On Steady Water Waves and Their Properties

Abstract: This thesis consists of four papers related to various aspects of steady water waves.


Paper I: Deep-water waves with vorticity: symmetry and rotational behaviour.

We show that for steady, periodic, and rotational gravity deep-water waves, a monotone surface profile between troughs and crests implies symmetry. It is observed that if the vorticity function has a bounded derivative, then it vanishes as one approaches great depths.


Paper II: Linear water waves with vorticity: rotational features and particle paths.

Steady linear gravity waves of small amplitude travelling on a current of constant vorticity are found. For negative vorticity we show the appearance of internal waves and vortices, wherein the particle trajectories are not any more closed ellipses. For positive vorticity the situation resembles that of Stokes waves, but for large vorticity the trajectories are affected.


Paper III: On the streamlines and particle paths of gravitational water waves.

We investigate steady symmetric gravity water waves on finite depth. For non-positive vorticity it is shown that the particles display a mean forward drift, and for a class of waves we prove that the size of this drift is strictly increasing from bottom to surface. This includes the case of particles within irrotational waves. We also provide detailed information concerning the streamlines and the particle trajectories.


Paper IV: Travelling waves for the Whitham equation.

The existence of travelling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surface water waves on finite depth. It is found that there exist small-amplitude periodic travelling waves with sub-critical speeds. As the period of these travelling waves tends to infinity, their velocities approach the limiting long-wave speed c0, and the waves approach a solitary wave. It is also shown that there can be no solitary waves with velocities much greater than c0. Finally, numerical approximations of some periodic travelling waves are presented.