The Bessel function at its upper limit becomes
which converges to
This is highly reminiscent of the Stokes layer on an oscillating flat plate, or the skin-depth penetration of an alternating magnetic field into an electrical conductor.
On the surface , but the exponential term becomes negligible once  becomes large, the velocity profile becomes almost constant and independent of the viscosity. Thus, the flow simply oscillates as a plug profile in time according to the pressure gradient,
However, close to the walls, in a layer of thickness , the velocity adjusts rapidly to zero. Furthermore, the phase of the time oscillation varies quickly with position across the layer. The exponential decay of the higher frequencies is faster.