🌀 nonlinear sciences

Stokes Waves in Water of Finite Depth

This paper investigates periodic Stokes waves in water of finite depth by applying established methods to analyze how various physical quantities evolve with increasing wave steepness, while also examining the Taylor sign condition and the location of complex singularities in the conformal mapping.

Original authors: Eleanor Byrnes, Bernard Deconinck, Anastassiya Semenova

Published 2026-04-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Eleanor Byrnes, Bernard Deconinck, Anastassiya Semenova

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the ocean as a giant, endless trampoline. When you jump on it, you create waves that travel across the surface. Scientists have been studying these waves for nearly 200 years, trying to understand exactly how they move, how much energy they carry, and what happens when they get so tall they almost break.

This paper is like a new, high-tech map that helps us understand these waves not just in the deep, endless ocean, but in shallow water too—like a wave traveling in a swimming pool or a harbor.

Here is a simple breakdown of what the authors did, using some everyday analogies:

1. The "Perfect" Wave (Stokes Waves)

The authors are studying a specific type of wave called a Stokes wave. Think of this as the "Goldilocks" wave: it's not crashing and breaking; it's a perfect, repeating loop that travels at a constant speed without changing shape.

The Challenge: In deep water, we've known how to calculate these waves for a long time. But in shallow water (like a pool), the bottom gets in the way, making the math much trickier. It's like trying to ride a bike on a smooth highway (deep water) versus riding it through a narrow, bumpy alleyway (shallow water). The rules change.

2. The "Magic Mirror" Trick

To solve the math, the authors used a clever trick called conformal mapping.

The Analogy: Imagine the water surface is a crumpled piece of paper. It's hard to draw straight lines on it. But if you could magically stretch and smooth that paper out into a perfect rectangle without tearing it, the math becomes easy.
They turned the messy, curvy shape of the wave into a neat, flat rectangle in a "mathematical world." They solved the equations there and then "folded" the paper back to see what the real wave looks like. This allowed them to handle the shallow water problem much more accurately than previous methods.

3. The "Oscillating Rollercoaster"

One of the most exciting discoveries is about what happens as you make the waves steeper and steeper.

The Old Idea: Scientists used to think that as a wave gets taller, its speed and energy just go up smoothly until it hits a maximum limit.
The New Discovery: The authors found that as the wave gets closer to its maximum height, its speed and energy don't just go up—they wiggle.
The Analogy: Imagine driving a car up a hill. You expect to just keep going faster. But instead, your speedometer starts to vibrate: fast, slow, fast, slow—like a rollercoaster going up a steep hill with tiny bumps. The authors confirmed that these "wiggles" (oscillations) happen in shallow water just like they do in deep water. They even found the exact spots where the wave speed hits a local high or low point before reaching the absolute limit.

4. The "Ghost Corners" and "Magic Points"

The paper also looks at the very top of the steepest possible wave.

The 120-Degree Corner: The authors confirmed that the steepest possible wave has a sharp point at the top, shaped like a 120-degree corner (think of the top of a house roof).
The "Ghost" Singularity: Here is the weird part. If you look at the math above the water (where there is no water, just air), there is a "ghost" point where the math breaks down.
The Analogy: Imagine the wave is a shadow cast by a light. The shadow is the water. But if you trace the light rays backward, they hit a "ghost" point in the air above the wave. In deep water, this ghost point is a "square-root" type break. The authors found that in shallow water, there is a mirror image of this ghost point hiding underneath the water floor, too! It's like the wave has a shadow above and a reflection below, and both are pulling on the math.

5. Why Does This Matter?

You might ask, "Who cares about wiggly waves in a pool?"

Safety: Understanding the exact limits of how tall a wave can get helps engineers design better sea walls, oil rigs, and ships.
Accuracy: Previous methods were like using a blurry camera to take a photo of a fast-moving wave. This paper uses a high-definition, slow-motion camera. They found that older methods were slightly off, especially for very shallow water.
The Future: By understanding these "ghost points" and the "wiggles," scientists can build better computer models to predict tsunamis, storm surges, and how energy moves through the ocean.

Summary

In short, the authors took a difficult math problem about waves in shallow water, used a "magic mirror" to simplify the geometry, and discovered that these waves behave in a surprisingly complex way: they wiggle in speed and energy as they get taller, and they have hidden mathematical "ghosts" both above and below the water surface. This gives us a much clearer picture of how nature's most powerful waves really work.

1. Problem Statement

The paper investigates Stokes waves—periodic, permanent-form traveling water waves—in fluids of fixed finite depth. While the existence and properties of these waves in infinite depth are well-established (including the existence of a limiting wave with a $120^\circ$ corner at the crest), their behavior in finite depth, particularly regarding high-amplitude limits, energy oscillations, and singularity structures, requires further rigorous analysis.

The study focuses on three main objectives:

Numerical Computation: Accurately computing Stokes waves across a wide range of finite depths and steepness values.
Asymptotic Analysis: Extending the perturbation theory of Longuet-Higgins and Fox (originally for infinite depth) to finite depth to analyze the oscillatory behavior of wave speed and energy as the limiting wave is approached.
Singularity Structure: Investigating the complex singularities of the conformal mapping associated with these waves, specifically examining the branch points above and below the fluid domain.

2. Methodology

A. Mathematical Formulation

The authors utilize a conformal mapping reformulation of the 2D Euler water wave equations.

