Power-law river kinematic wave ($A_t+[c{A}^{m+1}/(m+1){]}_s=M$)

With constant lateral supply $M$, a discharge law of the displayed form gives equilibrium area $[(m+1)(Q_{\mathrm{in}}+Ms)/c{]}^{1/(m+1)}$ and characteristic speed $c{A}^m$. Under a Chezy law, a wetted-perimeter relation $P_b\propto A^q$ gives $m=(1-q)/2$. Small area perturbations are transported downstream with amplitude inversely proportional to this characteristic speed; finite compressive disturbances can still form shocks because the flux is convex.