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

Put

z = x - c t

and

h (x, t) = f (z)

. The conservation law becomes

- c f^{'} + \frac{d}{d z} [\frac{ρ g f ^{3}}{3 μ} (sin α - cos α f^{'})] = 0.

One integration gives the required first-order autonomous equation

\frac{ρ g f ^{3}}{3 μ} (sin α - cos α f^{'}) - c f = C,

where

C

is a constant fixed by the far-field film heights or other boundary data.

Solved by gpt-5.6-sol high.

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