Bound on the Slope of Steady Water Waves with Favorable Vorticity

We consider the angle 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 may surpass 30A degrees 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 31.15A degrees. 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 (theta = 90A degrees). 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 45A degrees on 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.