Davies’ surface condition and singularities of deep water waves
Davies’ surface condition is an approximate free-surface condition on gravity waves progressing in permanent form on water of infinite depth. It is known that this condition preserves essential features of finite-amplitude waves including the highest one. This paper proposes a new surface condition that generalizes Davies’ idea of approximation and covers a fully nonlinear condition. Analytic continuation of the proposed surface condition allows us to explore singularities of solutions that dominate the flow. The results of singularity analysis elucidate the connection between Davies’ approximate solution and the fully nonlinear solution. In addition, it is shown that the nonmonotonic variation of wave speed with wave steepness can be predicted using a linear sum of a relatively small number of singularities. This suggests that a suitable choice of a parameter in the proposed surface condition can move singularities away from the flow field without changing their structure and may reduce numerical difficulties due to singularities for large-amplitude waves.
KeywordsAnalytic continuation Free-surface condition Singularity Water waves
The author thanks the anonymous referees for their helpful comments.
- 6.Murashige S (2012) Validity of Davies’ approximate solutions for water waves. In: Proceedings of 10th International Conference on Hydrodynamics 2: 214–219Google Scholar
- 17.Schwartz LW (1974) Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J Fluid Mech 62: 553–578Google Scholar