The Generation of Wind Waves on a Water Surface

Abstract

The statistical methods for treating continuous media developed elsewhere are applied to the generation of surface gravity waves. The wind stress is assumed to be everywhere normal to the disturbed surface of the water and caused by a known ensemble of ``gusts.'' Each gust is considered to be a center of high or low pressure, which moves with the mean wind speed, and has a radius L, and duration T. The rms value of the variable wind pressure is assumed to vary systematically from point to point in a storm. The calculations are made for an idealized, long continuing storm; before applying the statistical equations, the effect of a single gust is calculated. It is shown that the single gust produces a V-shaped wake, which, after the gust has blown itself out (t>T) can be considered as a packet of free gravity waves, which moves and spreads under the influence of divergence and dispersion. The results of the statistical calculation are compared with known general facts, and with an actual storm. From the fact that the storm waves have a non-sinusoidal character, and arrive at any one point in groups of 5 to 10 crests, it is deduced that the duration of a gust is from 15 to 30 seconds. From the fact that the majority of the wave energy travels in directions that are within 30° of the wind, and with phase velocities that are nearly equal to the wind speed, the mean radius of the gusts is deduced to be about 40 meters in the case of a wind speed of 20 meters per second. The crest length of the waves is proportional to the distance r from the storm, and inversely to its diameter, D; when r/D=10, the dominant length of the crests is 2.2 wavelengths. The rms displacement, H, of the sea-surface from its horizontal mean is a function of the parameters already mentioned and of P, the rms value of the variable component of the wind pressure at the center of the storm; for the idealized storm considered, and for r/D≫1, this is approximated by H2=10P2D2/rL,all quantities being in centimeters, including P, which is measured in centimeters on a water barometer. Comparison with data for an actual storm indicates that this theory requires a value of P that is possibly 10 times greater than its actual value. While the theory explains many of the phenomena of storm waves, it is therefore incapable of accounting quantitatively for the wave height. The possible reasons for this inadequacy are discussed. The relation of the theory to Jeffreys' sheltering theory is also discussed, and it is shown that both postulate that the variable component of the wind stress is normal to the water surface. The sheltering theory also postulates that the wind pressure lags 90° behind the surface disturbance. A similar phase lag is a consequence of the assumptions of the present theory.