  Motivated by wind blowing over water, we use asymptotic methods to study the
evolution of short wavelength interfacial waves driven by the combined action
of these flows. We solve the Rayleigh equation for the stability of the shear
flow, and construct a uniformly valid approximation for the perturbed
streamfunction, or eigenfunction. We then expand the real part of the
eigenvalue, the phase speed, in a power series of the inverse wavenumber and
show that the imaginary part is exponentially small. We give expressions for
the growth rates of the Miles (1957) and rippling (e.g., Young & Wolfe 2013)
instabilities that are valid for an arbitrary shear flow. The accuracy of the
results is demonstrated by a comparison with the exact solution of the
eigenvalue problem in the case when both the wind and the current have an
exponential profile.
