Beaufort Force and Wind Speed

The Beaufort scale ranks wind by force number, from force 0 (calm) to force 12 (hurricane force, 32.7 m/s and over). It was built from what the sea and the sails looked like before anyone had an anemometer, so the force numbers began as descriptions, not as speeds. This formula is the bridge: it gives the wind speed at 10 metres, in metres per second, from the force number, and the force number back from a measured speed. It is an empirical fit, not a physical law. It is a smooth curve that runs close to the published speed bands, and nothing in it explains why wind and sea are related this way.

In $v = 0.836\,B^{3/2}$, $B$ is the Beaufort force and $v$ the speed at 10 m. Because the exponent is above one, each step up the scale adds more speed than the step before it. The answer for a whole-number force falls inside that force's speed band, close to its middle, not a single true speed: force 4 covers 5.5 to 7.9 m/s, so any wind in that range is force 4. Run the other way, with $B = (v/0.836)^{2/3}$, a speed that falls between band centres returns a fractional force, which you can read as lying between the neighbouring whole forces.

The WMO and Met Office speed bands give a test of the fit. The centre of the force 4 band, 5.5 to 7.9 m/s, is 6.70 m/s; of force 8, 17.2 to 20.7, it is 18.95 m/s; of force 10, 24.5 to 28.4, it is 26.45 m/s. The fit lands within 0.2 % of each of those centres, and going backwards, 18.95 m/s returns force 8. Low on the scale, where the bands are narrow, it sits further from the middle of each band, though still inside it.

The relation is the empirical one in Beer (1997), Environmental Oceanography, 2nd edition, and it is meant for the scale it was fitted to, force 0 to 12. Treat the result as the middle of a band, not as a measurement.