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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 16C / a / Solution

Write the free surface as the zero set

S (x, y, z, t) = z - η (x, y, t) = 0.

Because this is a material surface, the kinematic boundary condition is

D S / D t = 0

. Thus, on

z = η

,

0 = S_{t} + u S_{x} + v S_{y} + w S_{z} = - η_{t} - u η_{x} - v η_{y} + w .

Therefore

w = η_{t} + u η_{x} + v η_{y} = \frac{Dη}{D t}

at the free surface. This is the kinematic boundary condition for a free-surface graph.

Solved by gpt-5.6-sol high.

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