Near-field energy imbalance in a spherical acoustic wave ($e_K-e_P=\frac{{\rho }_0}{2}\left(\frac{F^2}{r^4}-\frac{2F{F}^{\prime }}{r^3}\right)$)

For outgoing velocity potential $\varphi =F(r-ct)/r$, the radial velocity is $F^{\prime }/r-F/r^2$ and the pressure perturbation is ${\rho }_{0}c{F}^{\prime }/r$. The acoustic energy density separates into kinetic density ${\rho }_0(F^{\prime }/r-F/r^2{)}^2/2$ and compressional density ${\rho }_0(F^{\prime }/r{)}^2/2$. They are generally unequal near the source but agree at leading far-field order. The additional $r^{-2}$ velocity term is the spherical near field.