Head-loss correction to hydraulic control ($E+H+{\int }^x\gamma (s)ds=\mathrm{constant}$)

For a prescribed positive head-loss rate $\gamma (x)$, the steady energy balance is $(E+H{)}_x=-\gamma$. The accumulated loss is added to the bed elevation when applying hydraulic control compatibility: $H_x+\gamma =h_c{b}_x/b$. A local loss law alone does not determine the accumulated loss between a remote inlet and the control; that integral must also be specified.