We show that a spatially periodic solution to the irrotational two-dimensional gravity water wave problem, with the property that the horizontal velocity component at the surface, as well as the wave profile is symmetric, necessarily defines a traveling wave. The proof makes use of maximum principles for harmonic functions and structural properties of the governing equations for nonlinear water waves.
Kogelbauer, F. (2015). Symmetric irrotational water waves are traveling waves. Journal of Differential Equations, 259(10), 5271–5275. https://doi.org/10.1016/j.jde.2015.06.025