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An electromagnetic waveguide confines an electromagnetic wave across its axis while allowing it to travel along the axis. Conducting walls or changes of refractive index select transverse normal modes: reproducible cross-sectional field patterns with their own longitudinal wavenumbers. A hollow metal pipe and an optical fiber are different realizations of this principle. Their usefulness is to deliver electromagnetic energy along a chosen route instead of letting it spread freely.

For a uniform vacuum-filled conducting guide, write a normal mode's axial and time dependence as $e^{i(\beta z-\omega t)}$. Here $\omega >0$ is angular frequency, $\beta$ is the longitudinal wavenumber, and $k_{\perp }^2$ is the transverse eigenvalue selected by the wall boundary conditions. The electromagnetic wave equation separates intoThe geometry consumes part of the available squared wavenumber: when $\omega /c<k_{\perp }$, this branch is an evanescent wave rather than a propagating normal mode. The waveguide cutoff frequency is therefore a property of a specified branch and geometry. A transverse electromagnetic mode in a suitable two-conductor geometry can have $k_{\perp }=0$ and no positive geometric cutoff. An optical fiber needs its own dielectric boundary conditions, rather than the metal-wall formulas below.

The rectangular electromagnetic waveguide makes these ideas explicit with a single sine profile. The scalar-potential conducting waveguide modes give a broader construction. At an open aperture, a confined field can radiate into the electromagnetic aperture angular spectrum; it need not remain one guided normal mode.

The guided wavelength is the distance for a $2\pi$ change of axial wave phase in a propagating normal mode. For the vacuum-filled rectangular electromagnetic waveguide,for ${\mathrm{TE}}_{10}$, with ${\lambda }_0=c/f$ and $f>f_c$. It exceeds the vacuum wavelength at the same frequency. This longer axial spacing is consistent with $v_{\mathrm{ph}}>c$: multiplying ${\lambda }_g$ by $f$ gives the phase velocity, not the group velocity.

Two oppositely traveling fields at the same frequency form a standing wave. For a perfect electric short at $z=0$, the total electric field is proportional to $\sin (\beta z)\sin (\pi x/a)$, so successive axial electric nodes are separated byThus neighboring minima in a slotted line scan give half a guided wavelength. In the original $a=30\mathrm{mm}$, $f=1.25f_c$ example, $d=40\mathrm{mm}$ implies ${\lambda }_g=80\mathrm{mm}$, not $40\mathrm{mm}$. Combined with the chosen $f$, the inferred broad dimension is $a={\lambda }_0/[2\sqrt{1-({\lambda }_0/{\lambda }_g{)}^2}]=30\mathrm{mm}$. Below cutoff, $\beta$ is imaginary and $2\pi /\beta$ is not a physical periodic spacing.

For a hollow, air-filled rectangular electromagnetic waveguide with conducting walls, transverse confinement givesA transverse electric mode of a rectangular waveguide permits one zero index, but not two; a transverse magnetic mode of a rectangular waveguide requires both indices positive. Take $w=20\mathrm{mm}$ and $h=10\mathrm{mm}$. The dominant mode is ${\mathrm{TE}}_{10}$, withThe waveguide cutoff frequency is the limiting value: positive axial group velocity and propagating power require $f>f_c$. There is no transverse electromagnetic mode in this single hollow conducting pipe; otherwise a zero cutoff would incorrectly follow.

For the usual hollow rectangular electromagnetic waveguide, let the propagation axis be $z$ and the interior be $0<x<a$, $0<y<b$. A transverse magnetic polarization has $B_z=0$ and $E_z\ne 0$. At a perfect conductor wall the tangential electric field vanishes. Since $E_z$ is tangential to all four side walls, it satisfies a homogeneous Dirichlet boundary condition around the rectangle.

