An asymptotic interpretation for the maximum growth rate of the Miles instability of wind waves

In 1925, Jeffreys suggested that wind waves grow because the aerodynamic pressure is in phase with the wave slope; this is known as the sheltering hypothesis. Considering an inviscid parallel flow, $U(z)$, of air over water, Miles uncovered in 1957 an instability of the wind field in presence of surface waves. The origin of this instability was identified by Lighthill in 1962: there is a critical level $z_c$, at which the wind speed matches the phase speed of the surface waves and where the wind transfers energy to the waves. The point $z_c$ is a regular singularity of the Rayleigh equation, which describes the stability of the wind field. Here, we find asymptotic solutions of the Rayleigh equation for long waves, that is for waves whose wavelength is much larger than the length scale of the wind profile. We use those solutions to calculate analytically the growth rate of the Miles instability. In the strong wind limit, we show that the maximum growth rate occurs when the Jeffreys sheltering hypothesis holds.