The physical domain (fluid of depth $d$ ) is mapped to a rectangular conformal domain of width $2\pi$ and height $h$ in the complex plane.
The free surface is parameterized by $z(w) = x(u) + iy(u)$, where $w = u + iv$.
The governing equations are reduced to a pair of non-local, implicit PDEs on the interval $[-\pi, \pi]$ . In a traveling frame with speed $c$ , the surface elevation $y$ satisfies the Babenko equation:
$0 = \hat{S}y = (c^2 \hat{K} - g)y - g\left(y \hat{K}y + \frac{1}{2}\hat{K}y^2\right)$
where $\hat{K}$ and $\hat{R}$ are operators involving the finite-depth Hilbert transform generalization, dependent on the conformal depth $h$ .

B. Numerical Algorithm

Newton-Conjugate-Gradient Method: The authors solve the Babenko equation using a Newton-Conjugate-Gradient method.
Fixed Physical Depth Constraint: Unlike previous works that fixed the conformal depth $h$ , this study fixes the physical depth $d$ . Since $h$ and $d$ differ for non-flat surfaces ( $h = d + \langle y \rangle$ ), the authors employ a bisection algorithm to iteratively adjust the conformal depth $h$ for a given wave speed $c$ until the physical depth constraint is satisfied.
Precision: Computations are performed in double precision (Matlab) for general analysis. For high-amplitude waves near the limit, variable precision arithmetic (Julia's BigFloat) is used with a tolerance of $10^{-64}$ to mitigate numerical errors caused by the proximity of singularities.

C. Asymptotic Fitting

The authors fit numerical data to the asymptotic expansions derived by Longuet-Higgins and Fox (LHF). These expansions relate wave speed ( $c$ ), energy ( $H$ ), and steepness ( $S$ ) to a perturbation parameter $\epsilon$ (related to the particle speed at the crest):
$c^2 \sim c_{lim}^2 + a_2 \epsilon^3 \cos(3\mu \ln \epsilon + a_4) + \dots$
The study determines the coefficients ( $a_2, a_4, \dots$ ) for various finite depths to predict the properties of the limiting wave and local extremizers.

3. Key Contributions and Results

A. Numerical Discovery of Wave Profiles

Localization: In shallow water, high-amplitude Stokes waves become highly localized within their wavelength, resembling solitary waves. The authors quantify this using a modified wavenumber $\tilde{k}(d)$ , which is well-approximated by an inverse power of a shifted hyperbolic tangent, linking the localization to the dispersion relation.
Limiting Wave Properties: The paper provides precise values for the limiting steepness ( $S_{lim}$ ), speed ( $c_{lim}$ ), and energy ( $H_{lim}$ ) across a wide range of depths (from $d=0.01$ to $d=10$ ).
Scaling Laws: The limiting quantities decay as power laws of the modified dispersion relation $\tilde{\omega}(d)$ $\tilde{ω} (d)$ . Specifically:
- $c_{lim} \propto \tilde{\omega}^{-1}$
- $S_{lim} \propto \tilde{\omega}^{-2}$
- $H_{lim} \propto \tilde{\omega}^{-6}$

B. Oscillations in Speed and Energy

Verification of LHF Theory: The study confirms that as steepness increases toward the limit, the wave speed and total energy undergo infinite oscillations (log-periodic oscillations) in finite depth, just as in infinite depth.
Local Extremizers: By differentiating the asymptotic forms, the authors predict the steepness values where the speed and energy reach local maxima and minima. These extremizers interlace, and their spacing follows a geometric progression determined by the phase shift parameter $\mu$ .
Ratio of Extremizers: A limiting ratio $R_n$ is defined to measure the position of energy extremizers relative to speed extremizers. This ratio is found to be well-approximated by a power of the dispersion relation.

C. Singularity Structure

Branch Points: The authors analyze the analytic continuation of the wave profile into the complex plane.
- Above the fluid: A square-root type branch point ( $\beta = 1/2$ ) exists above the fluid. Its distance $v_c$ from the real axis decays as $(S_{lim} - S)^{1.5}$ as the wave steepness approaches the limit.
- Below the fluid: Due to the Neumann boundary condition at the bottom, a "mirror" singularity exists below the conformal domain. The study confirms that this lower singularity is also a square-root branch point, and its location is strictly linked to the upper singularity and the conformal depth via the mirror principle.
Riemann Sheets: The results support the hypothesis that the square-root branch point of steep waves transitions to a cube-root branch point for the limiting wave, a mechanism previously observed only in infinite depth.

4. Significance and Impact

Bridging Infinite and Finite Depth: This work successfully extends the sophisticated asymptotic theories and numerical techniques developed for infinite-depth Stokes waves to the more physically realistic finite-depth case.
High-Precision Data: The use of variable precision arithmetic provides a new benchmark for the accuracy of limiting wave parameters in finite depth, resolving discrepancies found in previous studies relying on Padé approximants with fewer Fourier modes.
Stability Implications: The identification of local extremizers of speed and energy is crucial for understanding the stability spectra of Stokes waves. These points often correspond to bifurcations or changes in the topology of the stability spectrum.
Algorithmic Improvement: By characterizing the singularity structure (branch points) in finite depth, the paper lays the groundwork for developing more efficient numerical methods (e.g., using auxiliary conformal mappings to push singularities away) for computing extreme waves in shallow water.

In summary, the paper provides a comprehensive numerical and theoretical framework for understanding the behavior of extreme Stokes waves in finite depth, confirming the universality of log-periodic oscillations and singularity structures across depth regimes.