Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 2 / c / Solution

Put $c=\sqrt{g^{\prime }h}$. The three real eigenvalues areso the system is strictly hyperbolic for $g^{\prime }h>0$. Along $dx/dt=u$,Left eigenvectors for ${\lambda }_{\pm }$ are $(\pm h/(2c),\pm c/h,1)$, hencewhere $D_{\pm }={\partial }_t+(u\pm c){\partial }_x$.

When $w_e,w_d\to 0$, reduced gravity is materially conserved. If it is initially uniform, it remains constant and the other two relations integrate to the standard Riemann invariants