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

The dimensionless equation is $H^3{H}^{\prime \prime \prime }=1-H^3$. At a crest, $H\gg 1$, soThe imposed far-field flux is negligible there; the dominant physical balance is between local gravity-driven drainage and the capillary-pressure gradient.

At a trough, $H\ll 1$, soLocal gravity-driven flux is negligible; the capillary-pressure gradient drives the fixed flux through the trough against viscous resistance. These alternating balances generate the strongly nonlinear capillary waves.