The dimensionless equation is $H^3H'''=1-H^3$. At a crest, $H\gg1$, so
$$
H'''\sim-1.
$$
The 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, $H\ll1$, so
$$
H^3H'''\sim1.
$$
Local 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 wave>[capillary waves].

Solution

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

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A capillary wave is a free-surface wave for which surface tension supplies an essential restoring stress.


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