= Near-field energy imbalance in a spherical acoustic wave
{title2=$e_K-e_P=\frac{\rho_0}2\left(\frac{F^2}{r^4}-\frac{2FF'}{r^3}\right)$}
For outgoing <velocity potential> $\phi=F(r-ct)/r$, the radial velocity is $F'/r-F/r^2$ and the pressure perturbation is $\rho_0cF'/r$. The <acoustic energy density> separates into kinetic density $\rho_0(F'/r-F/r^2)^2/2$ and compressional density $\rho_0(F'/r)^2/2$. 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.
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