New singularities for Stokes waves

TL;DR

This paper investigates the complex singularity structure of Stokes waves, revealing an infinite number of singularities that coalesce at the crest as the wave approaches its maximum height, advancing understanding of wave behavior.

Contribution

The study develops numerical methods to construct the Riemann surface of Stokes waves, uncovering a countably infinite set of singularities and their coalescence at the highest wave.

Findings

Infinite singularities exist on different branches of the solution

Singularities coalesce at the crest as the wave reaches maximum height

Numerical methods effectively construct the Riemann surface for analysis

Abstract

In 1880, Stokes famously demonstrated that the singularity that occurs at the crest of the steepest possible water wave in infinite depth must correspond to a corner of $120^\circ$. Here, the complex velocity scales like $f^{1/3}$ where $f$ is the complex potential. Later in 1973, Grant showed that for any wave away from the steepest configuration, the singularity $f = f^*$ moves into the complex plane, and is of order $(f-f^*)^{1/2}$ (J. Fluid Mech., vol. 59, 1973, pp. 257-262). Grant conjectured that as the highest wave is approached, other singularities must coalesce at the crest so as to cancel the square-root behaviour. Despite recent advances, the complete singularity structure of the Stokes wave is still not well understood. In this work, we develop numerical methods for constructing the Riemann surface that represents the extension of the water wave into the complex plane. We show that a countably infinite number of distinct singularities exists on other branches of the solution, and that these singularities coalesce as Stokes' highest wave is approached.