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Rigidity of symmetric doubly-periodic water waves near shear flows

Douglas Svensson Seth, Kristoffer Varholm, Erik Wahlén, Jörg Weber

TL;DR

The paper analyzes symmetric, doubly periodic capillary-gravity waves in finite depth near non-uniform shear flows. By flattening the free boundary, expanding in a small amplitude parameter, and performing a careful linear and quadratic solvability analysis, the authors show that the leading-order bifurcating waves are two-dimensional if the background shear $U(x_3)$ is non-constant and non-stagnant. The linear problem has a finite-dimensional kernel $N(U)$ determined by a dispersion relation, and the quadratic solvability imposes a resonance-type obstruction unless all modal amplitudes vanish, thereby precluding genuinely three-dimensional leading-order bifurcations. This rigidity contrasts with the uniform-flow case and clarifies when higher-dimensional wave structures may (or may not) arise under symmetric, doubly periodic settings. The approach also accommodates variations of the dynamic boundary condition and extends to other boundary phenomena such as hydroelastic waves.

Abstract

We prove that symmetric, doubly periodic, capillary-gravity water waves in finite depth bifurcating from non-uniform non-stagnant shear flows are necessarily two-dimensional to leading order. This is in stark contrast to the case of uniform background flows, where the existence of truly three-dimensional bifurcating waves is well known. While this paper focuses on capillary-gravity water waves, our proof also applies to other suitable types of dynamic boundary conditions, such as hydroelastic waves.

Rigidity of symmetric doubly-periodic water waves near shear flows

TL;DR

The paper analyzes symmetric, doubly periodic capillary-gravity waves in finite depth near non-uniform shear flows. By flattening the free boundary, expanding in a small amplitude parameter, and performing a careful linear and quadratic solvability analysis, the authors show that the leading-order bifurcating waves are two-dimensional if the background shear is non-constant and non-stagnant. The linear problem has a finite-dimensional kernel determined by a dispersion relation, and the quadratic solvability imposes a resonance-type obstruction unless all modal amplitudes vanish, thereby precluding genuinely three-dimensional leading-order bifurcations. This rigidity contrasts with the uniform-flow case and clarifies when higher-dimensional wave structures may (or may not) arise under symmetric, doubly periodic settings. The approach also accommodates variations of the dynamic boundary condition and extends to other boundary phenomena such as hydroelastic waves.

Abstract

We prove that symmetric, doubly periodic, capillary-gravity water waves in finite depth bifurcating from non-uniform non-stagnant shear flows are necessarily two-dimensional to leading order. This is in stark contrast to the case of uniform background flows, where the existence of truly three-dimensional bifurcating waves is well known. While this paper focuses on capillary-gravity water waves, our proof also applies to other suitable types of dynamic boundary conditions, such as hydroelastic waves.
Paper Structure (10 sections, 5 theorems, 79 equations)

This paper contains 10 sections, 5 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Flattening
  4. Periodicity and symmetry
  5. Expansion and main theorem
  6. Solvability at the linear level
  7. Recovering ℘¹ from 𝜂¹
  8. Recovering 𝒖¹
  9. Summing up at the linear level
  10. Solvability at the quadratic level and proof of the main theorem

Key Result

Lemma 2.1

If a vector field $\bm{u}$ is $(+)$-symmetric, $\Lambda$-periodic, and does not depend on $x_1$, then with $U$ being even and $\lambda_2$-periodic with respect to $x_2$. When $\eta \equiv 0$, all such $\bm{u} \in C_+^1(\Omega; \mathbb{R} ^3)$ are trivially solutions to eq:flatE1--eq:flatDyn with corresponding dynamic pressure $\wp \equiv 0$.

Theorems & Definitions (10)

  • Remark
  • Lemma 2.1
  • Theorem 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 5.1
  • proof
  • Remark 5.2