= Spherically symmetric vibration of a gas-filled elastic shell
Let a thin spherical shell have equilibrium radius $a_0$, mass $m$ per unit area, and radial restoring stiffness $\kappa$ per unit area. If it encloses gas of density $\rho_0$ and sound speed $c_0$, its spherically symmetric normal-mode frequencies obey
$$
\boxed{\theta^2\left(1+\frac\alpha{\theta\cot\theta-1}\right)
=\frac{\kappa a_0^2}{mc_0^2},
\qquad
\theta=\frac{\omega a_0}{c_0},
\quad
\alpha=\frac{\rho_0a_0}{m}.}
$$
The fraction is the frequency-dependent acoustic added mass of the enclosed gas.
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