On phase velocity and growth rate of wind-induced gravity-capillary waves

Abstract

The generation and growth of gravity–capillary waves (λ ≈ 1 cm) by wind are reconsidered using linear instability theory to describe the process. For all friction velocities we solve the resulting Orr–Sommerfeld equation using asymptotic methods. New elements in our theory, compared with the work of Benjamin (1959) and Miles (1962), are more stress on mathematical rigour and the incorporation of the wind-induced shear current. We find that the growth rate of the initial wavelets, the first waves to be generated by the wind, is proportional to u*3.We also study the effect of changes in the shape of the profiles of wind and wind-induced current. In doing this we compare results of Miles (1962), Larson & Wright (1975), Valenzuela (1976), Kawai (1979), Plant & Wright (1980) and our study. We find that the growth rate is very sensitive to the shape of the wind profile while the influence of changes in the current profile is much smaller. To determine correctly the phase velocity, the value of current and current shear at the interface are very important, much more so than the shape of either wind or current profile.