Complex singularity of a stokes wave
Abstract
Twodimensional potential flow of the ideal incompressible fluid with free surface and infinite depth can be described by a conformal map of the fluid domain into the complex lower halfplane. Stokes wave is the fully nonlinear gravity wave propagating with the constant velocity. The increase in the scaled wave height H/λ from the linear limit H/λ = 0 to the critical value H _{max}/λ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave. Here, H is the wave height and λ is the wavelength. We simulated fully nonlinear Euler equations, reformulated in terms of conformal variables, to find Stokes waves for different wave heights. Analyzing spectra of these solutions we found in conformal variables, at each Stokes wave height, the distance ν _{ c } from the lowest singularity in the upper halfplane to the real line which corresponds to the fluid free surface. We also identified that this singularity is the squareroot branch point. The limiting Stokes wave emerges as the singularity reaches the fluid surface. From the analysis of data for ν _{ c } → 0 we suggest a new power law scaling ν _{ c } ∝ ( H _{max}  H)^{3/2} as well as new estimate H _{max}/λ ≃ 0.1410633.
 Publication:

Soviet Journal of Experimental and Theoretical Physics Letters
 Pub Date:
 February 2014
 DOI:
 10.1134/S0021364013240077
 arXiv:
 arXiv:1311.1882
 Bibcode:
 2014JETPL..98..675D
 Keywords:

 Physics  Fluid Dynamics;
 Nonlinear Sciences  Pattern Formation and Solitons;
 Physics  Atmospheric and Oceanic Physics;
 Physics  Computational Physics
 EPrint:
 (5 pages, 4 figures, submitted to JETP Letters)