Separating the transverse Helmholtz equation therefore gives the transverse magnetic mode of a rectangular waveguideA zero value of either index makes $E_z$ identically zero, rather than producing a nonzero field uniform along that direction. The transverse electric and magnetic fields of this TM family are obtained from transverse derivatives of $E_z$, so they also vanish. Thus both TM indices must be nonzero: neither ${\mathrm{TM}}_{n0}$ nor ${\mathrm{TM}}_{0m}$ exists in this rectangular hollow-guide model. In contrast, transverse electric modes obey a Neumann boundary condition on $B_z$ and may have one zero index. The claim is about this waveguide geometry, not all possible guides.

Assume a hollow, perfectly conducting rectangular electromagnetic waveguide filled with air approximated by vacuum. Let its wider side be $a=0.040\mathrm{m}$ and its narrower side $b=0.020\mathrm{m}$. The longitudinal propagation constant of a mode isso the waveguide cutoff frequency isFor a mode with transverse electric polarization, one of $m,n$ may vanish, but they cannot both vanish. For a mode with transverse magnetic polarization, both must be positive. A single hollow conducting guide has no transverse electromagnetic mode with zero cutoff.

The lowest cutoff therefore belongs to the ${\mathrm{TE}}_{10}$ mode, varying across the wider side:Strictly, a traveling mode transporting energy down the guide requires $f>f_c$. At exactly $f_c$, $k_z=0$ and the axial group velocity vanishes. Below cutoff the field is evanescent. A different dielectric filling would replace $c$ by the wave speed in that material.

For the propagating mode of an air-filled or evacuated rectangular electromagnetic waveguide, the longitudinal dispersion relation isThe phase velocity and group velocity along the guide are consequentlyWith waveguide cutoff frequency $f_c=10\mathrm{GHz}$ and carrier frequency $f=30\mathrm{GHz}$, $\sqrt{1-(f_c/f{)}^2}=\sqrt{8}/3$. ThusThe pulse envelope travels at the group velocity for a sufficiently narrow range of frequencies. The phase velocity exceeding the speed of light does not describe superluminal signal transport. The check $v_p{v}_g=c^2$ follows directly from the dispersion relation. A different uniform filling medium would require replacing $c$ by its electromagnetic propagation speed; the stated numerical answer uses the usual empty-guide assumption.

Take a straight hollow rectangular electromagnetic waveguide with vacuum interior $0<x<a$, $0<y<b$ and axis $z$, where $a>b>0$. Its four walls are ideal perfect conductors. The time-dependent Maxwell equations require zero tangential electric field and zero normal oscillating magnetic field at these walls. Real metal has finite electrical conductivity, so real guides lose electromagnetic energy by Joule heating; the perfect-wall model isolates confinement and wave dispersion from that loss.

The simplest propagating transverse electric mode of a rectangular waveguide is ${\mathrm{TE}}_{10}$. “Transverse electric” means $E_z=0$, not that the magnetic field is also wholly transverse. With real peak scale $E_0$, its complex amplitudes, including the common axial factor, can be writtenThe physical fields are the real parts after multiplication by $e^{-i\omega t}$. In particular, $E_y$ is zero at $x=0,a$, maximal at $x=a/2$, and independent of $y$. At $y=0,b$ it is normal to the metal, so it need not vanish there. The ${\mathrm{TE}}_{10}$ electric field crosses the narrow gap, while its strength varies across the broad dimension.

For an original example choose $a=30\mathrm{mm}$, $b=12\mathrm{mm}$ and $f=1.25f_c$, where $f_c=c/(2a)$. With $c=299792458m/s$, this givesHere ${\lambda }_0=c/f$ is vacuum wavelength and ${\lambda }_g=2\pi /\beta$ is guided wavelength. The next cutoff is ${\mathrm{TE}}_{20}$ at $2f_c$; ${\mathrm{TE}}_{01}$ starts at $(a/b)f_c=2.5f_c$. Thus only ${\mathrm{TE}}_{10}$ can propagate at the chosen frequency, although a source or discontinuity can additionally excite local evanescent waves. These are calculated values, not specifications or measurements of a commercial guide.

