Small oscillation of a planar gas bubble in an annular liquid
= Small oscillation of a planar gas bubble in an annular liquid
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
$$
\omega^2=\frac{2p_\infty}{\rho a_0^2\log(b_0/a_0)}.
$$