Wind Power Density

Air arriving at speed $v$ carries kinetic energy $\tfrac{1}{2}mv^2$, and it arrives at a mass rate of $\rho A v$. Multiply the two and the area divides out: $P/A = \tfrac{1}{2}\rho v^3$. That is the whole derivation, and the third power is not a modelling choice — one factor of $v^2$ is the energy each parcel of air brings, and the remaining factor of $v$ is how fast the parcels keep coming.

The cube law is the single most consequential fact on this page. A wind speed 10% high overstates the power by 33%. A speed 20% high overstates it by 73%. A survey that reads 1 m/s optimistic at a 7 m/s site has overstated the resource by nearly half, and no amount of care downstream recovers it. This is why a year of anemometry at hub height is worth what it costs, why met masts are climbed rather than guessed at, and why "it seems windy up there" is not data.

The same exponent breaks an intuition worth breaking early: the average power is not the power at the average wind speed. Because the cube weights the strong hours so heavily, a site whose wind blows 5 m/s half the year and 15 m/s the other half delivers far more than a steady 10 m/s site of the same mean. A real assessment integrates $v^3$ over the measured speed distribution — usually fitted to a Weibull curve — and cubing a single annual mean typically understates a site by 50 to 100%. The ratio between the two is called the energy pattern factor, and for most inland sites it runs near 1.9.

Density is the other input and it is a variable here for a reason. The 1.225 kg/m³ standard is dry air at sea level and 15 °C. Climb to 1,000 m and it falls about 12%; to 2,000 m, about 21%. Warm it and it falls further. A turbine on a Colorado ridge therefore produces measurably less than the identical machine on the Danish coast in the identical wind — a real and frequently forgotten derate, and one that no amount of blade design recovers.