Improved bound in the Benjamin and Lighthill conjecture

TL;DR

This paper establishes a new explicit lower bound for the flow force constant in steady water waves, refining the Benjamin and Lighthill conjecture and narrowing the parameter space for steady solutions.

Contribution

It provides a novel, asymptotically sharp lower bound for the flow force constant, improving understanding of steady water wave solutions and their parameter constraints.

Findings

Derived a new explicit lower bound for the flow force constant.

Recovered the classical F<2 inequality for the Froude number.

Reduced the parameter region supporting steady waves.

Abstract

The classical Benjamin and Lighthill conjecture about steady water waves states that the non-dimensional flow force constant of a solution is bounded by the corresponding constants of the supercritical and subcritical uniform streams respectively. These inequalities determine a parameter region that covers all steady motions. In fact not all points of the region determine a steady wave. In this note we prove a new and explicit lower bound for the flow force constant, which is asymptotically sharp in a certain sense. In particular, this recovers the well known inequality F<2 for the Froude number, while significantly reducing the parameter region supporting steady waves.