Physical interpretation of the mathematical theory of wave generation by wind

Abstract

In the light of accumulated evidence in favour of Miles's mathematical theory of wave generation by wind, the author has thought it desirable to translate the theory into the form of a physical argument, which goes as follows. Travelling water waves in a wind produce, to a first approximation, airflow undulations with pressures least over crests and greatest over troughs. Hence, just above the ‘critical height’ where the airflow component in the direction of propagation equals the wave velocity, air after slowly overtaking a crest is turned back by the higher pressure near the trough, moves down to a lower level and back towards the crest. Similarly, behind crests, an upward movement at the critical height occurs. Quantitatively, the vertical velocity v is such that the ‘vortex force’ − ρων (where ρ is density and ω vorticity) balances the sinusoidal pressure gradient. Furthermore, since in turbulent boundary layers vorticity decreases with height, any downflow produces a local vorticity defect, and upflow a local vorticity excess, and hence the vortex force varies about a negative mean at the critical height (although at other levels, where air moves sinusoidally, with vertical displacement and velocity 90° out of phase, the mean vortex force is zero). The negative total mean force extracts wind energy, and transfers it to the wave, enabling it to grow exponentially. For pressure gradients adequate to initiate substantial energy transfer, the critical height must be fairly small compared with the wavelength, and waves can grow whenever their velocity and direction satisfies this condition, a conclusion supported by measurements (Longuet-Higgins 1962) of the directional spectrum of wind-generated waves.