Symmetric Water Waves with Surface Tension: Traveling Wave Behavior and Maximal Horizontal Velocity

Abstract We study spatially periodic solutions to the two-dimensional water wave problem with surface tension. Assuming symmetry of the wave profile and of the horizontal velocity component at the free surface, we show that the solution necessarily corresponds to a traveling wave. This result relies on maximum principles for subharmonic functions and the structural properties of the governing equations. As a further consequence, in the case of positive constant vorticity, we establish that the maximum of the horizontal velocity remains invariant in time.