= Solution
Substitute
$$
u=\chi(z)U,
\qquad
v=\chi(z)V,
\qquad
\phi=\chi(z)\Phi.
$$
The two horizontal momentum equations separate immediately because the same factor $\chi$ multiplies every term. Hydrostatic balance gives
$$
\boxed{\sigma=\chi'(z)\Phi}.
$$
The buoyancy equation then gives
$$
\boxed{w=-\frac{\chi'}{N^2}\Phi_t
=\frac{c^2\chi'}{N^2}(U_x+V_y)}.
$$
Differentiating this expression and using three-dimensional incompressibility together with the shallow-water mass equation yields
$$
\boxed{\chi''+\frac{N^2}{c^2}\chi=0}.
$$
Thus each vertical eigenfunction supplies an <equivalent depth> or wave speed $c$ to an equatorial shallow-water system.
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