_{10}$ the bottom rung unopposed.

Step 5. Orthogonality and completeness, in one breath. The catalog (1) is a complete orthogonal basis on the cross-section (the standard Sturm–Liouville facts, as in the E2 articles): any field configuration at a guide's mouth decomposes uniquely into modes, each then propagating with its own $k_{mn}$ . This is the licence for every "excite it and see what survives" argument in waveguide practice — a probe couples to whatever modes it overlaps; below their cutoffs, all but TE $_{10}$ die evanescently within a guide-width or so (closing article), and the surviving field is pure dominant mode regardless of how messily it was launched. Mode purity is not achieved by careful launching; it is enforced by the ladder. Verified: catalog and ladder match Griffiths §9.5.2 (Eqs. 9.186–9.190) and Jackson §8.4; the Neumann-via-slaving audit follows Jackson's Eq. 8.29 boundary treatment; Sturm–Liouville completeness per the E2 separation-of-variables articles; TE00 exclusion consistent with the no-TEM theorem of the previous article.

Go deeper · Derivation B · 2 minTE10 in full — fields, wall currents, and the slot rule

Derivation BTE10 in full — fields, wall currents, and the slot rule

The goal is the working anatomy (3): every nonzero component of the dominant mode, the surface currents it drives, and the slot rule that turns the current map into instruments. The strategy: feed $B_z = B_0\cos(\pi x/a)$ , $m = 1$ , $n = 0$ into the framework's slaving relations and read everything off, then apply the PEC surface-current condition from the boundary-conditions article.

Step 1. Slave the transverse fields. With $E_z = 0$ and $\partial_y B_z = 0$ (no $y$ -structure), the slaving relations collapse to one electric and one magnetic transverse component:

E_y = -\frac{i\omega}{\gamma^2}\,\partial_x B_z = \frac{i\omega a}{\pi}\,B_0\sin\frac{\pi x}{a}, \qquad B_x = \frac{ik}{\gamma^2}\,\partial_x B_z = -\frac{ika}{\pi}\,B_0\sin\frac{\pi x}{a},

with $\gamma = \pi/a$ . Renaming the amplitude ( $E_0 \equiv i\omega a B_0/\pi$ ) gives (3). Note the structure: $E_y$ and $B_x$ share the $\sin(\pi x/a)$ profile and travel in phase (their ratio is the real number $-k/\omega$ ) — these two do the transporting — while $B_z$ carries the $i$ : a quarter-cycle offset, standing rather than traveling, closing the field loops. Why $E_y/B_x = -\omega/k$ instead of $-c$ : the mode's transverse fields mimic a plane wave whose effective impedance is enhanced by $\omega/kc = 1/\sqrt{1 - f_{10}^2/f^2}$ — the wave impedance of the guide, another quantity that diverges at cutoff.

Step 2. Audit the walls. Side walls $x = 0, a$ : $E_y \propto \sin(\pi x/a)$ vanishes ✓ (tangential E), $B_x \propto \sin$ vanishes ✓ (normal B). Broad walls $y = 0, b$ : $E_y$ is normal there (legal — it terminates on surface charge), $E_z = 0$ ✓, and both B components are tangential (legal — they ride on surface currents). The mode fits the box exactly as the PEC package demands, with charges and currents as the bookkeepers.

Step 3. Map the currents. The surface current is $\mathbf{K} = \hat{\mathbf{n}}\times\mathbf{H}$ (boundary package, recalled). On the broad wall $y = 0$ ( $\hat{\mathbf{n}} = +\hat{\mathbf{y}}$ , pointing into the guide): $\mathbf{K} = \hat{\mathbf{n}}\times\mathbf{H} = \hat{\mathbf{y}}\times(H_x\hat{\mathbf{x}} + H_z\hat{\mathbf{z}}) = -H_x\,\hat{\mathbf{z}} + H_z\,\hat{\mathbf{x}}$ — an axial piece from $B_x$ (peaking at the centerline $x = a/2$ , where $\sin = 1$ ) and a transverse piece from $B_z$ (vanishing at the centerline, where $\cos = 0$ ). At the broad-wall centerline the current is purely axial; toward the edges it bends transverse and climbs the side walls (which carry pure $K_y$ from $B_z$ ).

