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An improved upper bound for the Froude number of irrotational solitary water waves

Evgeniy Lokharu, Jörg Weber

TL;DR

The paper addresses refining the analytic upper bound on the Froude number $Fr$ for two-dimensional, irrotational solitary gravity waves, a quantity historically bounded by $Fr < \sqrt{2}$ but with numerical evidence suggesting tighter limits. It introduces a novel flow-force based framework that avoids integral identities, constructing bounds via the flow force function $F$, the derived quantity $S$, and auxiliary functions $J$ and \bar J$ evaluated across three regimes of crest height relative to stagnation. The main result is a rigorous bound $Fr < 1.37838$, supported by detailed case analyses (very far from stagnation, moderately close, and near stagnation) and corroborated by numerical interpolation tables. This tightens the gap between analytic theory and numerical observations (which favor $Fr \le 1.294$) and informs subharmonic bifurcation considerations for Stokes waves, with potential implications for stability and bifurcation analyses in irrotational gravity waves.

Abstract

A classical and central problem in the theory of water waves is to classify parameter regimes for which nontrivial solitary waves exist. In the two-dimensional, irrotational, pure gravity case, the Froude number $Fr$ (a non-dimensional wave speed) plays the central role. So far, the best analytic result $Fr < \sqrt{2}$ was obtained by Starr (1947), while the numerical evidence of Longuet-Higgins & Fenton (1974) states $Fr \le 1.294$. On the other hand, as shown recently by Kozlov (2023), the upper bound $Fr < 1.399$ is related to the existence of subharmonic bifurcations of Stokes waves. In this paper, we develop a new strategy utilizing the flow force function and rigorously establish the improved upper bound $Fr < 1.37838$.

An improved upper bound for the Froude number of irrotational solitary water waves

TL;DR

The paper addresses refining the analytic upper bound on the Froude number for two-dimensional, irrotational solitary gravity waves, a quantity historically bounded by but with numerical evidence suggesting tighter limits. It introduces a novel flow-force based framework that avoids integral identities, constructing bounds via the flow force function , the derived quantity , and auxiliary functions and \bar JFr < 1.37838Fr \le 1.294$) and informs subharmonic bifurcation considerations for Stokes waves, with potential implications for stability and bifurcation analyses in irrotational gravity waves.

Abstract

A classical and central problem in the theory of water waves is to classify parameter regimes for which nontrivial solitary waves exist. In the two-dimensional, irrotational, pure gravity case, the Froude number (a non-dimensional wave speed) plays the central role. So far, the best analytic result was obtained by Starr (1947), while the numerical evidence of Longuet-Higgins & Fenton (1974) states . On the other hand, as shown recently by Kozlov (2023), the upper bound is related to the existence of subharmonic bifurcations of Stokes waves. In this paper, we develop a new strategy utilizing the flow force function and rigorously establish the improved upper bound .

Paper Structure

This paper contains 8 sections, 4 theorems, 52 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some lemmas
  4. A proof of Theorem \ref{['thm:main']}
  5. The case η<=0.95r
  6. The case 0.95r<η<=(1-β)r and β<=0.05.
  7. The case η>(1-β)r and β<=0.05.
  8. Acknowledgments

Key Result

Theorem 1

The Froude number of any solitary wave is bounded from above by $1.37838$.

Figures (1)

  • Figure 1: Plot of $\partial J/\partial\hat{\eta}$.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof