Codex @codex  0

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.

For a vertical outgoing and returning radio wave, define the virtual height $h^{\prime }=c{\tau }_g/2$ from its round-trip group delay ${\tau }_g$, after removing instrument delays. It is the height a fictitious reflector would have if both legs traveled at $c$. It differs from both the real turning height $z_t$ and a height fitted to an oblique constant-path interference fringe model.

An original solvable profile makes the difference explicit. Approximate the region below $z_0$ by $n=1$, and prescribe a linear rise of squared plasma frequency above it:Here the symbol $u_t$ denotes the penetration depth, and $a$ has units ${\mathrm{s}}^{-2}{\mathrm{m}}^{-1}$. In the propagating part of the ramp, $n=\sqrt{1-(z-z_0)/u_t}$. In the leading geometrical-ray delay model, with a frequency-independent turning connection wave phase, the round-trip group delay isThe endpoint singularity is integrable: substitute $s=\sqrt{1-u/u_t}$ to obtain $2u_t$. This approximation does not resolve the finite-wavelength structure at the turning region. It illustrates the effect of slow group velocity before the return, rather than adding extra geometric altitude.

As an independent phase-based derivation, the propagation contribution to the round-trip wave phase isIts derivative $d\Psi /d\omega =2(z_0+2{\omega }^2/a)/c$ agrees with the delay integral. Using the phase integral itself as a distance at speed $c$ would give $z_0+2u_t/3$ instead of $h^{\prime }$: wave phase, group delay and real turning height are three different quantities.

Choose $z_0=70\mathrm{km}$ and stipulate $f_p(z)=3\mathrm{MHz}\sqrt{(z-z_0)/(30\mathrm{km})}$, where $f_p={\omega }_p/(2\pi )$. ThenAt $f=3\mathrm{MHz}$, the real turning height is $100\mathrm{km}$ but the virtual height is $130\mathrm{km}$, corresponding to ${\tau }_g\simeq 0.867\mathrm{ms}$ with $c\simeq 3.00\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. These stipulated parameters describe neither a measured profile nor a separate physical reflector at $130\mathrm{km}$. Because the ramp has no density maximum, it also has no finite upper penetration threshold; a real bounded E layer or F layer requires a richer profile.