Bound on the Slope of Steady Water Waves with Favorable Vorticity

We consider the angle θ of inclination (with respect to the horizontal) of the profile of a steady two dimensional inviscid symmetric periodic or solitary water wave subject to gravity. Although θ may surpass 30° for some irrotational waves close to the extreme wave, Amick (Arch Ration Mech Anal 99(2):91–114, 1987) proved that for any irrotational wave the angle must be less than 31.15°. Is the situation similar for periodic or solitary waves that are not irrotational? The extreme Gerstner wave has infinite depth, adverse vorticity and vertical cusps (θ = 90°). Moreover, numerical calculations show that even waves of finite depth can overturn if the vorticity is adverse. In this paper, on the other hand, we prove an upper bound of 45° on θ for a large class of waves with favorable vorticity and finite depth. In particular, the vorticity can be any constant with the favorable sign. We also prove a series of general inequalities on the pressure within the fluid, including the fact that any overturning wave must have a pressure sink.