Step 4. The slot rule, cashed three ways. A narrow slot cut along the local current direction interrupts nothing: the currents flow past it and the mode never notices — hence the slotted line (a lengthwise slot at the broad-wall centerline through which a probe rides to sample the interior standing-wave pattern: the classic microwave measurement) and hence, inverted, the slot antenna: cut the slot across the current (transverse slot in the broad wall, or a slanted slot displaced from the centerline) and the severed current must divert around it, driving the slot as a radiating aperture. Ship radar antennas — long waveguides with computed arrays of tilted slots — are Step 3's current map, monetized. Third cash-out: the microwave oven door's viewing mesh works partly because its fine grid leaves wall-current paths intact at 2.45 GHz. Verified: fields match Griffiths Eq. 9.176-family and Pozar Table 3.2 (TE10 entries); wall-current map and slot-coupling rules per Pozar §4.8/slot-array practice; the wave-impedance reading $Z_{TE} = Z_0/\sqrt{1 - f_c^2/f^2}$ standard.

Go deeper · Derivation C · 2 minPower down the pipe — the Poynting integral and the megawatt ceiling

Derivation CPower down the pipe — the Poynting integral and the megawatt ceiling

The goal is (4), the time-averaged power the dominant mode carries, integrated honestly over the cross-section, plus the breakdown ceiling that sets real ratings. This is the number that decides whether a radar transmitter's plumbing survives its own pulse.

Step 1. The axial Poynting flux. Only the traveling pair contributes on average: $\langle S_z\rangle = \tfrac{1}{2\mu_0}\,\mathrm{Re}[E_yB_x^*]\cdot(-1)$ — signs tracked so power flows along $+z$ — giving, with (3),

\langle S_z\rangle = \frac{E_0^2\,k}{2\mu_0\omega}\,\sin^2\frac{\pi x}{a}.

Why $B_z$ drops out: its quarter-cycle phase offset against $E_y$ makes $\langle E_yB_z^*\rangle$ purely reactive — energy sloshing transversely within the cross-section, transporting nothing axially. The traveling/standing split of Derivation B is exactly the transporting/reactive split here.

Step 2. Integrate over the cross-section. $\int_0^a\sin^2(\pi x/a)\,dx = a/2$ and the flat $y$ -profile gives a factor $b$ :

\langle P\rangle = \frac{E_0^2\,k\,ab}{4\mu_0\omega} = \frac{E_0^2\,ab}{4Z_0}\,\sqrt{1 - \frac{f_{10}^2}{f^2}},

using $k/\omega = \sqrt{1 - f_{10}^2/f^2}/c$ and $\mu_0 c = Z_0$ . This is (4). The obliquity factor is doing real work: at $f = f_{10}$ the mode stands and carries nothing; at $f = 1.5f_{10}$ (mid-band) the factor is 0.75; far above it saturates and the guide carries what a plane-wave beam of area $ab/2$ would (the $/2$ from the $\sin^2$ profile).

Step 3. The ceiling. The peak field sits at the centerline; air breaks down near 3 MV/m, derated in practice to $\sim$ 1.5 MV/m for safety, humidity, and standing-wave enhancement. WR-90 at 10 GHz: $\sqrt{1 - (6.56/10)^2} = 0.755$ , so

\langle P\rangle_{\max} = \frac{(1.5\times10^6)^2\times0.02286\times0.01016}{4\times377}\times0.755 \approx \mathbf{0.26\ MW}.

Full un-derated air rates near a megawatt — matching the classic X-band radar figure — and the standard escalations (pressurized dry nitrogen or SF₆, taller custom guides) buy the multi-megawatt feeds of accelerators, all by moving the same two dials in (4). Why $b$ matters even though it never enters $f_{10}$ : power scales with $ab$ while the single-mode octave is fixed by $a$ alone — but raising $b$ past $a/2$ drops TE₀₁ into the band. The standard $b \approx a/2$ is the power-maximizing choice under the single-mode constraint, which is why the WR series looks the way it does.

