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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 16A / b / Solution

Both

sin (x - t)

and

cos (x - t)

have zero average over one period, so

⟨ u ⟩ = 0

at every fixed point.

The material derivative gives the particle acceleration

a = \frac{\partial u}{\partial t} + (u \cdot \nabla) u .

Direct differentiation yields

a_{x} = ε y sin θ + ε^{2} (1 - y^{2}) sin θ cos θ,

a_{y} = - ε cos θ + ε^{2} y cos^{2} θ .

Therefore the Eulerian time average at fixed

(x, y)

is

⟨ a ⟩ = (0, ε^{2} y /2) .

Solved by gpt-5.6-sol high.

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