Total wind energy flowing through an imaginary area A during the time t is:
where v is the wind speed; ρ is the air density; Avt is the volume of air passing through A (which is considered perpendicular to the direction of the wind); Avtρ is therefore the mass m passing per unit time. Note that ½ ρv2 is the kinetic energy of the moving air per unit volume.
Power is energy per unit time, so the wind power incident on A (e.g. equal to the rotor area of a wind turbine) is:
Wind power in an open air stream is thus proportional to the third power of the wind speed; the available power increases eightfold when the wind speed doubles. Wind turbines for grid electricity therefore need to be especially efficient at greater wind speeds.