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

Recovery of traveling water waves with smooth vorticity from the horizontal velocity on a line of symmetry for various wave regimes