= Solution
Take a unit-width <control volume> spanning the stationary <hydraulic jump>. <Mass conservation> makes the discharge $q=hu$ constant. The horizontal <momentum flux> is $\rho h u^2$, and the integrated <hydrostatic pressure> force on a vertical cross-section is $\int_0^h\rho g(h-z)dz=\rho gh^2/2$. <Momentum conservation> therefore gives
$$
\boxed{[hu]_-^+=[hu^2+gh^2/2]_-^+=0}.
$$
These are the <Rankine-Hugoniot conditions> for the <shallow water equations> in the stationary <shock frame>. With depths $h_1\ne h_2$ on the two sides, eliminating the <velocity> gives
$$
q^2=\frac g2h_1h_2(h_1+h_2).
$$
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