 Solution

The physically meaningful extension requires $h\ge 0$: negative net input removes water where wet and cannot keep removing nonexistent surface water where dry. Keeping the algebraic source $E(t)<0$ unchanged in a dry zone would instead produce negative depths, so the drying-front description implicitly includes this availability condition.

Use the same $H(t)$, increasing until $t^{\ast }$ and decreasing afterwards. While $H(t)>0$, there is one earlier wetting time ${\tau }_d(t)\in [0,t^{\ast }]$ such that $H({\tau }_d(t))=H(t)$. The boundary-emitted characteristic with this label has just exhausted its water. Its position is the drying front:The integrand is nonnegative because $H(r)\ge H(t)$ between these two equal-height times. At $t=t^{\ast }$ the interval collapses and $s_d=0$; as net input becomes negative the front moves downstream. Differentiating gives $s_d^{\prime }=-(10/9)E(t){\int }_{{\tau }_d(t)}^t[H(r)-H(t){]}^{-1/3}dr>0$, where endpoint singularities are integrable away from the turning-time limit.

The wet solution ahead of the front still follows unexhausted characteristics. With $S_0(t)=\frac{5}{3}{\int }_0^{t}H(r{)}^{2/3}dr$,where $0\le \tau <{\tau }_d(t)$ is determined by the same characteristic-position integral as in (iv). If cumulative loss reaches $H(t)=0$, all remaining initial-line water has also vanished and the surface is dry thereafter while $E<0$. If the total later loss is smaller, the formula applies for all later times with $H(t)>0$. This is the drying front in a rainfall-driven Manning kinematic wave.