## Abstract

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.

Original language | English |
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Pages (from-to) | 1555-1580 |

Number of pages | 26 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 222 |

Issue number | 3 |

Early online date | 20 Jul 2016 |

DOIs | |

Publication status | Published - 31 Dec 2016 |

## ASJC Scopus subject areas

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering