= Solution
For the supplied hillslope values,
$$
\boxed{u_0=\frac{0.01^{2/3}\sqrt{0.01}}{0.04}\simeq0.116\ {\rm m\,s^{-1}},\quad\epsilon=0.01,\quad F\simeq0.367,\quad\epsilon F^2\simeq1.35\times10^{-3}.}
$$
The <Froude number> is below one, but it is not extremely small numerically; the much smaller <products> actually multiplying inertia and the depth-gradient term justify their leading neglect. Positive flow then satisfies $u=h^{2/3}$ and
$$
\boxed{h_t+(h^{5/3})_s=E.}
$$
In overland flow the source is net <rainfall> input after <soil infiltration>[infiltration] into soil, with <evaporation> or other local losses included when important. Lateral inflow can also contribute. Negative net input must be limited by water availability once the surface dries.
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