MUF and critical frequency relation

Maximum usable frequency

The maximum usable frequency (MUF) is the highest frequency that is returned to earth by the ionosphere for a given angle of incidence (that is, for a given distance between transmitter and receiver).

Relation between critical frequency and MUF

Critical frequency (fc) is the highest frequency reflected at vertical incidence.

                 ~~~~~~~ ionised layer ~~~~~~~
                      \  normal  /
                   θi  \   |    /
                        \  |   /
   Tx ___________________\_|__/_______ Rx

The refractive index of a layer with electron density N (per m³) at frequency f (Hz) is

n = √(1 − 81N / f²)

Using Snell's law, a ray entering from free space (n = 1) at angle θi is bent until its angle of refraction is 90° (ray horizontal), where it starts to return:

n = √(1 − 81N / f²)           (refractive index)
Snell's law at return point: sin θi = n
sin² θi = 1 − 81Nmax / f²
cos² θi = 81Nmax / f²
f = 9√Nmax / cos θi
Vertical incidence (θi = 0):  fc = 9√Nmax
Hence  f_MUF = fc sec θi      (secant law)

So f_MUF = fc sec θi = fc / cos θi. This is the secant law. Since sec θi ≥ 1, the MUF is always higher than the critical frequency, and it increases as the path becomes longer (more oblique).

For flat earth with virtual height h and distance D: sec θi = √(1 + D²/4h²), so f_MUF = fc√(1 + D²/4h²).

Example: fc = 6 MHz, θi = 60°: f_MUF = 6 × sec 60° = 6 × 2 = 12 MHz.