Outgoing acoustic field of a pulsating sphere

For $R(t)=a+\mathrm{Re}(ϵe^{i\omega t})$, put $k=\omega /c_0$. To first order in $ϵ/a$, the exterior outgoing velocity-potential amplitude isIt satisfies ${\hat{u}}_r(a)={\hat{\varphi }}^{\prime }(a)=i\omega ϵ$.

The mean power radiated by the pulsating sphere is

At $r=a$,For $\omega a/c_0\gg 1$, kinetic and compressional energies are equal and the field is locally radiative. For $\omega a/c_0\ll 1$, reactive kinetic energy dominates and the mean radiated flux is small.