The growth of waves on water due to the action of the wind

The behaviour of the surface of water over which a wind is blowing was considered mathematically by Kelvin. Assuming the air and the water to be perfect liquids moving irrotationally, he found that the motion is governed by the following relation between U' the velocity of the wind relative to the water, λ the wave-length and c the wave-velocity. c2 = gλ/2π ρ - ρ'/ρ + ρ' + 2πT/(ρ + ρ') λ - ρρ'/(ρ + ρ')2 U'2, (1) where ρ, ρ' are the densities of the water and air respectively and T is the surface tension of the water-air boundary. In any actual case the air will not be moving irrotationally; also, it is difficult to specify what is to be considered as the velocity of the air, owing to the considerable velocity gradient which exists near any fixed boundary, so that the equation (1) cannot be confirmed experimentally.