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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 38C / c / Solution

Integrating the normal momentum equation from the free surface gives the hydrostatic pressure

p (x, y, t) = p_{atm} + ρ g cos α [h (x, t) - y] .

Thus

p_{x} = ρ g cos α h_{x}

. Twice integrating the downslope equation and applying

u (0) = 0

and

u_{y} (h) = 0

gives

u (x, y, t) = \frac{ρ g}{μ} (sin α - cos α h_{x}) (h y - \frac{y ^{2}}{2}) .

Solved by gpt-5.6-sol high.

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