Recovery of Traveling Water Waves with Smooth Vorticity from the Horizontal Velocity on a Line of Symmetry for Various Wave Regimes

In the general context of rotational water waves with a given smooth vorticity, it is shown that the wave profile can be recovered from the horizontal component of the velocity field on a line of symmetry. The method, which applies to waves on water of finite and infinite depth, uses only the values of the horizontal velocity of particles located on the line of symmetry that are close to the wave surface. In fact, together with the wave surface, we also recover the velocity field in a suitable surface layer. The explicit recovery formula is valid under the assumption that there are no stagnation points in the fluid for both periodic and solitary waves in each of the three regimes of gravity, capillary–gravity, and capillary waves. The efficiency of this method is illustrated in the context of the explicit solutions provided by Crapper for periodic capillary waves and Gerstner for periodic gravity waves.