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

TL;DR

This paper presents a method to recover the entire wave profile and velocity field of rotational water waves with smooth vorticity from horizontal velocity data on a symmetry line, applicable to various wave regimes.

Contribution

It introduces an explicit recovery formula for wave profiles and velocity fields from symmetry line velocity data, valid for finite/infinite depth and multiple wave types.

Findings

Method successfully recovers wave profiles in theoretical examples.

Applicable to gravity, capillary-gravity, and capillary waves.

Works for both periodic and solitary waves.

Abstract

In the general context of rotational water waves with a 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 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 recover also 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.