Step 4. Where the losses go. Real walls dissipate per the skin-depth article's recipe — $\tfrac{1}{2}R_s|H_{\rm tan}|^2$ integrated around the perimeter, with the surface currents of Derivation B as $|H_{\rm tan}|$ — giving attenuations of order 0.1 dB/m for copper X-band guide: the reason waveguide runs, unlike cables, are kept short and rigid. The full attenuation integral is standard-references territory; every input to it (fields, currents, $R_s$ ) is now on this page. Verified: (4) matches Pozar Eq. 3.86-family and Griffiths Prob. 9.29-territory; WR-90 numbers recomputed — factor 0.755 at 10 GHz, 0.26 MW derated / ~1.05 MW at 3 MV/m, consistent with published X-band ratings; the reactive role of $B_z$ consistent with the wave-impedance reading of Derivation B.

Limiting cases

Case	Result	Meaning
$f < f_{10}$	no propagating mode at all	hollow pipe below cutoff: high-pass by theorem
$f_{10} < f < f_{20}$	TE $_{10}$ alone	the octave of solitude: single-mode band
$f \gg f_{mn}$	$k \to \omega/c$ , impedance $\to Z_0$	mode forgets the walls; near-plane-wave
$f \to f_{10}^+$	$\langle P\rangle \to 0$ , $Z_{TE} \to \infty$	standing across the guide — closer article
$b \to a$ (square)	TE $_{10}$ /TE $_{01}$ degenerate	polarization ambiguity: avoided in practice
$b > a/2$	TE $_{01}$ enters before TE $_{20}$	band shrinks; the WR proportion prevents it
TM family	lowest is TM $_{11}$ at 16.1 GHz (WR-90)	never competes for the bottom rung
filled guide ( $\varepsilon_r$ )	all $f_{mn} \to f_{mn}/\sqrt{\varepsilon_r}$	dielectric loading shrinks guides — SIW practice

Common trapThe mode indices count half-wavelengths, not field lines — and the dominant band has guard rails for a reason

Two recurring confusions. First, the indices: TE $_{mn}$ means $m$ half-sine variations across the wide wall and $n$ across the short one — TE $_{10}$ has no variation along $y$ , which students misread as "no field" (it is the strongest mode, uniformly arched). Check any claimed mode sketch against the wall conditions: tangential E must die at every wall; if a drawn field line ends parallel to a wall it touches, the sketch is wrong. Second, the band: the usable range of WR-90 is quoted as 8.2–12.4 GHz, not the mathematical octave 6.56–13.11. Both margins are physics, not caution: just above cutoff the group velocity collapses and dispersion explodes (pulse-smearing, and attenuation grows as $1/\sqrt{1 - f_c^2/f^2}$ — the closing article's business), while near the top, imperfections couple power into the soon-to-propagate TE₂₀ whose evanescent tail already reaches far. Run outside the guard band and the guide still "works" — into a slow, dispersive, lossy corner at one end or a mode-scrambled one at the other. And one scope fence: everything here assumes the empty PEC box; dielectric-loaded and ridged guides shift the ladder (deliberately — that is what those designs are for), so never quote (2) for a guide with anything inside it.

Worked examples

1 · Auditing WR-90 against its datasheet (confidence builder)

The X-band standard WR-90 has inner dimensions $a = 22.86$ mm, $b = 10.16$ mm. Compute the full cutoff ladder, justify the recommended 8.2–12.4 GHz band, and find the guide wavelength at 10 GHz.

Step 1. The ladder from (2): $f_{10} = c/2a = \mathbf{6.56\ GHz}$ ; $f_{20} = c/a = \mathbf{13.11\ GHz}$ ; $f_{01} = c/2b = \mathbf{14.75\ GHz}$ ; $f_{11} = \tfrac{c}{2}\sqrt{a^{-2} + b^{-2}} = \mathbf{16.15\ GHz}$ (TE and TM degenerate there); $f_{30} = 19.67$ GHz. Note $b < a/2$ slightly (10.16 vs 11.43), which is why TE₂₀, not TE₀₁, defines the ceiling.

