Normal water elevation is reach when the water line is parallel to the bottom, the head is therefore parallel to the water line and then head loss is egual to bottom slope : $I_f = J$

With :
If : Bottom slope en m/m
J : Head loss en m/m

The head loss J is calculated with the Manning-Strickler formula : $J=\frac{U^2}{K^{2}R^{4/3}}=\frac{Q^2}{S^2K^{2}R^{4/3}}$

With :
K : the Strickler coefficient en m1/3/s

In uniform flow, one obtains the formula :

$Q=KR^{2/3}S\sqrt{I_f}$

From which, one can directly calculate the discharge $Q$, the bottom slope $I_f$ and the Strickler $K$.

For calculating normal water elevation $h_n$, one can solve $f(h_n)=Q-KR^{2/3}S\sqrt{I_f}=0$

using the Newton method : $h_{k+1} = h_k - \frac{f(h_k)}{f’(h_k)}$ avec :
$f(h_k) = Q-KR^{2/3}S\sqrt{I_f}$
$f’(h_k) = -K \sqrt{I_f}(\frac{2}{3}R’R^{-1/3}S+R^{2/3}S’)$

For calculating geometric parameters of the section, the calculator uses the discharge equation and solves the problem with the dichotomia method.