Bound on the slope of steady water waves with favorable vorticity

Abstract

We consider the angle $\theta$ of inclination (with respect to the horizontal) of the profile of a steady 2D inviscid symmetric periodic or solitary water wave subject to gravity. Although $\theta$ may surpass 30${}^{\circ }$ for some irrotational waves close to the extreme wave, Amick [Ami87] proved that for any irrotational wave the angle must be less than 31.15${}^{\circ }$. 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 ($\theta =90^{\circ }$). 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${}^{\circ }$ on $\theta$ 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.

21 pages, manuscript completed February 2015

References in corpus (1)

• The Froude number for solitary water waves with vorticity

Cited by in corpus (3)

• Large-amplitude steady gravity water waves with general vorticity and critical layers
• On the amplitude of steady water waves with positive constant vorticity
• The dynamic pressure in deep-water extreme Stokes waves