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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.

For surface phase speed $c$ on a spherical Earth, the conserved angular ray parameter is $p=R_E/c$. The local horizontal slowness at radius $r$ is $p/r$, giving squared radial shear wavenumber $q_r^2={\omega }^2[\beta (r{)}^{-2}-p^2/r^2]$ in ray theory. The displayed inequality makes it negative. A jump in shear speed can turn a propagating wave above the interface into an evanescent wave below it. This is a local condition, not proof of evanescence at every greater depth.

A stratified layer between unstratified regions can trap internal gravity waves, because the exterior disturbance is an evanescent wave. Its discrete normal modes obey Robin boundary conditions representing the exterior layers. Periodic boundary forcing at a trapped normal mode frequency produces resonance in the ideal undamped model.

Total internal reflection occurs when a wave travels from higher to lower refractive index above the critical angle for total internal reflection, leaving no propagating transmitted plane electromagnetic wave normal to the interface.

At an infinite planar interface, total internal reflection returns all incident normal power in the reflected plane electromagnetic wave. Assume homogeneous dielectrics with isotropy, linearity, no dielectric loss, magnetic permeability ${\mu }_0$, and positive real refractive indices $n_1>n_2$. Above the critical angle for total internal reflection, both TE polarization and TM polarization have unit optical reflectance despite the external evanescent wave.^[ref]

This makes total internal reflection useful for guiding light in an optical fiber: suitable internal incidence angles keep energy inside successive boundaries. Its more surprising feature is that perfect optical reflectance coexists with an external electric field. The evanescent field in total internal reflection can interact with a nearby object, as the frustrated total internal reflection experiment demonstrates.

For an original idealized example, choose $n_1=1.5$, $n_2=1$ at vacuum wavelength ${\lambda }_0=633\mathrm{nm}$. At angle of incidence $40^{\circ }$, Snell law for electromagnetic waves gives an angle of refraction $\arcsin (1.5\sin 40^{\circ })\approx 74.6186^{\circ }$, so a transmitted light ray still emerges. At ${\theta }_c\approx 41.8103^{\circ }$, its direction becomes tangent to the boundary. At $50^{\circ }$, the proposed sine of a real angle of refraction would be $1.5\sin 50^{\circ }\approx 1.14907>1$: the transmitted wavevector must instead have an imaginary normal component. These are predictions for the stipulated media, not measured properties of a particular block.

A real semicircular-block experiment isolates this threshold. The University of Maryland laboratory instructions place a laser, motor-driven rotation stage and photodetector together; an aperture limits the detector voltage. The semicircular block is centered on the rotation axis, with the photodetector aligned along its flat face. The instructions identify a strong grazing-beam signal and a weaker signal just before the critical angle for total internal reflection. The readout is detector voltage versus stage angle. Retracing the incident laser beam identifies normal incidence for alignment.^[ref]

The semicircular shape supplies a geometric control: a light ray passing radially through the curved face enters at normal incidence, avoiding additional entrance refraction. The weaker signal near the flat face locates a limiting angle of refraction; a vanishing remote-detector signal alone would not establish that the external electric field vanishes. Beam width and alignment limit angular resolution; no measured threshold or decay depth is supplied.^[ref]

The tunnelling decay constant is the inverse length controlling the exponential decay of an evanescent wave in a constant potential barrier with $E<V_0$. The wavefunction amplitude decay length is $1/\kappa$; the squared-amplitude decay length is $1/(2\kappa )$.

For constant buoyancy frequency $N$ and $U=U_0+Sz$ with $S>0$, a stationary component of horizontal wavenumber $k$ turns at $z_t=(N/k-U_0)/S$ if it propagates at the source. Its vertical wavenumber vanishes there; WKB approximation fails and an Airy function transition connects reflection to an upper evanescent wave.

A vertically propagating stationary topographic internal gravity wave meets a turning level when its squared vertical wavenumber changes from positive to negative. The stationary internal-wave WKB solution then fails locally. A simple zero is resolved by an Airy function transition connecting incident and reflected waves below to a decaying evanescent wave above. For negligible wind curvature the condition is $|kU(z_t)|=N(z_t)$, distinct from the zero-intrinsic-frequency critical layer in a shear flow.

Fix horizontal slowness $p_{\alpha }$ and define $T_{ip}=c_{i3p3}$, $R_{ip}=c_{i3p\alpha }{p}_{\alpha }$ and $Q_{ip}=c_{i\alpha p\beta }{p}_{\alpha }{p}_{\beta }$. Under strong ellipticity in elasticity, $T$ is positive definite, so the vertical-slowness determinant has degree six with real coefficients. Real roots are intersections of a vertical line with the elastic slowness surface; complex-conjugate pairs give evanescent waves.

Here $m=\sqrt{1-p^2-q^2}$ has nonnegative imaginary part. This representation of the outgoing Helmholtz equation kernel separates propagating plane waves from evanescent waves. The absolute vertical separation is required for fields on either side of a source; choosing its sign fixes the upward or downward angular spectrum.