  When wind blows over water, ripples are generated on the water surface. These
ripples can be regarded as perturbations of the wind field, which is modelled
as a parallel inviscid flow. For a given wavenumber $k$, the perturbed
streamfunction of the wind field and the complex phase speed are the
eigenfunction and the eigenvalue of the so-called Rayleigh equation in a
semi-infinite domain. Because of the small air-water density ratio,
$\rho_{\rm{a}}/\rho_{\rm{w}}\equiv\epsilon<<1$, the wind and the ripples are
weakly coupled, and the eigenvalue problem can be solved perturbatively. At the
leading order, the eigenvalue is equal to the phase speed $c_0$ of surface
waves. At order $\epsilon$, the eigenvalue has a finite imaginary part, which
implies growth. Miles (1957) showed that the growth rate is proportional to the
square modulus of the leading-order eigenfunction evaluated at the so-called
critical level $z=z_c$, where the wind speed is equal to $c_0$ and the waves
extract energy from the wind. Here, we construct uniform asymptotic
approximations of the leading-order eigenfunction for long waves, which we use
to calculate the growth rate as a function of $k$. In the strong wind limit, we
find that the fastest growing wave is such that the aerodynamic pressure is in
phase with the wave slope. The results are confirmed numerically.
