Waves and Curves

Water waves are perhaps the most notable features of the planet, and they have occupied the attention of scientist since the birth of civilization. Yet they remain incompletely understood. Despite recent theoretical advances, the generic mathematical behavior of water waves eludes description. I will present a different view of water waves traveling in two-dimensions, one based on the relationship between the curvature and the arclength. There is a strikingly simple connection associated with the formation of breaking waves. It is based on the presence of pole singularities in the complex arclength plane that approach the real axis. A companion square-root singularity in the surface height as a function of the horizontal coordinate reaches the real axis in finite time when the slope becomes vertical. The presence of this singularity affects the wave spectra directly.