= Solution
For a plane acoustic wave propagating along $\widehat{\mathbf k}$, the linear momentum equation and dispersion $\omega=c_0k$ give $\widetilde p=\rho_0c_0u_\parallel$ and $\mathbf u=u_\parallel\widehat{\mathbf k}$. Therefore the instantaneous acoustic intensity is
$$
\boxed{\widetilde p\,\mathbf u=\rho_0c_0|\mathbf u|^2\widehat{\mathbf k}.}
$$
For harmonic complex amplitudes the time average instead has the factor one half, $\langle\mathbf I\rangle=\tfrac12\rho_0c_0|\widehat{\mathbf u}|^2\widehat{\mathbf k}$.
For a radial velocity potential, set $v=r\phi$. The spherical wave equation reduces to $v_{tt}=c_0^2v_{rr}$, so
$$
\boxed{\phi(r,t)=\frac{F(t-r/c_0)+G(t+r/c_0)}r.}
$$
No incoming radiation from infinity means $G=0$. Linearizing the moving-surface condition at $r=a$ gives $\phi_r(a,t)=i\omega a\epsilon e^{i\omega t}$, with real parts understood. Write the outgoing solution as $\phi=A e^{i\omega(t-(r-a)/c_0)}/r$. Its radial derivative at $a$ is $-A(1+i\omega a/c_0)e^{i\omega t}/a^2$. Therefore the <outgoing acoustic field of a pulsating sphere> is
$$
\boxed{\phi(r,t)=\operatorname{Re}\left\{-\frac{i\omega a^3\epsilon}{1+i\omega a/c_0}
\frac{e^{i\omega(t-(r-a)/c_0)}}r\right\}.}
$$
In the far field the velocity amplitude has magnitude $(\omega/c_0)|A|/r$. Multiply its average plane-wave intensity by the spherical area to obtain the <mean acoustic power of a pulsating sphere>:
$$
\boxed{\langle P\rangle=\frac{2\pi\rho_0\omega^2}{c_0}|A|^2
=\frac{2\pi\rho_0\epsilon^2a^6\omega^4}{c_0[1+(\omega a/c_0)^2]}.}
$$
This is positive outward radiated power, evaluated to leading quadratic order in the small real displacement amplitude.
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