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Write $R=k_R\sqrt{A}$, $\bar{h}=k_H\sqrt{A}$, where the given geometry givesTake a Chézy drag law $\tau =\rho c_{d}u|u|$ and restrict to downstream flow $u>0$. Choose an area scale $A_0$, length scale $L$, and speed scale $U_0$ satisfyingThis makes gravitational driving and drag equal in the reference uniform steady flow. Dividing the printed momentum equation by $gS$ givesMass conservation retains $A_t+(Au{)}_x=0$. In this notation $F$ is the square of the velocity-to-gravity-wave-speed ratio based on the chosen depth scale; authors sometimes instead denote its square root by Froude number. If the actual triangular pressure head from the preceding solution is used, replace $k_H$ by $2k_H$ in both parameter definitions. The displayed dimensionless form remains the same, with consistently redefined parameters.