Travel coordinate for a conservative transport perturbation ($a(s,t)=\varphi (\xi (s)-t)/V(s),\xi (s)={\int }_0^{s}V(r{)}^{-1}dr$)

For $a_t+(V(s)a{)}_s=0$ with $V>0$, the flux perturbation $b=Va$ solves $b_t+V{b}_s=0$. The travel coordinate transforms it into unit-speed transport. Thus the flux perturbation is constant on characteristics while area amplitude changes as $1/V$. In a supplied power-law river, $A_{\mathrm{eq}}^{\prime }=M/V$, giving travel time $[A_{\mathrm{eq}}(l)-A_{\mathrm{eq}}(0)]/M$. At a zero-speed head, this time can remain integrable even though the amplitude formula is not uniform.