= Solution
For $\mathbf u=u(y,t)\mathbf e_x$, <incompressibility> is automatic and the nonlinear advection term is zero. With no <pressure> gradient the streamwise momentum equation is $\boxed{\rho u_t=\mu u_{yy}}$. Seek its periodic response as $u=\operatorname{Re}[Uf(y)e^{i\Omega t}]$. The amplitude satisfies $f''=i\rho\Omega f/\mu$ with $f(\pm d)=1$. Since $[k(1+i)]^2=i\rho\Omega/\mu$ for $k=\sqrt{\rho\Omega/(2\mu)}$, even symmetry gives
$$
\boxed{f(y)=\frac{\cosh[k(1+i)y]}{\cosh[k(1+i)d]}.}
$$
Odd components vanish because the two boundary amplitudes are equal. This is the unique periodic solution. Without a specified initial profile, decaying zero-wall diffusion transients may be added, so the formula should not be read as a claim that every transient is absent.
As $kd\to\infty$, the wall motions penetrate only a distance $k^{-1}=\sqrt{2\mu/(\rho\Omega)}$. Near either wall, with inward distance $s=d-|y|$, the leading <velocity> is
$$
\boxed{u\sim Ue^{-ks}\cos(\Omega t-ks).}
$$
The two oscillatory viscous layers are separated by an almost motionless core; the phase lag increases inward. This is <in-phase oscillating parallel-plate flow>.
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