EW Math Guide Module

Propagation and Link Geometry

From HF skywave to microwave line-of-sight, ducts, diffraction, Fresnel clearance, rain loss and atmospheric absorption.

Propagation Is the Tactical Environment of RF

A link budget starts with free-space spreading, but real RF energy travels through ionosphere, troposphere, terrain, sea surface, rain, oxygen, water vapor and platform geometry. The same transmitter and antenna can behave very differently at HF, VHF, UHF, X band or millimeter wave because each band couples to the environment differently.

For radar and electronic warfare, propagation controls intercept range, jammer effectiveness, radar horizon, low-altitude coverage, multipath fading, beyond-line-of-sight paths and atmospheric loss. A formula is useful only when the propagation regime behind it is understood.

Learning path: start with HF ionospheric propagation, then move upward through ground wave, VHF/UHF line of sight, microwave Fresnel/two-ray behavior, diffraction, ducting, rain and atmospheric absorption.

Animated HF skywave: the ionosphere bends energy back toward Earth, making beyond-line-of-sight communication possible.

HF Skywave and Ionospheric Propagation

HF, roughly 3 to 30 MHz, is special because the ionosphere can refract radio waves back toward the Earth. This is why HF communications can reach hundreds or thousands of kilometers without satellites. The usable path depends on frequency, time of day, solar activity, ionospheric layer height, takeoff angle and absorption.

Critical Frequency and MUF

fc ≈ 9Nmax
MUF ≈ fccosψ
LUF < fop < MUF

Formula reading: fc is critical frequency, the highest frequency returned at vertical incidence. Nmax is peak electron density; in practical ionospheric formulas it is scaled so fc is in MHz. MUF is maximum usable frequency for an oblique path. ψ is the incidence angle relative to vertical at the ionosphere. LUF is lowest usable frequency, set mainly by absorption and required signal-to-noise ratio. HF works best when the operating frequency is above absorption-dominated LUF and below MUF.

Worked Example: choosing an HF operating frequency

Vertical critical frequencyfc = 7 MHz
Oblique factor1 / cosψ = 2.2
MUFMUF ≈ 7 × 2.2 = 15.4 MHz
Practical choiceUse roughly 0.8 × MUF, about 12 MHz, if absorption and noise allow it.

Ground Wave, Surface Wave and Low-Frequency Coverage

Below HF and in the lower part of HF, energy can follow the Earth as a ground or surface wave. This is useful for maritime, navigation and some long-range communication systems, but attenuation depends heavily on ground conductivity, polarization and frequency.

Surface-wave intuition

Vertical polarization over seawater propagates better than over dry ground because seawater has higher conductivity. As frequency increases, surface-wave attenuation generally increases, so skywave and line-of-sight mechanisms become more important.

E(d) ≈ E0A(d,f,\sigmag,\epsilonr)
Lground,dB = -20log10A

Formula reading: E(d) is received field strength at distance d. A is an attenuation factor controlled by distance, frequency, ground conductivity σg and relative permittivity εr. The loss form converts that field attenuation into decibels. The key engineering point is that ground is part of the antenna system at these frequencies.

EW relevance

Surface-wave systems can be difficult to hide from wide-area receivers because they illuminate large regions. Conversely, low-frequency receive antennas are physically large, inefficient when shortened, and vulnerable to environmental noise. Intercept range is often noise-limited rather than thermal-noise-limited.

Rule of thumb

Lower frequencyBetter terrain following, larger antennas, higher atmospheric/man-made noise.
Higher frequencySmaller antennas, less surface wave, more line-of-sight behavior.

VHF/UHF Line of Sight and Radio Horizon

From VHF upward, many tactical links become primarily line-of-sight. The Earth blocks low-altitude paths, while atmospheric refraction slightly extends the radio horizon. This is central to radar coverage, ESM intercept geometry, datalink range and jammer placement.

Radio horizon

RLOS,km ≈ 4.12(√h1,m + √h2,m)
ke ≈ 43

Formula reading: RLOS is approximate radio line-of-sight distance in kilometers. h1 and h2 are antenna heights in meters. The coefficient 4.12 includes the common 4/3 effective-Earth approximation for standard refraction. If the atmosphere is nonstandard, this coefficient can be wrong: super-refraction extends range, while sub-refraction reduces it.

