= Solution
At larger frequencies the denominator
$$
D(\alpha)=\cos\alpha+\frac{\sin\alpha}{\alpha}
$$
can vanish. Its zeros satisfy
$$
\tan\alpha=-\alpha
$$
and are the normal-mode frequencies for a pressure node at the outer sphere and zero radial velocity at the inner sphere. Near one of these <radial acoustic resonance in a spherical annulus>[radial acoustic resonances], the ideal undamped harmonic response becomes very large; exactly at resonance the linear lossless initial-value problem develops secular growth rather than a bounded periodic solution.
In a physical system, fluid viscosity, thermal losses, acoustic radiation, and damping in the elastic shell keep the amplitude finite and shift its phase. A nonzero pressure-velocity phase component then supplies positive mean power. At sufficiently large amplitude the linear acoustic approximation can also fail, leading to nonlinear frequency shifts or shock formation.
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