 Solution

For the established time-periodic response, take a complex separated solution $v=X(x)e^{i\omega t}$ and set $u=\mathrm{Re}v$. The diffusion equation gives $X^{\prime \prime }=(i\omega /\nu )X$. For $\omega >0$, define the penetration depth $\delta =\sqrt{2\nu /\omega }$; the spatial roots are $\pm (1+i)/\delta$. Decay at infinity selectsIts real part is the solution of Stokes' second problem: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.

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.