 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 obeyThe fraction is the frequency-dependent acoustic added mass of the enclosed gas.