Singularities in the complex physical plane for deep water waves

Abstract Deep water waves in two-dimensional flow can have curvature singularities on the surface profile; for example, the limiting Stokes wave has a corner of $2\lrm{\pi} / 3$ radians and the limiting standing wave momentarily forms a corner of $\lrm{\pi} / 2$ radians. Much less is known about the possible formation of curvature singularities in general. A novel way of exploring this possibility is to consider the curvature as a complex function of the complex arclength variable and to seek the existence and nature of any singularities in the complex arclength plane. Highly accurate boundary integral methods produce a Fourier spectrum of the curvature that allows the identification of the nearest singularity to the real axis of the complex arclength plane. This singularity is in general a pole singularity that moves about the complex arclength plane. It approaches the real axis very closely when waves break and is associated with the high curvature at the tip of the breaking wave. The behaviour of these singularities is more complex for standing waves, where two singularities can be identified that may collide and separate. One of them approaches the real axis very closely when a standing wave forms a very narrow collapsing column of water almost under free fall. In studies so far, no singularity reaches the real axis in finite time. On the other hand, the surface elevation $y(x)$ has square-root singularities in the complex $x$ plane that do reach the real axis in finite time, the moment when a wave first starts to break. These singularities have a profound effect on the wave spectra.