= Drying front in a rainfall-driven Manning kinematic wave
{title2=$h_t+(h^{5/3})_s=E(t),\quad h\geq0$}
Let $H(t)=\int_0^tE(r)\,dr$, increasing until $t^*$ and decreasing afterwards. For dry initial data and zero upstream depth, a <characteristic> emitted at time $\tau$ has $h=H(t)-H(\tau)$ and position $s=\frac53\int_\tau^t[H(r)-H(\tau)]^{2/3}\,dr$, for as long as its depth stays nonnegative. The height follows by integrating the source along a <characteristic>, and the position follows from the <derivative> of the Manning flux.
When $E<0$ and $H(t)>0$, define the earlier emission time $\tau_d(t)\leq t^*$ by $H(\tau_d)=H(t)$. The front where this water is just exhausted has
$$
s_d(t)=\frac53\int_{\tau_d(t)}^t[H(r)-H(t)]^{2/3}\,dr.
$$
It moves downstream because $s_d'=-(10/9)E(t)\int_{\tau_d}^t[H(r)-H(t)]^{-1/3}\,dr>0$. Behind it depth is zero; ahead, unexhausted <characteristic> labels determine the wet profile. This construction presumes removal limited by water availability: continuing a negative source unchanged at zero depth would violate nonnegativity. If $H$ reaches zero, the remaining initially generated water has also vanished.
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