 Solution

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.