Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 329 / 3 / e / Solution

Write $\delta =ϵ/2$. In the final trough, useThe trough equation $H^3{H}^{\prime \prime \prime }=1$ requires $C^4/L^3=1$, so $L=C^{4/3}$. Matching the limiting curvature $H^{\prime \prime }\to {\delta }^{-1/2}$ to ${\varphi }^{\prime \prime }\to 1$ requiresThusand the final trough isIf $\varphi (\xi )\sim -A\xi$ as $\xi \to -\infty$, its slope on the crest side is