= Solution
For inviscid barotropic flow, conservation gives $\rho_t+\nabla\cdot(\rho u)=0$ and $\rho(u_t+u\cdot\nabla u)=-\nabla p$. Linearizing about uniform rest $\rho_0,p_0$ gives
$$
\rho'_t+\rho_0\nabla\cdot u=0,\qquad
\rho_0u_t=-\nabla p',\qquad
p'=c_0^2\rho',\qquad
\boxed{c_0^2=(dp/d\rho)_{\rho_0}},
$$
with the <derivative> adiabatic for ordinary sound.
Taking curl gives $\partial_t(\nabla\times u)=0$. Therefore the acoustic velocity is irrotational if the initial <vorticity> is zero. Uniform rest as the background does not alone forbid a superposed stationary vortical perturbation: for example $u=\varepsilon(\sin y,0,0)$, $\rho'=p'=0$ solves the linearized equations but has nonzero curl. Thus the printed irrotational conclusion requires the usual restriction to the longitudinal acoustic mode. In that mode set $u=\nabla\phi$ and absorb a time-only gauge into $\phi$, so $p'=-\rho_0\phi_t$. <Continuity> gives
$$
\boxed{\phi_{tt}-c_0^2\Delta\phi=0}.
$$
This separates the sound wave from the <stationary vorticity mode in linear acoustics>.
Define the quadratic <acoustic energy density> and <acoustic energy flux> by
$$
e=\frac{\rho_0}2|u|^2+\frac{(p')^2}{2\rho_0c_0^2},\qquad
j=p'u.
$$
Multiplying momentum by $u$ and the pressure form of <continuity> by $p'/(\rho_0c_0^2)$ gives
$$
\boxed{\partial_t e+\nabla\cdot j=0}.
$$
For any progressive <plane wave>, not necessarily sinusoidal, let $\phi=F(\xi)$ with $\xi=\widehat k\cdot x-c_0t$. Then $u=F'(\xi)\widehat k$ and $p'=\rho_0c_0F'(\xi)$. Its kinetic and compressional <energy> densities both equal $\rho_0(F')^2/2$, so
$$
\boxed{j=c_0\widehat k\,e}.
$$
The <energy> is transported at the <sound speed> in the propagation direction. A superposition of oppositely traveling waves can form a standing wave and need not have this pointwise equality; the statement concerns a single progressive wave.
For a spherical outgoing wave, $\phi=F(r-c_0t)/r$. Its radial velocity and pressure are
$$
u_r=\frac{F'}r-\frac F{r^2},\qquad p'=\rho_0c_0\frac{F'}r.
$$
Therefore
$$
\boxed{e_K=\frac{\rho_0}2\left(\frac{F'}r-\frac F{r^2}\right)^2,\qquad
e_P=\frac{\rho_0}2\left(\frac{F'}r\right)^2}.
$$
They are generally unequal because of the additional near-field velocity term; their difference is $\rho_0[F^2/r^4-2FF'/r^3]/2$. They agree only to leading far-field order, or for special profile values. This is the <near-field energy imbalance in a spherical acoustic wave>.
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