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In the sharp planar, nonmagnetic ionosphere model with frequency above the plasma frequency, the refractive index is $n=\sqrt{1-{\omega }_p^2/{\omega }^2}<1$. From air, the Brewster angle satisfies $\tan {\theta }_B=n$. At that angle the reflected p-polarized wave vanishes, while s polarization retains a nonzero reflected component. A ground-to-ground single reflection from height $h$ spans $D=2h\tan {\theta }_B=2hn$. This is below the critical angle for total internal reflection for total internal reflection, which reflects both polarizations in this lossless isotropic model. The distinction diagnoses ordinary Brewster reflection rather than total internal reflection.

Take a cold, unmagnetized, collisionless plasma with uniform equilibrium electron density $n_e$, fixed neutralizing ions, and a small electromagnetic wave that is a transverse wave. Cold means that thermal-motion corrections are neglected. For electron mass $m_e$, elementary charge magnitude $e$ and electric field $\mathbf{E}=\mathrm{Re}(\hat{\mathbf{E}}{e}^{-i\omega t})$, the linear Newton's second law equation is $m_e\dot{\mathbf{v}}=-e\mathbf{E}$. Thus $\hat{\mathbf{v}}=-ie\hat{\mathbf{E}}/(m_e\omega )$ and the current density is $\hat{\mathbf{J}}=i{n}_e{e}^2\hat{\mathbf{E}}/(m_e\omega )$. Writing the induced electric polarization as $\hat{\mathbf{J}}=-i\omega \hat{\mathbf{P}}$ gives a negative electric susceptibility. With vacuum permittivity ${ϵ}_0$,The plasma frequency ${\omega }_p$ controls the response. For $\omega >{\omega }_p$, the propagating branch has real refractive index $0<n<1$. For $\omega <{\omega }_p$, this homogeneous model has an evanescent wave, not an ordinary propagating wave with a negative real refractive index. In a slowly varying vertical profile, a normally incident ray approaches a turning region where ${\omega }_p(z)=\omega$; the geometrical-ray approximation fails right at that region. At oblique incidence the conserved horizontal wave vector changes the turning condition. A smooth density gradient can therefore return a skywave without a mirror or a layer of metal.

For a homogeneous propagating section, direct differentiation of the dispersion relation givesA phase velocity greater than $c$ does not send information faster than $c$. Here a narrow wave packet has group velocity $v_g$, while the full causal response must still respect special relativity. Even this stripped-down plasma is strongly dispersive near its plasma frequency. In the actual ionosphere, collisions produce absorption, and the magnetic field of Earth changes the response with direction and wave polarization; a single real scalar $n$ cannot capture all of that behavior.

The E layer is a main region of the ionosphere below the F layer. The 1924 Appleton-Barnett radio interference experiment concerns this lower region.^[ref]

The F layer is a higher main region of the ionosphere, above the E layer. Its different electron density and persistence can permit skywave propagation when a probing radio wave passes through the lower E layer.

The ionosphere is a region of the upper atmosphere of Earth with substantial partial ionization. Its mobile Electrons and ions constitute a plasma mixed with neutral gas. For radio communication, the crucial question is whether a chosen radio wave returns toward the ground, is absorbed, or passes through. A skywave can connect stations beyond their direct view across the curved surface of Earth, while wave interference between a ground wave and a skywave can make reception fade even when the radio transmitter remains steady.

The ionosphere is useful both as a route for radio communication and as a remote laboratory: varying the probing frequency changes how deeply the radio wave penetrates the plasma. The Appleton-Barnett radio interference experiment turned an unwanted fluctuation in reception into a measurement of the overhead route. Cold-plasma radio dispersion explains why this is a tunable medium, while ionospheric virtual height explains why a measured delay is not automatically an altitude.

Consider an original two-path model with a locally flat ground baseline $d$ and a symmetric equivalent return height $h$. Replace the skywave by two straight segments traveling at the speed of light $c$, and replace the ground wave by a straight segment of length $d$. Their lengths and positive excess distance areThis is an equivalent geometric construction, not a metallic layer overhead. It neglects the curvature of Earth, refraction within the return region, ground-induced wave phase shifts and optical dispersion.

For received complex amplitudes $A_g$ and $A_s$, absorb their constant relative wave phase into ${\varphi }_0$ and choose their magnitudes nonnegative. An ideal detector reading proportional to squared received field hasAssume $\Delta L$ and ${\varphi }_0$ are independent of frequency across the sweep, with slowly changing amplitudes. Consecutive maxima differ by $2\pi$ in phase difference, soThe often-used approximation $\Delta f\simeq c/\Delta L$ therefore carries a frequency-independent-path assumption. Counting $N$ complete fringe periods between $f_1$ and $f_2$ gives $\Delta L\simeq Nc/|f_2-f_1|$; equivalently, $N=\Delta L|1/{\lambda }_2-1/{\lambda }_1|$ when the vacuum wavelength is $\lambda =c/f$. Count whole periods, not maxima plus minima as though each were a whole period.

Choose $d=150\mathrm{km}$ and $h=100\mathrm{km}$, with $c\simeq 3.00\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. Then $L_s=250\mathrm{km}$, $\Delta L=100\mathrm{km}$ and $\Delta f=3.00\mathrm{kHz}$. A $12.0\mathrm{kHz}$ sweep covers four full fringe periods. Conversely, the inferred equivalent height isThese chosen values illustrate the inversion; they are not a transcription of the 1924 measurements. With $A_g=1$ and $A_s=0.7$, the normalized reading ranges from $(1-0.7{)}^2=0.09$ to $(1+0.7{)}^2=2.89$: incomplete cancellation does not remove the distance information.

