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An equivalent depth is the shallow-water depth whose gravity-wave speed matches one vertical mode of a continuously stratified fluid. Each vertical mode then obeys its own horizontal shallow-water wave equations.

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.