  We investigate kinetic energy properties of an irrotational deep-water Stokes
wave. By introducing a conformal hodograph transform, we perform complex
analysis in the new variables, and show that the existence of the streamline
time-period for deep-water Stokes waves. Relying on the conformal mappings, we
prove some qualitative results about the kinetic energy such as the convexity
property of integral means of kinetic energy, the streamline time-period being
non-increasing and independent of initial data and the monotonicity and
logarithmic convexity of the total kinetic energy over a streamline
time-period. Moreover, taking advantage of the non-increasing nature of the
streamline time-period, we show that the streamline time-period strictly larger
than the wave period, which suggests the fact that there are no closed paths
for all particles in a deep-water Stokes flow and the drift of any streamline
is positive. These results allow us to quantify the observation that the
kinetic energy and the streamline time-period are larger near the free surface
and decrease with increasing the depth.
