Electromagnetic Theory

Critical Frequency

Q: What is the difference between critical frequency and maximum usable frequency?

The critical frequency (foF2 for the F2 layer) is the highest frequency reflected straight up at vertical incidence (90 degree elevation). The maximum usable frequency (MUF) is the highest frequency reflected back at an oblique angle for a given path. Because oblique rays travel more horizontally, MUF is always higher than foF2 by the secant of the incidence angle: MUF = foF2 times sec(theta_i). For a low-angle path the obliquity factor is typically 3 to 3.5, so an foF2 of 8 MHz can support an MUF near 25 to 28 MHz on a long single-hop circuit. Operators choose a working frequency around 80 to 90 percent of the MUF, called the FOT or optimum traffic frequency, to keep a margin against ionospheric variability.

Q: How is critical frequency measured?

Critical frequency is measured with a vertical-incidence ionosonde, a swept-frequency HF radar that transmits straight up and records the echo delay versus frequency on an ionogram. As the transmitted frequency rises, the virtual reflection height climbs, and at the critical frequency the echo trace shoots up vertically because the wave just barely fails to be reflected. The frequency at that cusp is foE, foF1, or foF2 depending on the layer. Modern digisondes sweep roughly 1 to 30 MHz every few minutes, and global networks such as the GIRO publish near-real-time foF2 maps used for HF propagation prediction.

Q: Why does critical frequency change between day and night?

Critical frequency tracks the peak electron density of the layer, and that density is driven by solar EUV ionization balanced against recombination. During the day, solar radiation keeps electron density high, so foF2 can reach 10 to 15 MHz at midday near solar maximum. At night the ionizing source is gone, the D and F1 layers vanish, and the F2 layer decays through recombination, dropping foF2 to roughly 3 to 8 MHz. This is why long HF circuits shift to lower bands such as 40 and 80 meters after dark; the higher daytime bands fall below their MUF and the signal punches through the ionosphere into space instead of being reflected.

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In ionospheric propagation, the highest frequency a given layer reflects back to Earth when the wave is launched straight up at vertical incidence. Written foE, foF1, or foF2 for each layer, the critical frequency equals the plasma frequency at the layer's electron-density peak, so it scales as roughly 9√N_max Hz. A vertical wave above foF2 punches through into space, while an oblique ray can still be returned at a higher frequency, which is why the critical frequency sets the lower anchor of the maximum usable frequency for any HF skywave circuit. Typical midday foF2 ranges from about 5 to 15 MHz, dropping to 3 to 8 MHz after dark, and it controls the usable band on long-haul radio wave paths.

Symbol: f_o (foF2, foE)

Typical foF2: 3 to 15 MHz

Set by: Peak electron density N_max

How Electron Density Fixes the Reflection Limit

When a radio wave enters the ionosphere it encounters a region of free electrons whose density rises with altitude up to a peak, then falls. The refractive index seen by the wave drops below unity as it climbs into denser plasma, and the ray bends back toward Earth. Reflection occurs at the height where the local plasma frequency equals the wave frequency. The critical frequency is simply the plasma frequency at the layer's density maximum: any vertically launched wave below it is turned back, while any wave above it reaches a height of maximum density without total internal refraction and escapes into space.

Because the critical frequency depends only on the peak electron density N_max, it is a direct probe of ionization state. The F2 layer carries the highest daytime density and therefore the highest critical frequency, foF2, which is the parameter most HF circuits live or die by. The E and F1 layers have lower densities and lower critical frequencies, foE and foF1, and they support shorter skip distances. Solar activity, season, latitude, and the day-night cycle all shift N_max, so foF2 can swing by a factor of three or more over a solar cycle.

For oblique paths the geometry helps. A wave that strikes the layer at a shallow angle is reflected at a frequency well above the vertical-incidence critical frequency, scaled by the secant of the incidence angle. This secant law is what lets a station with an foF2 of 8 MHz still work a distant correspondent on 21 or 28 MHz, and it ties the critical frequency directly to the operational MUF and the achievable skip distance.

Governing Equations

Critical (plasma) frequency:
f_c ≈ 8.98√N_max Hz (≈ 9√N_max)
N_max = peak electron density (m⁻³)

Maximum usable frequency (secant law):
MUF = f_c × secθ_i = f_c / cosθ_i
θ_i = angle of incidence at the layer

Skip distance (single hop, flat-Earth approx.):
d_skip ≈ 2h′ × tanθ_i,max

Where h′ = virtual reflection height, f_c in Hz with N_max in electrons/m³. Example: N_max = 1×10¹² m⁻³ → f_c ≈ 9 MHz; at θ_i = 74°, MUF ≈ 9 / cos74° ≈ 32 MHz.

Critical Frequency by Ionospheric Layer

Layer	Height	f_c (day)	f_c (night)	Symbol	Role
D layer	60 to 90 km	~0.5 MHz	Vanishes	—	Absorbs MF/low HF
E layer	100 to 120 km	3 to 4 MHz	~0.5 MHz	foE	Short single hops
F1 layer	150 to 250 km	5 to 7 MHz	Merges into F2	foF1	Daytime only
F2 layer	250 to 400 km	5 to 15 MHz	3 to 8 MHz	foF2	Primary HF DX
Sporadic E	100 to 120 km	5 to 50 MHz	Variable	foEs	Intense, transient

Common Questions

Frequently Asked Questions

What is the difference between critical frequency and maximum usable frequency?

The critical frequency (foF2) is the highest frequency reflected straight up at vertical incidence; the MUF is the highest reflected at an oblique angle for a given path. Because oblique rays travel more horizontally, MUF = foF2 × secθ_i and is always higher. With an obliquity factor of 3 to 3.5, an foF2 of 8 MHz can support an MUF near 25 to 28 MHz. Operators work at about 80 to 90% of the MUF, the FOT, to keep margin against variability.

How is critical frequency measured?

A vertical-incidence ionosonde sweeps HF straight up and records echo delay versus frequency on an ionogram. As frequency rises the virtual height climbs, and at the critical frequency the trace shoots up vertically because the wave nearly fails to reflect. That cusp gives foE, foF1, or foF2. Modern digisondes sweep roughly 1 to 30 MHz every few minutes, and networks like GIRO publish near-real-time foF2 maps for propagation prediction.

Why does critical frequency change between day and night?

It tracks peak electron density, which is driven by solar EUV ionization balanced against recombination. By day, foF2 can reach 10 to 15 MHz near solar maximum. At night the source is gone, the D and F1 layers vanish, and F2 decays, dropping foF2 to roughly 3 to 8 MHz. That is why long circuits shift to 40 and 80 meters after dark; the higher daytime bands fall below their MUF and signals punch through into space.

Related Terms

← Critical Field Critical Temperature →

HF and Microwave Systems

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