MIT's documented slotted-line waveguide measurement shorts one end of a rectangular electromagnetic waveguide and excites it through a probe coupled by capacitance. A second such probe moves through a longitudinal slot, aligned with the wall electric current to reduce disturbance. Its output locates axial standing wave nulls. The book's standing-wave setup chooses a frequency above the first waveguide cutoff frequency and below higher-mode cutoffs. Lowering it below the first cutoff instead reveals spatial decay away from the source. The text supplies apparatus and an observable comparison, rather than a table of measured null positions.^[ref]

The inference can be checked without assuming that detector output is proportional to electric field amplitude. At fixed measured frequency, record several minima at coordinates $z_0,\dots ,z_N$; then estimate ${\lambda }_g=2(z_N-z_0)/N$ when the minima are consecutive. This uses positions alone. Compare the result with the guided wavelength prediction using independently measured $a$ and $f$. The $40\mathrm{mm}$ spacing above is a synthetic example, not a reading from the MIT apparatus.

Additional proposed controls are to keep probe insertion fixed, repeat the scan in both directions, recheck source frequency, and compare shallow probe settings for shifts in the minima. These test probe loading, mechanical backlash and drift. A deliberately reflecting termination supplies the standing wave; varying its position should translate the pattern without changing adjacent-null spacing in a uniform single-mode region. Source-adjacent evanescent waves, unintended reflections, additional propagating normal modes, detector noise and finite positional resolution can spoil that inference. A disappearing remote signal alone does not distinguish below-cutoff decay from loss or poor excitation, and a scan at one frequency does not measure a pulse's group velocity.

A transverse electric mode of a rectangular waveguide has $E_z=0$ while its axial magnetic field is generally nonzero. For time dependence $e^{-i\omega t}$ and axial dependence $e^{i\beta z}$, set $\Delta =(\omega /c{)}^2-{\beta }^2$. The Maxwell equations reconstruct the transverse components from $B_z$:The axial complex amplitude satisfies a transverse Helmholtz equation, $({\partial }_x^2+{\partial }_y^2+\Delta )B_z=0$. Zero tangential electric field at each perfect conductor wall becomes a Neumann boundary condition on $B_z$. Separation of variables givesThe index $m$ counts half-periods across $a$, and $n$ across $b$. Either index can vanish, allowing ${\mathrm{TE}}_{10}$; both cannot vanish. Indeed a spatially constant axial oscillating magnetic field would give nonzero changing magnetic flux through the cross-section, whereas the line integral of tangential electric field around its metal boundary is zero. The integral form of Faraday's law excludes this ${\mathrm{TE}}_{00}$ candidate. Dividing the reconstruction formulas by $\Delta =0$ would be invalid and cannot produce the separate transverse electromagnetic mode.

For $m=1,n=0$, choose $B_{\ast }=-i(\pi /a)E_0/\omega$. The formulas recover exactly the $E_y$, $B_x$ and $B_z$ in rectangular electromagnetic waveguide. In particular, the transverse magnetic field accompanies $B_z$ and ensures zero divergence; an oscillating longitudinal magnetic field on its own is not the guided solution.

In a hollow rectangular electromagnetic waveguide with perfect conductor walls and axis $z$, a transverse magnetic polarization has $B_z=0$ and a nonzero longitudinal electric field $E_z$. The wall boundary conditions impose $E_z=0$ around the transverse rectangle. Separation of variables therefore gives $E_z=E_0\sin (n\pi x/a)\sin (m\pi y/b)e^{i(k_{z}z-\omega t)}$, with $n,m\ge 1$. Setting either index to zero makes the entire field vanish. Thus there is no ${\mathrm{TM}}_{n0}$ or ${\mathrm{TM}}_{0m}$ branch. The transverse eigenvalue is $k_{\perp }^2=(n\pi /a{)}^2+(m\pi /b{)}^2$, with dispersion relation ${\omega }^2/c^2=k_z^2+k_{\perp }^2$. This exclusion is specific to this geometry and these boundary conditions.

