= Pressure-forced standing acoustic wave in a spherical annulus
For $R<r<2R$, a rigid inner sphere and harmonic pressure
$$
p'(2R,t)=\varepsilon p_0\cos\omega t
$$
produce a standing radial field. With $k=\omega/c_0$ and $\alpha=kR$,
$$
\phi(r,t)
=-\frac{2R\varepsilon p_0}{\rho_0\omega r}
\frac{\cos k(r-R)+(kR)^{-1}\sin k(r-R)}
{\cos\alpha+\alpha^{-1}\sin\alpha}
\sin\omega t.
$$
The pressure and radial velocity are in temporal quadrature, so the period-averaged <acoustic energy conservation>[acoustic intensity] vanishes.
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