A sharp version of the Benjamin and Lighthill conjecture for steady waves with vorticity

TL;DR

This paper proves the Benjamin and Lighthill conjecture for all 2D steady water waves with vorticity, establishing sharp bounds on flow force and surface profile without geometric restrictions, and characterizing equality cases.

Contribution

It provides a general proof of the Benjamin and Lighthill conjecture for waves with arbitrary vorticity, extending previous results and characterizing equality cases in a broad setting.

Findings

Flow force constant bounded by conjugate laminar flows

Sharp bounds for surface profile established

Equality cases fully characterized

Abstract

We prove the Benjamin and Lighthill conjecture for all two-dimensional steady water waves with an arbitrary vorticity distribution. We show that the flow force constant of an arbitrary smooth wave is bounded by the corresponding flow force constants for conjugate laminar flows. We prove these inequalities without any assumptions on the geometry of the surface profile and put no restrictions on wave's amplitude. Furthermore, we give a complete description of cases when equalities can occur. Our results are new already for Stokes waves with vorticity, while the case of equalities is new even in the irrotational setting. Beside proving the Benjamin and Lighthill conjectrure, we establish sharp bounds for the surface profile, extending previous results on two-dimensional steady water waves.