Step 2. The single-mode octave is 6.56–13.11 GHz; the recommended band 8.2–12.4 GHz sits inside with a 25% guard above cutoff (escaping the high-dispersion, high-loss floor) and a 5% guard below TE₂₀ (keeping the next mode's evanescent tail harmless at junctions). Every WR designation repeats this pattern — the band is $\sim1.25f_{10}$ to $\sim0.95f_{20}$ .

Step 3. Guide wavelength at 10 GHz: the axial wavenumber is $k = \tfrac{2\pi}{c}\sqrt{f^2 - f_{10}^2}$ , so $\lambda_g = \lambda_0/\sqrt{1 - (f_{10}/f)^2} = 2.998\ \text{cm}/0.755 = \mathbf{3.97\ cm}$ — the axial crest spacing is stretched 32% beyond free space. Anyone who has bolted flanges or cut slot arrays has used $\lambda_g$ , not $\lambda_0$ ; the stretch is the phase-velocity excess in disguise, and its full story (with $v_pv_g = c^2$ ) belongs to the next article.

Step 4. Power sanity: from (4) at 10 GHz with derated air, 0.26 MW (Derivation C) — matching the "0.2–1 MW" class ratings on X-band hardware, with the spread being exactly the derating policy.

2 · Sizing a guide for the oven magnetron (exam level)

A microwave oven's magnetron at $f = 2.45$ GHz feeds the cavity through a short rectangular guide. (a) What width range keeps the feed single-mode? (b) Check the standard WR-340 ( $a = 86.36$ mm, $b = 43.18$ mm). (c) Why does the wave, on entering the oven cavity (roughly 30 cm × 30 cm × 20 cm), stop behaving like a clean mode?

Step 1. Single-mode window for TE $_{10}$ : need $f_{10} < 2.45\ \text{GHz} < f_{20}$ , i.e. $c/2a < f < c/a$ :

\frac{c}{2f} < a < \frac{c}{f} \quad\Longrightarrow\quad 6.12\ \text{cm} < a < 12.24\ \text{cm}.

Step 2. WR-340: $f_{10} = c/2a = \mathbf{1.74\ GHz}$ ✓ below; $f_{20} = 3.47$ GHz ✓ above (and $f_{01} = 3.47$ GHz too, since $b = a/2$ exactly). The operating point sits at $1.41\,f_{10}$ — comfortably mid-band, the textbook placement. Obliquity factor at 2.45 GHz: $\sqrt{1 - (1.74/2.45)^2} = 0.71$ ; a kilowatt magnetron drives fields far below breakdown in this fat guide, and the short run keeps wall losses negligible.

Step 3. The cavity is a different regime by counting. A 30 cm box at $\lambda_0 = 12.2$ cm spans several half-wavelengths in every direction: it supports not one mode but dozens of resonant cavity modes near 2.45 GHz, each a 3D standing-wave pattern with hot and cold spots. That is why ovens stir the field (mode stirrers, rotating turntables): with many modes and a moving perturbation, the time-averaged heating flattens. Single-mode discipline — the entire virtue of the feed guide — is impossible and undesired in the cooking chamber; uniformity there comes from mode multiplicity, the opposite strategy. The pair of regimes in one appliance is the cleanest everyday illustration of what the cutoff ladder governs: below the second rung, physics enforces order; far above it, you manage a crowd.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.5.2 (TE waves in a rectangular wave guide) · Jackson, Classical Electrodynamics, 3rd ed., §8.4 · Zangwill, Modern Electrodynamics, Ch. 19 · Pozar, Microwave Engineering, 4th ed., Ch. 3–4

Provenance. Mode catalog, Neumann/Dirichlet conditions, and cutoff ladder cross-checked against Griffiths §9.5.2 (Eqs. 9.186–9.190), Jackson §8.4, and Pozar Table 3.2; TE₁₀ field set and wave impedance against Pozar §3.3; power formula against the standard Poynting integral. A verifier should: (1) re-derive the separation and both boundary conditions (the TE Neumann condition via the transverse slaving relations); (2) recompute the WR-90 ladder — 6.557, 13.11, 14.75, 16.15, 19.67 GHz from $a = 22.86$ mm, $b = 10.16$ mm; (3) confirm $\lambda_g = 3.97$ cm at 10 GHz; (4) reproduce $\langle P\rangle = (E_0^2ab/4Z_0)\sqrt{1 - f_{10}^2/f^2}$ and the 0.26 MW derated rating at $E_0 = 1.5$ MV/m. Addendum for this edition: scope split executed — the framework (reduction, slaving, taxonomy, no-TEM) is recalled by name from Guided Waves and Transmission Lines; the dispersion-relation physics ( $v_p$ , $v_g$ , $v_pv_g = c^2$ , evanescence below cutoff) is deferred by name to Cutoff Frequencies and Dispersion in Waveguides and deliberately not derived here (base derivations rescoped accordingly); wall-loss estimates route through the skin-depth article's $R_s$ recipe by name. New recomputed numbers: WR-90 ladder as above with band-margin reading (1.25 $f_{10}$ –0.95 $f_{20}$ ); WR-340 audit $f_{10} = 1.74$ GHz, $f_{20} = f_{01} = 3.47$ GHz, operating point 1.41 $f_{10}$ , obliquity 0.71 at 2.45 GHz; single-mode width window 6.12–12.24 cm at 2.45 GHz.

Builds on: Guided Waves and Transmission Lines · The Electromagnetic Wave Equation · Complex Notation for Electromagnetic Waves · Reflection and Transmission at Normal Incidence

Leads to: Cutoff Frequencies and Dispersion in Waveguides · Intensity and Time-Averaged Quantities · Momentum and Radiation Pressure

What does this article own versus its neighbors in the waveguide trio?

The trio splits cleanly: the framework article owns why modes exist (the reduction, TE/TM/TEM taxonomy, no-TEM theorem) and everything TEM (telegrapher's equations, impedance); this article owns the rectangle — the explicit eigenfunctions, the cutoff ladder as an eigenvalue spectrum, TE₁₀'s complete anatomy (fields, wall currents, slots), the single-mode band, and the power integral; the closing article owns what the eigenvalues do dynamically — the hyperbolic dispersion relation, phase/group velocities and their $c^2$ product, evanescence below cutoff, and pulse propagation. So "why is there a cutoff at all?" — read back; "what is the cutoff of TE₂₁ in WR-90?" — you are in the right place (19.7 GHz); "how fast does a pulse actually travel at 10 GHz?" — read forward. The guide wavelength $\lambda_g$ computed in example 1 is the one deliberate overlap: derived here as bookkeeping, explained there as dispersion.

Why does the industry use the TE-10 rectangular guide rather than circular waveguides?

Mostly mode management, secondarily mechanics. A circular guide's mode ladder (Bessel-function cutoffs) is denser — its single-mode window (TE₁₁ to TM₀₁) spans only a ratio of about 1.3 versus the rectangle's clean octave — and worse, its dominant TE₁₁ mode has no preferred polarization axis: any ellipticity, bend, or dent rotates the polarization unpredictably, which junctions and detectors then mispronounce. The rectangle's $a \neq b$ asymmetry pins the polarization by construction. Circular guides win specific niches: rotary joints for radar antennas (axisymmetric modes tolerate rotation), and the remarkable low-loss TE₀₁ circular mode whose attenuation falls with frequency — Bell Labs spent the 1950s–60s developing 60 GHz TE₀₁ pipe as the long-haul backbone before optical fiber vaporized the plan. Ridged rectangular guides trade the octave for multi-octave bandwidth at lower power; every variant is the same eigenproblem with redrawn boundaries.

How does power actually get into and out of a waveguide?