The waveguide cutoff frequency separates oscillatory axial propagation from exponential decay for a particular normal mode. For the vacuum-filled rectangular electromagnetic waveguide, the dispersion relation isFor a transverse electric mode of a rectangular waveguide, $m,n$ are nonnegative integers but not both zero. For a mode with $B_z=0$ and nonzero $E_z$, both indices must be positive. The lowest cutoff for $a>b$ is consequently ${\omega }_{10}=\pi c/a$, or ordinary frequency $f_c=c/(2a)$.

A square cross-section instead gives equal ${\mathrm{TE}}_{10}$ and ${\mathrm{TE}}_{01}$ cutoffs. The absence of a transverse electromagnetic mode is specific to this hollow pipe, rather than all electromagnetic waveguides.^[ref]

For $\omega >{\omega }_{mn}$, choose real $\beta >0$ for propagation toward $+z$. For $0<\omega <{\omega }_{mn}$, a field decaying into a semi-infinite guide hasA finite section can contain both growing and decaying axial solutions set by its two ends; the single decaying solution is not a complete finite-section transmission calculation. At exact cutoff $\beta =0$, the two axial solutions become a constant and a linear function of $z$: there is no finite axial wavelength for a traveling sinusoid.

In the chosen $a=30\mathrm{mm}$ guide at $f=0.8f_c=3.99723277\mathrm{GHz}$, the ${\mathrm{TE}}_{10}$ branch has $\alpha =20\pi {\mathrm{m}}^{-1}$. Its amplitude decay length is $1/\alpha \simeq 15.9155\mathrm{mm}$; after $50\mathrm{mm}$, its normalized wave amplitude is $e^{-\pi }\simeq 0.0432139$. This is geometric evanescence even with ideal walls, not Joule heating. A single decaying branch has zero cycle-averaged axial Poynting vector: substituting $\beta =i\alpha$ makes $B_x$ a quarter-cycle out of phase with $E_y$. Coupling across a finite section requires the complete boundary-value solution.

For a propagating branch of an ideal vacuum-filled rectangular electromagnetic waveguide, the phase velocity tracks constant wave phase along the axis, whereas the group velocity describes a sufficiently narrow, weakly distorted wave packet. Using ${\omega }^2=c^2{\beta }^2+{\omega }_c^2$ givesThus $v_{\mathrm{ph}}>c$, $0<v_g<c$, and $v_{\mathrm{ph}}{v}_g=c^2$. Differentiating the dispersion relation proves the second formula; dividing it by ${\beta }^2$ proves the first. This is the same mathematical structure as Klein-Gordon packet velocities. In the stipulated $f=1.25f_c$ example,Over $0.30\mathrm{m}$, the corresponding narrow-envelope delay is $(0.50\mathrm{m})/c$, approximately $1.668\mathrm{ns}$. Timing a carrier wave crest would give $0.6004\mathrm{ns}$; it does not time the arrival of a newly introduced message.

There is an independent energy check. For the forward ${\mathrm{TE}}_{10}$ fields, the average axial Poynting vector is $⟨S_z⟩=\beta |E_y{|}^2/(2{\mu }_0\omega )$. Integrating across the guide gives average power $P$ and average electromagnetic energy per unit length $U^{\prime }$:The transverse sine and cosine squared each integrate to $a/2$; ${\beta }^2+(\pi /a{)}^2={\omega }^2/c^2$ then makes the integrated electric and magnetic contributions to $U^{\prime }$ equal. The axial energy transport therefore agrees with the group velocity in this ideal model.

As $\omega \downarrow {\omega }_c$, $v_g\to 0$ and $v_{\mathrm{ph}}\to \infty$; exactly at cutoff $\omega /\beta$ is undefined. Strong wave dispersion close to cutoff also limits the narrow-packet approximation. Neither a large phase velocity nor a pulse peak establishes superluminal information velocity; front velocity asks a separate causal question. In a uniform filling with constant positive permittivity $\epsilon$ and magnetic permeability $\mu$, replace $c$ by $u=1/\sqrt{\epsilon \mu }$, giving $v_{\mathrm{ph}}{v}_g=u^2$. Frequency-dependent material response or significant loss requires differentiating the actual dispersion relation, so this product is not universal.