On the symmetry of spatially periodic two-dimensional water waves

Abstract

We show that a spatially periodic solution to the irrotational two-dimensional gravity water wave problem, with the propertythat the horizontal velocity component at the flat bed is symmetric, while the acceleration at the flat bed is anti-symmetric with respectto a common axis of symmetry, necessarily constitutes a traveling wave. The proof makes use complex variables and structural properties of the governing equations for nonlinear water waves.