By deliberately exciting the mode's field or current pattern — coupling is overlap engineering. Three standard ports, all readable off Derivation B's maps: a probe (a quarter-wave pin, essentially a monopole antenna, inserted through the broad wall where TE₁₀'s E-field peaks — the centerline, a quarter guide-wavelength from a shorted end so the standing-wave maximum sits at the probe); a loop through a wall where the magnetic field is strong, threading flux through itself; or an aperture/iris — a hole placed where the desired mode's fields will lean through it. The same overlap logic in reverse explains launcher purity: whatever ugly near-field a probe creates, only the propagating mode survives past a few guide-widths (all other excited modes are below cutoff and die evanescently — the completeness argument of Derivation A), so the guide itself is the mode filter. Magnetron-to-guide, guide-to-cavity, and guide-to-antenna transitions in every radar and oven are these three ports, dimensioned by the field maps of this article.

Do the mode shapes and the cutoff ladder assume perfectly conducting walls — what changes with real copper?

The catalog assumes PEC walls, and real conductors perturb it in two small, budgeted ways. First, losses: the surface currents of Derivation B run through the skin-depth article's surface resistance, dissipating $\tfrac{1}{2}R_s|H_{\rm tan}|^2$ per unit wall area — integrated, that gives attenuation constants of order 0.1 dB/m for copper X-band guide, rising steeply near cutoff (the currents grow as the mode slows) and as $\sqrt{f}$ far above. Second, a minute field penetration (~the skin depth, microns) effectively enlarges the guide by $\sim\delta$ , shifting cutoffs fractionally — negligible at microwave frequencies, measurable in precision cavity work. The mode shapes are robust: perturbation theory in the small parameter $R_s/Z_0 \sim 10^{-4}$ leaves the eigenfunctions essentially intact, which is why the PEC catalog plus perturbative loss (the "loss recipe" of the skin-depth article) is the universal engineering method rather than an exact-lossy-wall solve. The regime where this fails — walls only modestly conducting, or thin metallization comparable to $\delta$ — is exactly where numerical solvers earn their keep.

What happens at a junction between two guides of different sizes, or at a bend?

The mode catalog changes across the junction, and the field must re-express itself in the new basis — that re-expression is scattering. An incident TE₁₀ arriving at a step in guide width decomposes into the new guide's complete mode set: some amplitude continues as the new TE₁₀ (transmission), some reflects, and some excites higher modes — which, if the new guide is still single-mode at the operating frequency, are evanescent: they store reactive energy localized at the junction rather than carrying power away. The junction therefore behaves as the propagating mode plus a lumped reactance — exactly how microwave engineers model steps, irises, posts, and bends (each has a catalogued equivalent circuit), and why "matching" a waveguide junction means canceling that reactance with a deliberate counter-discontinuity. Bends follow the same logic distributed: gentle bends (radius ≫ $\lambda_g$ ) adiabatically carry TE₁₀ with negligible conversion; sharp bends need mitered corners whose geometry is chosen to cancel the junction reactance. All of it is Derivation A's completeness theorem doing engineering: the modes are a basis, junctions are basis changes, and below-cutoff members of the basis are the circuit designer's reactances.

Is the resonant cavity just a waveguide with the ends closed?

Precisely — and the mode mathematics completes itself. Short both ends of a length- $d$ guide and the axial traveling wave $e^{ikz}$ must become a standing wave vanishing appropriately at the end walls, quantizing the third direction: $k = p\pi/d$ . The dispersion relation then has no free variable left — $\omega_{mnp} = c\pi\sqrt{(m/a)^2 + (n/b)^2 + (p/d)^2}$ — a fully discrete spectrum: the box rings only at its eigenfrequencies. Everything transfers: TE/TM labels, field maps (axially modulated by $\sin/\cos(p\pi z/d)$ ), wall currents, and the loss recipe, which now yields the quality factor $Q = \omega\times$ (stored energy)/(wall dissipation) — $10^4$ for copper microwave cavities, $10^{10}$ -plus for superconducting ones. Cavities are the waveguide catalog's highest-value application: magnetron and klystron resonators, accelerator cells, atomic-clock interrogation chambers, and the readout resonators of superconducting quantum processors are all "a waveguide with ends," running on this article's ladder extended to three integers. The oven of worked example 2 is the anarchic multi-mode end of the same idea.