Wind-generated waves on a water layer of finite depth

In this paper, we study the linear stability of a two-dimensional shear flow an air layer overriding water finite depth. The is considered to be infinite extent with exponential velocity profile. Three different background conditions are in finite-depth layer: quiescent background, profile and quadratic It known that cases allow for analytical treatment. We further provide solution case field: specifically consider flow-reversal profile, although result could generalized other profiles as well. role depth on growth rate Miles rippling instabilities studied each three cases. Using asymptotic analysis, air–water density ratio being small parameter, obtain expression mode discuss condition existence long-wave cutoff these profiles. expressions boundary parameter space inverse squared Froude number wavenumber. scenarios where does not exist, have carried out behaviour regime.