No steady water waves of small amplitude are supported by a shear flow   with still free surface
  The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadric ones with arbitrary positive coefficients. 
Vladimir Kozlov, Nikolay Kuznetsov
math-ph math.MP