7. Wave Propagation

We have built the antenna; now let's see how the wave actually gets from antenna to antenna in the real world. The medium between transmitter and receiver is rarely the empty vacuum of the Friis equation. Instead, it might be the Earth's surface, the ionosphere, the troposphere, or some combination, and each of these has interesting effects.

7.1 Frequency bands and dominant modes

Different frequency ranges propagate by different mechanisms. A rough map:

Three propagation modes do most of the work: ground wave, sky wave, and space wave. The frequency of the signal is the strongest predictor of which mode dominates.

7.2 Ground wave propagation

Below about 2 MHz, the wave can travel along the surface of the Earth, partly because the conducting Earth supports a "creeping wave" along its surface. The wave has:

• A direct wave (from antenna to receiver, line of sight or close to it).
• A ground-reflected wave (bouncing off the ground).
• A surface wave that is guided along the conducting Earth.

At low frequencies the surface wave dominates. AM broadcast at 1 MHz can reach 100 to 200 km easily, sometimes much further over saltwater (which has higher conductivity than land). Above 2 MHz, the surface wave attenuates rapidly because the Earth is no longer a "good enough" conductor at those frequencies, and the ground wave becomes ineffective beyond a few tens of km.

The attenuation depends on soil conductivity. Saltwater (high conductivity) supports ground waves better than dry desert. This is why many AM broadcasters site transmitters near coasts and run very tall vertical towers (a tall vertical excites the surface wave more efficiently than a short one).

7.3 Sky wave propagation: bouncing off the ionosphere

Above 3 MHz, the surface wave is dead, but a new mode opens up: the sky wave. The wave radiates into the ionosphere, gets refracted (or reflected, depending on how you analyze it) back to Earth, and lands hundreds or thousands of km away.

The ionosphere is the upper atmosphere (60 km to 500 km altitude), where solar UV ionizes oxygen and nitrogen, leaving free electrons. Free electrons make the medium dispersive (refractive index depends on frequency). At low enough frequencies, the ionosphere bends radio waves enough to send them back down. At high enough frequencies, the wave punches through into space.

                   ▲
   Ionosphere ─ ─ ╱ ─ ─ ─ ╲ ─ ─ ─ ─ ─
                ╱           ╲
              ╱               ╲
            ╱                   ╲
   ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Earth
        TX                       RX
        ←   skip distance   →

Layers

• D-layer (60–90 km): forms by day, dies at night. Absorbs low-HF (rather than reflecting), which is why daytime AM listening is shorter range than nighttime.
• E-layer (90–150 km): reflects MF and lower HF. Sporadic E ("Es") events open up 50 MHz propagation in summer.
• F1-layer (150–250 km): forms by day.
• F2-layer (250–500 km): the main HF reflector, present day and night, denser by day.

Key parameters

• Critical frequency $f_0$: the highest frequency that bounces back at vertical incidence. Above $f_0$, the wave punches through.
• Maximum Usable Frequency (MUF): the highest frequency that successfully bounces for a given path geometry. $\text{MUF} = f_0 / \cos(i)$ where $i$ is the angle of incidence at the layer. Higher elevation (more oblique) means higher MUF. So shorter hops use higher MUF.
• Lowest Usable Frequency (LUF): the lowest frequency that gets through D-layer absorption. Below LUF, the signal is absorbed before it gets refracted.
• Skip distance: the minimum surface distance between transmitter and the first ionospheric return, set by the geometry and the wave's takeoff angle. Inside the skip distance, the sky wave is unavailable (you only get ground wave, or nothing).

Operators choose frequencies between LUF and MUF for a given path and time of day. Shortwave broadcasters announce schedules that switch frequencies twice or more per day to follow the ionosphere.

Ionospheric variations

• Diurnal: D-layer fades at night, F-layer thins. Long-distance HF propagation often improves at night (less D-absorption, more F-reflection).
• Seasonal: sun angle changes, ionization patterns shift. Some bands are summer bands; some, winter.
• Solar cycle (sunspot cycle): roughly 11 years. Solar maximum brings lots of UV, dense ionosphere, high MUF. Solar minimum brings sparse ionosphere, low MUF, and reduced HF DX.
• Geomagnetic storms: solar mass ejections disturb the ionosphere, sometimes causing radio blackouts.

