= Solution
For the established time-periodic response, take a complex separated solution $v=X(x)e^{i\omega t}$ and set $u=\operatorname{Re}v$. The <diffusion equation> gives $X''=(i\omega/\nu)X$. For $\omega>0$, define the penetration depth $\delta=\sqrt{2\nu/\omega}$; the spatial roots are $\pm(1+i)/\delta$. Decay at infinity selects
$$
v(x,t)=U_0e^{-(1+i)x/\delta}e^{i\omega t}.
$$
Its real part is the solution of <Stokes' second problem>:
$$
\boxed{u(x,t)=U_0e^{-x/\delta}\cos(\omega t-x/\delta),\qquad\delta=\sqrt{\frac{2\nu}{\omega}}.}
$$
The velocity amplitude decays exponentially, while the phase lags by $x/\delta$. Direct substitution verifies the <diffusion equation>, and the two <boundary conditions> follow immediately.
\b[These data select a unique periodic response, but do not specify an arbitrary initial transient.] With no initial velocity supplied, one may add any suitable solution of the homogeneous-boundary diffusion problem. The displayed solution is the usual established oscillatory flow after transients have decayed.
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