Wind-generated waves on a water layer of finite depth: A linear stability study

In this work, we study the linear stability of an air-water two-phase system. Motivated by recent experiments, we consider a quadratic velocity profile in the finite-depth water layer and an exponential velocity profile in the air layer. In the inviscid stability problem, it is known that the exponential velocity profile is amenable to analytical calculation. In this work, we provide an analytical solution to the Rayleigh equation with the quadratic velocity profile in terms of spheroidal wave functions. A comparison with recent experiments shows a good match with the rippling instability (the instability due to shear flow in the water layer) growth rates, indicating that rippling instability is more important than the Miles instability (the instability due to shear flow in the air layer) in parameter regimes corresponding to the experiment. Further, we provide analytical expressions for the neutral stability curve and long-wave limit growth rate asymptote. We perform a viscous stability study to support the inviscid stability analysis results. Interestingly, it reveals that the energy contribution to the viscous versions of the instability is different than the inviscid ones (rippling and Miles instabilities).