Ionospheric refraction: the Appleton-Hartree formula

The refractive index of an ionized plasma is, in simplified form,

$n^2 = 1 - \frac{f_p^2}{f^2}$

where $f_p$ is the plasma frequency, set by electron density: $f_p = \sqrt{N e^2 / (4\pi^2 \varepsilon_0 m_e)}$. For $f < f_p$, $n^2 < 0$, the wave is evanescent: total reflection. For $f$ slightly above $f_p$ at oblique incidence, the wave bends like Snell's law into a denser-to-thinner refraction and curves back to Earth. For $f \gg f_p$, $n \to 1$ and the wave passes through (this is what your GPS signal does).

Used in: shortwave broadcasting (BBC World Service, VOA, Radio Havana), amateur DX (long-distance contacts), over-the-horizon (OTH) radar (uses ionospheric reflection to detect targets beyond the radio horizon), backup military communications.

7.4 Space wave / line-of-sight propagation

Above 30 MHz, the ionosphere is mostly transparent. Propagation is essentially line of sight. Three components matter:

• Direct wave (TX antenna to RX antenna, straight line through atmosphere).
• Ground-reflected wave (bouncing off the ground between them).
• Diffracted wave (slipping over hills and around buildings, especially at lower VHF).

Radio horizon

The Earth is curved, so even at perfect visibility the geometric horizon limits range. With antenna heights $h_t$ and $h_r$ (in meters):

$d_{horizon} \approx 4.12(\sqrt{h_t} + \sqrt{h_r}) \text{ km}$

The factor 4.12 (not 3.57, the geometric horizon constant) reflects atmospheric refraction. The atmosphere's refractive index decreases gently with altitude, bending radio rays slightly back toward the Earth's surface, effectively extending the horizon by about 15%. This is sometimes captured as the 4/3 effective Earth radius rule: replace the Earth's true radius with $4/3$ times its value, then use straight-line geometry.

For 100 m towers: $d \approx 82$ km. For two aircraft at 10 km altitude: $d \approx 825$ km (which is why AWACS planes can see far). For ground-level cell phones: just a few km, which is why cellular networks have base stations on every corner.

Fresnel zones

Even when there is unblocked line of sight, partial obstruction by terrain or buildings can degrade the signal. Fresnel zones are nested ellipsoids around the LOS path; the first Fresnel zone has radius

$F_1 = \sqrt{\frac{\lambda d_1 d_2}{d_1 + d_2}}$

where $d_1$ and $d_2$ are the distances from each end to the obstruction. For 5 GHz over 5 km, $F_1$ at the midpoint is about 8.7 m. A clear first Fresnel zone is the working rule for microwave links: ensure the first Fresnel zone is unobstructed, and you get full Friis-like performance.

Tropospheric ducting

The troposphere (lowest ~15 km) sometimes forms layers of unusual refractive index that trap RF energy, propagating it like a leaky waveguide for hundreds of km. Common conditions: temperature inversions over warm seas, cold fronts. Causes: surprise long-distance VHF/UHF reception (TV stations from neighboring countries appearing on your TV), unexpected radar clutter or interference, fade events on microwave links.

Multipath fading

When direct and reflected (or diffracted) waves arrive with different phases, they may add constructively (signal boost) or destructively (deep fade). In an indoor Wi-Fi environment, you get hundreds of multipath components, and the fading varies dramatically over wavelength-scale movements. Two antennas a few cm apart often have very different signal levels, the basis for spatial diversity reception.

Atmospheric absorption

Above about 10 GHz, atmospheric gases start to absorb. The absorption peaks:

• 22 GHz: water vapor.
• 60 GHz: oxygen molecular resonance, very strong (~10 dB/km at sea level).
• 118 GHz: another oxygen line.
• 183 GHz: water vapor.

The 60 GHz peak is exploited for short-range, high-bandwidth links: the atmospheric absorption naturally limits range to a few hundred meters, providing physical-layer privacy and frequency reuse. WiGig and IEEE 802.11ad/ay use 60 GHz. Inter-satellite links use 60 GHz between LEO satellites because the atmosphere does not block the link (no atmosphere up there).

Rain attenuation is also significant above 10 GHz (heavy rain can cause 5 dB/km at Ka-band), which is why satellite links at Ka-band include "rain margin" of 5 to 15 dB.

7.5 Tropospheric scattering

Beyond LOS, you can sometimes get a signal by forward scattering off tropospheric inhomogeneities (small fluctuations in refractive index). The scattered signal is weak (~70 dB below LOS Friis), but with high-power transmitters and high-gain antennas, troposcatter links can carry kilobit data over 200 to 800 km, beyond the radio horizon. Used in military communications between mountaintops, oil-platform links, and historical NATO and Soviet networks. Modern systems mostly use satellites instead.