Pressure-forced standing acoustic wave in a spherical annulus

For $R<r<2R$, a rigid inner sphere and harmonic pressureproduce a standing radial field. With $k=\omega /c_0$ and $\alpha =kR$,The pressure and radial velocity are in temporal quadrature, so the period-averaged acoustic intensity vanishes.

The ideal pressure-forced response becomes resonant whenequivalently $\tan \alpha =-\alpha$. Damping regularizes the divergent linear response and introduces a nonzero phase lag and mean supplied power.