 Solution

A fluid particle follows a Lagrangian trajectory, so its position satisfies $\dot{x}=u(x,t)$ rather than the velocity evaluated permanently at its initial position. Assume $\omega \ne 0$, and writeSet $\alpha =k{X}_0$ and $\theta =\alpha -\omega t$. A Taylor expansion of the velocity about $X_0$ givesThus $\dot{a}=\sin \theta$, andAt the next order,so the required second-order trajectory in a sinusoidal velocity field isBoth corrections vanish at $t=0$. All retained terms have units of length. The secular contribution gives the Stokes driftalthough the Eulerian time average of the given velocity is zero. The particle samples the wave at its displaced position, producing this second-order drift. The expansion is in the dimensionless parameter $|k{u}_0/\omega |\ll 1$, for fixed times; it ceases to be uniform when the accumulated drift changes the wave phase by order unity.