Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 3 / c / Solution

With a long uniform corridor, constant depth, and negligible entrainment, the pressure head $g^{\prime }h$ driving the gravity current does not change as the nose advances. There is no growing geometric length in the local front balance, so dimensional analysis gives the constant velocity $U_f\propto \sqrt{g^{\prime }h}$ after the short release transient.