Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 334 / 1 / g

Since $U_2=1/2$, the speed enhancement isThe strict inequality follows from $\sinh d>d$. It tends to one as the wall recedes, while ${\sinh }^{2}d-d^2\sim d^4/3$ gives ${\overline{U}}_2\sim 3/d^2$ in a narrow gap. The wall enhances the mean pumping generated by transverse boundary motion. This small-amplitude limit requires $ϵ\ll d$; letting the wall collide with a wave crest falls outside the expansion.