Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 1 / i / Solution

The first two equations are horizontal momentum balances: local acceleration plus the Coriolis acceleration equals the pressure-gradient force. The third is linearized mass conservation: convergence raises the free surface, represented by $\varphi$, while divergence lowers it. The constant $c$ is the long-wave gravity-wave speed.