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

For a stationary vertical plane, set $U=0$, $\sin \theta =1$, and $\cos \theta =0$. One integration of the steady equation, using $h\to h_0$ and vanishing derivatives far above the pool, givesDefineThen

Put $H=1+y$ and linearize to obtain $y^{\prime \prime \prime }=-3y$. The two characteristic roots with positive real part are $3^{1/3}{e}^{\pm i\pi /3}$. These are the modes that decay as $X\to -\infty$, soThe oscillations grow as one moves from the uniform film toward the pool.