Worked Example: low-altitude target

Radar heighth1 = 25 m
Target heighth2 = 100 m
Radio horizonRLOS ≈ 4.12(5 + 10) = 61.8 km
EW meaningA powerful jammer below the horizon may be geometrically unavailable, regardless of transmitter power.

Free-Space Path Loss

Free-space path loss is the reference loss before terrain, atmosphere, antenna pattern, polarization and multipath are applied. It is not a complete propagation model, but it is the baseline that tells you how severe pure geometric spreading is.

FSPL equation

LFSPL,dB = 92.45 + 20log10(Rkm) + 20log10(fGHz)
PRX,dBm = PTX,dBm + GT + GR - Lpath

Formula reading: Rkm is range in kilometers and fGHz is frequency in gigahertz. The constant 92.45 exists only because those units are chosen. The two 20log terms show that doubling range adds 6 dB loss and doubling frequency also adds 6 dB loss for isotropic antennas. The receive-power equation then adds transmit power and antenna gains, and subtracts path loss and other losses.

Worked Example

InputsR = 50 km, f = 10 GHz
FSPL92.45 + 20log10(50) + 20log10(10) = 146.4 dB
MeaningBefore antenna gain, the one-way spreading loss is already enormous.

Fresnel Zone and Two-Ray Multipath

A link can have visual line of sight and still lose margin if the first Fresnel zone is blocked. Near the ground or sea, the direct and reflected paths can add or cancel depending on phase, creating deep fading.

Fresnel clearance

r1 = λd1d2d1 + d2
ΔL = Lreflected - Ldirect
Δφ = 2πΔLλ + φrefl

Formula reading: r1 is first Fresnel-zone radius at the obstacle point. d1 and d2 are distances from that point to transmitter and receiver. The two-ray equations compare reflected and direct path lengths. If phase difference Δφ is near 180 degrees, the paths cancel and the received signal fades.

Diffraction and Terrain Masking

When terrain blocks the direct ray, energy can still diffract over the obstacle edge. Diffraction is why a signal may be detectable behind a ridge, but it usually arrives with significant additional loss.

Knife-edge parameter

ν = h2(d1 + d2)λd1d2
LdB ≈ 6.9 + 20log10(√((ν - 0.1)2 + 1) + ν - 0.1)

Formula reading: h is obstacle height above the straight line between antennas. If h is positive, the obstacle protrudes into the path. ν normalizes that height by wavelength and path geometry. Larger positive ν means deeper obstruction and higher diffraction loss. The loss approximation is commonly used for a single dominant ridge or edge.

Tropospheric Refraction, Ducting and Anomalous Propagation

The troposphere changes refractive index with height. Standard refraction bends rays gently downward, extending the radio horizon. Strong gradients can create ducts that trap microwave energy and carry it far beyond normal line of sight, especially over sea surfaces.

Effective Earth and refractivity

N = (n - 1)106
Re,eff = kRe
k ≈ 43 for standard atmosphere

Formula reading: n is refractive index and N is radio refractivity in N-units. k is the effective-Earth-radius factor. Standard atmosphere uses k about 4/3, but ducts and strong gradients can make propagation very nonstandard. In EW this can create unexpected radar detections, radar holes, false coverage assumptions or beyond-horizon intercept opportunities.

Microwave, Millimeter Wave, Rain and Atmospheric Absorption

At microwave and millimeter-wave frequencies, gas absorption and hydrometeors matter. Oxygen and water vapor create frequency-dependent absorption. Rain becomes a major loss mechanism as wavelength approaches the size of raindrops.

Specific attenuation

Latm,dB = γgasR
γrain = kRrainα
Ltotal = LFSPL + Lgas + Lrain + Lmisc

Formula reading: γ is specific attenuation in dB/km. R is path length in km. Rrain is rain rate in mm/h. k and α are frequency- and polarization-dependent coefficients. At X band rain may be manageable for many radar paths; at Ka band and above, rain fade can dominate availability and must be treated as a design driver.