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.