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 . Here, the complex velocity scales like where is 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 of order (Grant 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 squareroot 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 exist on other branches of the solution, and that these singularities coalesce as Stokes’ highest wave is approached.
Original language  English 

Pages (fromto)  256283 
Number of pages  28 
Journal  Journal of Fluid Mechanics 
Volume  798 
Early online date  31 May 2016 
DOIs  
Publication status  Published  10 Jul 2016 
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Phil Trinh
 Department of Mathematical Sciences  Lecturer in Applied Mathematics
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Centre for Nonlinear Mechanics  CoDirector
 EPSRC Centre for Doctoral Training in Advanced Automotive Propulsion Systems (AAPS CDT)
Person: Research & Teaching, Affiliate staff