### Oscillatory boundary layer

This was addressed by Lamb, and G.K. Batchelor, and many others. For shallow water wave, linear theory predicts a uniform oscillatory movement of water particles along depth. Assuming a impermeable plane ocean bottom, the boundary layer behavior is similar to that in the fluid with oscillating bottom ($y=0$) and rest at infinity ($y=\infty$). Solution of a 2D case shows a wave-like behavior for horizontal velocity varying in vertical direction within boundary layer, with corresponding wave number $\kappa=\sqrt{\omega/2\nu}$, where $\omega$ is frequency of outer flow, and $\nu$ is kinematic viscosity. Solution also shows a exponential decay of this wave’s amplitude as $y$ increases. Dimension argument gives the effective range of viscosity at bottom for this kind of flow, for common ocean wave period, as of order of 1cm.

Horizontal velocity agains time

Horizontal velocity profile within quarter of outer flow period