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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.

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.

A slotted line samples an electromagnetic field along a guide with a movable probe introduced through a longitudinal slot. Positions of standing wave minima determine the guided wavelength; a detector response need not be a calibrated field amplitude.

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.