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Transverse Instability of Stokes Waves at Finite Depth

Ryan P. Creedon, Huy Q. Nguyen, Walter A. Strauss

Abstract

A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. In 1981 McLean discovered via numerical methods that Stokes waves are unstable with respect to transverse perturbations. In \cite{CreNguStr} for the case of infinite depth we proved rigorously that the spectrum of the water wave system linearized at small Stokes waves, with respect to transverse perturbations, contains unstable eigenvalues lying approximately on an ellipse. In this paper we consider the case of finite depth and prove that the same spectral instability result holds for all but finitely many values of the depth. The computations and some aspects of the theory are considerably more complicated in the finite depth case.

Transverse Instability of Stokes Waves at Finite Depth

Abstract

A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. In 1981 McLean discovered via numerical methods that Stokes waves are unstable with respect to transverse perturbations. In \cite{CreNguStr} for the case of infinite depth we proved rigorously that the spectrum of the water wave system linearized at small Stokes waves, with respect to transverse perturbations, contains unstable eigenvalues lying approximately on an ellipse. In this paper we consider the case of finite depth and prove that the same spectral instability result holds for all but finitely many values of the depth. The computations and some aspects of the theory are considerably more complicated in the finite depth case.
Paper Structure (13 sections, 16 theorems, 195 equations, 3 figures)

This paper contains 13 sections, 16 theorems, 195 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Transverse perturbations of Stokes waves
  3. Stokes waves
  4. Transverse perturbations, linearization, and transformations
  5. The transformed Dirichlet-Neumann operator
  6. The reformulated instability problem
  7. Resonance condition and Kato's perturbed basis
  8. Resonance condition
  9. Kato's perturbed basis
  10. Third-order expansions
  11. Third-order expansions of $\mathcal{L}_{\varepsilon,\beta}$
  12. Third-order expansions of $\textrm{L}_{\varepsilon, \delta}$
  13. Proof of Theorem \ref{['theo:main']}

Key Result

Theorem 1.1

Let the depth $h\in (0,\infty)$ be given. Except for a finite number $N$ of values of $h$, the following instability statement is true. There exist $\varepsilon_{\textrm{max}} > 0$ and $\delta_{\text{max}}>0$ such that for all $\varepsilon \in (- \varepsilon_{\textrm{max}}, \varepsilon_{\textrm{max where $T$ and $\Delta$ are real-valued, real-analytic functions such that $T(\varepsilon,\delta) =

Figures (3)

  • Figure 1: A comparison of the transverse instability isola obtained in this work (orange curves) and numerical computations of the unstable eigenvalues of the transverse instability (blue dots) for a Stokes wave with amplitude $\varepsilon = 0.01$ in water of depth $h = 1$ (left), $h = 3/2$ (middle), and $h = 2$ (right). The center of the isola is subtracted from its imaginary component to show a sense of scale. In each plot the distance between the orange curves and blue dots is $O\left(\varepsilon^4\right)$.
  • Figure 2: A plot of $\beta_*$ as a function of $h$ (solid blue) along with its asymptotic expansions as $h \rightarrow 0^+$ and $h \rightarrow \infty$ (dashed orange) according to Proposition \ref{['prop:rescond']}.
  • Figure 3: A plot of $b_{3,0}$ as a function of $h$. As $h \rightarrow 0^+$, we have $b_{3,0} \rightarrow +\infty$. As $h \rightarrow \infty$, we have $b_{3,0} \rightarrow -0.49476...$, in agreement with CreNguStr. At $h_{crit} = 0.25065....$, we have $b_{3,0} = 0$. Thus, for this depth, we observe no transverse instability at $O(\varepsilon^3)$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 25 more