= Solution
Put $c=\sqrt{g'h}$. The three real eigenvalues are
$$
\boxed{\lambda_0=u,\qquad\lambda_\pm=u\pm c},
$$
so the system is strictly hyperbolic for $g'h>0$. Along $dx/dt=u$,
$$
\boxed{\frac{D_0g'}{Dt}=-g'\frac{w_e}{h}}.
$$
Left eigenvectors for $\lambda_\pm$ are $(\pm h/(2c),\pm c/h,1)$, hence
$$
\boxed{
D_\pm u\pm\frac chD_\pm h\pm\frac h{2c}D_\pm g'
=\pm\frac ch\left(\frac{w_e}{2}-w_d\right)-u\frac{w_e}{h}},
$$
where $D_\pm=\partial_t+(u\pm c)\partial_x$.
When $w_e,w_d\to0$, reduced gravity is materially conserved. If it is initially uniform, it remains constant and the other two relations integrate to the standard <Riemann invariants>
$$
\boxed{D_\pm(u\pm2\sqrt{g'h})=0}.
$$
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