Improved bound in the Benjamin and Lighthill conjecture

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.

Global bifurcation and highest waves on water of finite depth

“…This new formulation in terms of flow force function F is of special interest as shown recently in [33]. However, in the present paper we only need F as a comparison function for proving the next result.…”

“…The latter expression is independent of x, which makes it of special importance for spatial dynamics (see [19] and references therein). Beside, it plays an important role in classification of steady waves; see [8,29,33].…”

“…Now using (2.5b) we conclude ψ y (x, 0) ≤ S −1 2r. It is left to use inequality S > 1 2 r 2 obtained in [33].…”