Abstract

Consider the angle of inclination of the profile of a steady 2D (inviscid,  symmetric, periodic or solitary) water wave subject to gravity. Although  the angle may surpass 30 degrees for some irrotational waves close to the  extreme Stokes wave, Amick proved in 1987 that the angle must be less than  31.15 degrees if the wave is irrotational.  However, for any wave that is  not irrotational, the question of whether there is any bound on the angle  has been completely open. An example is the extreme Gerstner wave, which  has adverse vorticity and vertical cusps. Moreover, numerical calculations  show that waves of finite depth with adverse vorticity can overturn,  so the angle can be 90 degrees.  On the other hand, Miles Wheeler and I  prove that there is an upper bound of 45 degrees for a large class of  waves with favorable vorticity and finite depth.  Seung Wook So and I  prove a similar bound for waves with small adverse vorticity.