= 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 $(a\omega/k)e^{kz}\sin\chi$. Therefore
$$
u_1=a\omega e^{kz}\cos\chi,\qquad
w_1=a\omega e^{kz}\sin\chi.
$$
Relative to mean parcel labels, integrate at fixed zeroth-order position to obtain the oscillatory <fluid displacement>
$$
\xi_1=-ae^{kz}\sin\chi,\qquad \zeta_1=ae^{kz}\cos\chi.
$$
The <Stokes drift> is the period mean of the displacement correction to the first-order <velocity field>:
$$
\begin{aligned}
u_s&=\left\langle\xi_1u_{1x}+\zeta_1u_{1z}\right\rangle
=a^2k\omega e^{2kz}\langle\sin^2\chi+\cos^2\chi\rangle,\\
w_s&=\left\langle\xi_1w_{1x}+\zeta_1w_{1z}\right\rangle=0.
\end{aligned}
$$
Thus $\boxed{u_s=\eta_0^2k\omega e^{2kz}}$, 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, $k\eta_0\ll1$, relative to the <Eulerian mean velocity>.
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