Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 40A / c / Solution

At larger frequencies the denominatorcan vanish. Its zeros satisfyand 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 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.

Solved by gpt-5.6-sol high.