= Solution
At every radius, the pressure perturbation has the form
$$
\widetilde p(r,t)
=-\rho_0\omega F(r)\cos\omega t,
$$
whereas the radial velocity is
$$
u_r(r,t)=F'(r)\sin\omega t.
$$
Define the average over one forcing period $T=2\pi/\omega$ by
$$
\langle I_r\rangle
=\frac1T\int_0^T
\widetilde p(r,t)u_r(r,t)\,dt.
$$
Then
$$
\langle I_r\rangle
=-\rho_0\omega F(r)F'(r)
\frac1T\int_0^T\cos\omega t\sin\omega t\,dt
=\boxed{0}.
$$
The same is true for the net flux through every concentric sphere.
Pressure and velocity are in temporal quadrature, so this is a <reactive acoustic energy flux>. The forcing establishes a standing wave in the closed, lossless annulus: compressional and kinetic energy exchange during each cycle, but no energy is transported outward on average. This is the <pressure-forced standing acoustic wave in a spherical annulus>.
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