= Solution
Since $U_2=1/2$, the speed enhancement is
$$
\boxed{\frac{\overline U_2}{U_2}=\frac{\sinh^2d+d^2}{\sinh^2d-d^2}=1+\frac{2d^2}{\sinh^2d-d^2}>1\quad(d>0).}
$$
The strict inequality follows from $\sinh d>d$. It tends to one as the wall recedes, while $\sinh^2d-d^2\sim d^4/3$ gives $\overline U_2\sim3/d^2$ in a narrow gap. The wall enhances the mean pumping generated by transverse boundary motion. This small-amplitude limit requires $\epsilon\ll d$; letting the wall collide with a wave crest falls outside the expansion.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-334-wall-enhancement.png]
{title=Wall enhancement of transverse-sheet swimming}
{description=The leading swimming-speed ratio is greater than one for every finite gap and approaches one for a distant wall. The narrow-gap divergence is used only while the wave amplitude remains much smaller than the gap.}
Back to article page
