Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 333 / 4 / a

SubstituteThe two horizontal momentum equations separate immediately because the same factor $\chi$ multiplies every term. Hydrostatic balance givesThe buoyancy equation then givesDifferentiating this expression and using three-dimensional incompressibility together with the shallow-water mass equation yieldsThus each vertical eigenfunction supplies an equivalent depth or wave speed $c$ to an equatorial shallow-water system.