For a sweep with linearly changing frequency $f(t)=f_0+\dot{f}t$ and two constant delays ${\tau }_s>{\tau }_g$, the received instantaneous frequency difference has magnitude $|\dot{f}|({\tau }_s-{\tau }_g)$. In this model ${\tau }_s-{\tau }_g=\Delta L/c$, giving a beat frequency $|\dot{f}|\Delta L/c$. Equivalently, traversing each $\Delta f$ of the sweep crosses one fringe. This calculation requires a sufficiently slow sweep that propagation, tuning and amplitude changes can be treated consistently; it is not a beat between two independent transmitters.

More generally, measure the total phase difference $\Phi (f)$, including propagation and any reflection or apparatus contribution. Define the differential group delayFor a locally monotone wave phase with slowly varying derivative, the local spacing is $\Delta f\simeq 1/|\Delta {\tau }_g|$. If $\Phi =2\pi fD(f)/c+{\varphi }_r(f)$, this becomesHere $D(f)$ is a phase-equivalent optical path difference, not necessarily the geometric excess distance $\Delta L$. A drifting ionosphere, several simultaneous skywaves, changing wave polarization, a frequency-dependent reflection wave phase or significant loss can invalidate the simple constant-spacing inversion.

Awarded to Edward Victor Appleton for upper-atmosphere investigations, especially the Appleton layer. The ionosphere connects this work to radio propagation.^[ref]

Two interface effects. For $n_2<n_1$, Snell's law gives $\sin {\theta }_t=(n_1/n_2)\sin {\theta }_i$. If $\sin {\theta }_i>n_2/n_1$, no propagating transmitted ray exists: there is total internal reflection, with an evanescent wave on the far side. An optical fibre uses this to guide light.

For ordinary isotropic nonmagnetic media, $\tan {\theta }_i=n_2/n_1$ is the Brewster angle. The reflected p-polarized component, whose electric field lies in the incidence plane, vanishes; the reflected s-polarized component generally remains. Reflection polarizers and polarized glare reduction exploit this selectivity.

The plasma response. Neglect collisions, thermal-pressure corrections and background magnetic field. Let the wave vary as $e^{-i\omega t}$. An electron displacement obeysIts dipole moment is $-e{\mathbf{x}}_e$, so the electron electric polarization is ${\mathbf{P}}_e=-n{e}^2\mathbf{E}/(m_e{\omega }^2)$. Ions of charge $Ze$ give a further contribution ${\mathbf{P}}_i=-nZ{e}^2\mathbf{E}/(m_i{\omega }^2)$, smaller by $Z{m}_e/m_i$. Thus heavy ions move negligibly at these electron-wave frequencies. Using $\mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}$ gives the cold collisionless plasma dielectric response,For this model ${\mu }_r\simeq 1$, so $n_{\mathrm{refr}}^2={\epsilon }_r$ and $c^2{k}^2={\omega }^2-{\omega }_p^2$ for the transverse electromagnetic wave.

The vertical cutoff and density. Below the plasma frequency, the squared refractive index is negative: the wave is an evanescent wave instead of propagating through a layer hundreds of kilometres thick. The observed $5\mathrm{MHz}$ cutoff therefore identifies $f_p=5.0\times 10^6\mathrm{Hz}$, with ${\omega }_p=2\pi f_p$. ConsequentlyThis uses ${\epsilon }_0=8.854\times 10^{-12}\mathrm{F}{\mathrm{m}}^{-1}$, $m_e=9.109\times 10^{-31}\mathrm{kg}$ and $e=1.602\times 10^{-19}\mathrm{C}$. The density is also $3.10\times 10^5\mathrm{c}{\mathrm{m}}^{-3}$.

The transmission elevation. At $f=8\mathrm{MHz}$, the ionosphere has refractive indexIf $\alpha$ is elevation above horizontal, incidence at the layer is ${\theta }_i=\pi /2-\alpha$. Avoiding total internal reflection requires $\sin {\theta }_i=\cos \alpha <n_i$. The limiting elevation is thereforeStrictly, it must point above this limiting angle to carry upward power through the finite layer: at equality the transmitted wave is grazing. With a suitable satellite position, the stated orbital altitude imposes no additional angular constraint in this flat-layer approximation.

Polarization-selective ground reflection. The ray's incidence plane is vertical. Horizontal electric polarization is s polarization; the other transverse polarization lies in the vertical incidence plane and is p polarization. Reflection of s but not p identifies the Brewster angle, not total internal reflection, which would reflect both in this lossless isotropic model. ThusFor a reflecting height $h=200\mathrm{km}$, the horizontal range consists of two equal legs. The Brewster reflection from an ionospheric interface givesAt this angle ${\theta }_B$ is below the critical incidence angle, so a transmitted ray also exists; the detected horizontal component is an ordinary partial reflection. The s-polarized reflected power fraction is approximately $[(1-n_i^2)/(1+n_i^2){]}^2\simeq 0.059$, whereas the ideal p-polarized reflected fraction is zero. These results use the question's sharp planar interface model and exclude upper-interface reflections.

A skywave is a radio wave returned toward Earth through interaction with the ionosphere. In a smoothly varying plasma, the ray can bend continuously through refraction before returning: there need be no sharp material surface at the apparent point of reflection. Different frequencies, launch directions and wave polarizations can have different paths and losses.

The upper part of the terrestrial atmosphere includes regions where ionization makes the response to electromagnetic waves depend strongly on mobile Electrons. The ionosphere describes that electrical structure; its boundaries need not coincide with the boundaries defined by temperature in the mesosphere and thermosphere.