= Second-order trajectory in a sinusoidal velocity field
{title2=$\overline{u_L}=ku_0^2/(2\omega)$}
For prescribed one-dimensional <velocity> $u=u_0\sin(kx-\omega t)$, let $\alpha=kX_0$, $\theta=\alpha-\omega t$ and expand the <Lagrangian trajectory> as $x=X_0+u_0a+u_0^2b$. The <Taylor expansion> of $\dot x=u(x,t)$ gives $\dot a=\sin\theta$ and $\dot b=ka\cos\theta$. Initial position $X_0$ requires $a=(\cos\theta-\cos\alpha)/\omega$ and
$$
b=\frac k\omega\left[\frac t2+\frac{\sin2\alpha-\sin2\theta}{4\omega}-\frac{\cos\alpha(\sin\alpha-\sin\theta)}\omega\right].
$$
The secular <velocity> $ku_0^2/(2\omega)$ is <Stokes drift>, even though the <Eulerian time average> vanishes. This fixed-time expansion requires $|ku_0/\omega|\ll1$ and is not uniform once the secular displacement changes the phase by order unity.
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