Solution

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

For $a=a_0(1+\epsilon)$, equilibrium outer radius $b_0$, and a bubble gas obeying $p_0\pi a^2=\text{constant}$, the <linearization> is
$$
\rho a_0^2\log\frac{b_0}{a_0}\,\ddot\epsilon
=-2p_\infty\epsilon.
$$
The bubble therefore undergoes <simple harmonic motion> with
$$
\omega^2=\frac{2p_\infty}{\rho a_0^2\log(b_0/a_0)}.
$$

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 Incoming links: Small oscillation of a planar gas bubble in an annular liquid