New singularities for Stokes waves

Samuel C. Crew, Philippe H. Trinh

Published 2015-10-14Version 1

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. Even today, it is not well understood how this process occurs, nor is it known what other singularities may exist. In this work, we explain how to construct the Riemann surface that represents the extension of the water wave into the complex plane. We develop numerical methods for achieving this analytic continuation, and demonstrate that the singularity structure of a finite amplitude wave is much more complicated than previously anticipated. In particular, a countably infinite number of singularities is shown to exist on other branches of the solution; these singularities coalesce as Stokes' highest wave is approached. The recombination of all such singularities in the steepest limit is shown to be a daunting prospect.