= Solution
When $\Delta\ll1$, the small-argument expansion $\sinh z=z+O(z^3)$ gives
$$
f(y)=\frac{(1+i)\Delta\widehat y+O(\Delta^3)}
{(1+i)\Delta+O(\Delta^3)}
=\widehat y+O(\Delta^2).
$$
Therefore
$$
\boxed{
u(y,t)\simeq\frac{U_0y}{L_0}\cos\omega t
}.
$$
The <viscous diffusion time> $L_0^2/\nu$ is then short compared with the oscillation period. Viscosity communicates the wall motion across the entire layer almost instantaneously, producing the linear profile of quasi-steady <Couette flow>.
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