Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 71 / 4 / a / Solution

Let $a={\eta }_0$ and $\chi =kx-\omega t$. For a deep-water gravity wave, ${\omega }^2=gk$ and the first-order velocity potential is $(a\omega /k)e^{kz}\sin \chi$. ThereforeRelative to mean parcel labels, integrate at fixed zeroth-order position to obtain the oscillatory fluid displacementThe Stokes drift is the period mean of the displacement correction to the first-order velocity field:Thus $\boxed{u_s={\eta }_0^{2}k\omega e^{2kz}}$, in the direction of propagation. A parcel samples faster forward motion near the top of its orbit and weaker backward motion lower down. This is the leading second-order drift of small-steepness waves, $k{\eta }_0\ll 1$, relative to the Eulerian mean velocity.