OCIAM Group Meeting - New singularities for Stokes waves

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°. Here, the complex velocity scales like the one-third power of the complex potential. Later in 1973, Grant showed that for any wave away from the steepest configuration, the singularity moves into the complex plane, and is instead of order one-half. 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 talk, we shall explain how we have been able to construct the Riemann surface that represents the extension of the water wave into the complex plane. We shall also demonstrate the existence of a countably infinite number of singularities, never before noted, which coalesce as Stokes' highest wave is approached. Our results demonstrate that the singularity structure of a finite amplitude wave is much more complicated than previously anticipated,