= Solution
Both $\sin(x-t)$ and $\cos(x-t)$ have zero average over one period, so $\langle u\rangle=0$ at every fixed point.
The <material derivative> gives the particle acceleration
$$
a=\frac{\partial u}{\partial t}+(u\mathbin{\cdot}\nabla)u.
$$
Direct differentiation yields
$$
a_x=\varepsilon y\sin\theta
+\varepsilon^2(1-y^2)\sin\theta\cos\theta,
$$
$$
a_y=-\varepsilon\cos\theta
+\varepsilon^2y\cos^2\theta.
$$
Therefore the <Eulerian time average> at fixed $(x,y)$ is
$$
\boxed{\langle a\rangle=(0,\varepsilon^2y/2)}.
$$
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