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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.

A small pressure disturbance propagates because compression raises the pressure, the pressure gradient accelerates nearby fluid, and the induced convergence compresses the next region. For a uniform fluid at rest, linearizing mass and momentum conservation and using the adiabatic pressure-density response givesEliminating ${\rho }^{\prime }$ and ${\mathbf{u}}^{\prime }$ gives ${\partial }_t^2{p}^{\prime }=c_s^2{\nabla }^2{p}^{\prime }$. A plane wave has ${\omega }^2=c_s^2|\mathbf{k}{|}^2$ and longitudinal velocity parallel to $\mathbf{k}$. In a perfect gas,The adiabatic response is appropriate when heat cannot diffuse appreciably out of each compressed region during a period; an isothermal response would give a different derivative. In a moving uniform medium, the acoustic characteristics travel at $U\pm c_s$ along the flow, rather than at $\pm c_s$ in the laboratory.

For a progressive wave, $p^{\prime }={\rho }_0{c}_s{u}_{\parallel }^{\prime }$; pressure, density and longitudinal velocity are in phase. The acoustic energy density is the sum of kinetic and compression energies,Sound transports energy while fluid elements execute small oscillations about their mean positions. The ideal uniform-fluid acoustic mode is nondispersive. Spherical spreading reduces pressure amplitude as $1/r$ when total power is conserved; viscosity and heat conduction attenuate waves, while sufficiently large compressions can steepen into shocks. In a stratified astrophysical gas the uniform result is local: wavelengths short compared with background scale heights justify it, whereas long waves can couple to buoyancy and acoustic-cutoff behavior.

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 ${\rho }_0,p_0$ giveswith 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 $u=\epsilon (\sin y,0,0)$, ${\rho }^{\prime }=p^{\prime }=0$ 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 $u=\nabla \varphi$ and absorb a time-only gauge into $\varphi$, so $p^{\prime }=-{\rho }_0{\varphi }_t$. Continuity givesThis separates the sound wave from the stationary vorticity mode in linear acoustics.

Define the quadratic acoustic energy density and acoustic energy flux byMultiplying momentum by $u$ and the pressure form of continuity by $p^{\prime }/({\rho }_0{c}_0^2)$ gives

For any progressive plane wave, not necessarily sinusoidal, let $\varphi =F(\xi )$ with $\xi =\hat{k}\cdot x-c_{0}t$. Then $u=F^{\prime }(\xi )\hat{k}$ and $p^{\prime }={\rho }_0{c}_0{F}^{\prime }(\xi )$. Its kinetic and compressional energy densities both equal ${\rho }_0(F^{\prime }{)}^2/2$, soThe 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, $\varphi =F(r-c_{0}t)/r$. Its radial velocity and pressure areThereforeThey are generally unequal because of the additional near-field velocity term; their difference is ${\rho }_0[F^2/r^4-2F{F}^{\prime }/r^3]/2$. 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.

Write the density and pressure as $\rho ={\rho }_0+{\rho }^{\prime }$ and $p=p_0+p^{\prime }$, and let $\mathbf{u}$ be the small fluid velocity. The linear homentropic acoustic equations arewhere the adiabatic sound speed isIf the flow is irrotational, write $\mathbf{u}=\nabla \varphi$. The momentum equation integrates spatially toa purely time-dependent integration term having been absorbed into $\varphi$. Substitution into continuity gives the wave equationso the wave speed is $c_0$.

To obtain energy conservation, dot the momentum equation with $\mathbf{u}$ and use the product rule:The continuity and pressure-density relations giveTherefore the acoustic energy conservation law iswith acoustic energy density and intensity

Consider a uniform tube $0\le x\le L$ with rigid ends, constant undisturbed mass density ${\rho }_0$ and speed of sound $c$. Neglect losses and transverse structure. A free normal mode hasand a consistent choice of sound pressure, acoustic particle velocity and acoustic displacement isHere $P>0$ is a peak sound pressure. Direct differentiation verifies ${\rho }_0{\partial }_{t}u=-{\partial }_x{p}^{\prime }$ and ${\partial }_t{p}^{\prime }=-{\rho }_0{c}^2{\partial }_{x}u$, the one-dimensional linear homentropic acoustic equations. The rigid ends enforce $u=0$ there. A driven experiment only approximates this free-mode geometry; the formulas do not predict a finite resonant amplitude for an undamped driven tube.

The maxima of sound pressure amplitude occur at $kx=j\pi$, where acoustic particle velocity vanishes. The maxima of acoustic particle velocity amplitude instead occur at $kx=(j+1/2)\pi$, where sound pressure vanishes. The two families are displaced by $\lambda /4$, but each family repeats after $\lambda /2$. At a position where both are nonzero, the time dependence is also a quarter cycle apart. Using consecutive maxima of sound pressure amplitude therefore givesThe relation also applies to consecutive minima of sound pressure amplitude, provided the same family is used throughout. Mixing maxima and minima would halve the spacing and give an erroneous speed of sound.

The local acoustic energy density is $p^{\prime 2}/(2{\rho }_0{c}^2)+{\rho }_0{u}^2/2$. Averaging over one wave period gives $P^2/(4{\rho }_0{c}^2)$, independent of $x$. Likewise $⟨p^{\prime }u⟩=0$: this ideal standing wave has no mean longitudinal acoustic energy flux, although its pressure and fluid velocity oscillate. Energy transport into a real lossy tube is a different question from the free-mode pattern.