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WriteConservation of the two-dimensional volume $V=\int hdx$ suggestsSubstitution into the scalar conservation law $h_t+(C{h}^3{)}_x=0$ givesThe integration constant is zero at the dry rear, soThusThe jump from $h_N$ to the dry region is a shock. Its Rankine-Hugoniot condition gives ${\dot{x}}_N=C{h}_N^2$, which is also satisfied by the similarity profile.

Volume conservation determines the front:Therefore the finite-volume gravity-driven thin-film current hasand its head height is