Growth of surface wind-waves in water of finite depth. A theoretical approach

The Miles' theory of wave amplification by wind is extended to the case of finite depth. A depth-dependent wave growth rate is derived from the dispersion relation of the wind/water interface. For different values of the dimensionless water depth parameter δ = gh/U12, with h the depth, g the gravity and U1 a characteristic wind speed, a family of wave growth curves is plotted as a function the dimensionless parameter θfd=cU1, with c the wave phase velocity. The model provides a fair agreement with the data and empirical relationships obtained from the Lake George experiment, as well as with the data from the Australian Shallow Water Experiment. Two major results are obtained: (i) for small θfd the wave growth rates are comparable to those of deep water and (ii) for large θfd a finite-depth limited growth is reached, with wave growth rates going to zero.