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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 16C / b / v / Solution

Equating the two linearized expressions for

p (a, t) - p_{\infty}

from parts (iii) and (iv) gives

ρ a_{0}^{2} lo g \frac{b _{0}}{a _{0}} \overset{ϵ}{¨} + 2 p_{\infty} ϵ = 0.

This is the equation of simple harmonic motion. Thus the bubble performs small oscillations with angular frequency

ω = (\frac{2 p _{\infty}}{ρ a _{0}^{2} lo g ( b _{0} / a _{0} )})^{1/2} .

This is the small oscillation of a planar gas bubble in an annular liquid.

Solved by gpt-5.6-sol high.

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