= 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 gives
$$
\frac{\partial\rho'}{\partial t}+\rho_0\nabla\cdot\mathbf u'=0,\qquad\rho_0\frac{\partial\mathbf u'}{\partial t}=-\nabla p',\qquad p'=c_s^2\rho',\qquad c_s^2=\left(\frac{\partial p}{\partial\rho}\right)_s.
$$
Eliminating $\rho'$ and $\mathbf u'$ gives $\partial_t^2p'=c_s^2\nabla^2p'$. A <plane wave> has $\omega^2=c_s^2|\mathbf k|^2$ and longitudinal <velocity> parallel to $\mathbf k$. In a <perfect gas>,
$$
\boxed{c_s=\sqrt{\gamma p_0/\rho_0}=\sqrt{\gamma(R^*/\mu)T_0}}.
$$
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, $p'=\rho_0c_su'_\parallel$; <pressure>, density and longitudinal <velocity> are in phase. The <acoustic energy density> is the sum of kinetic and compression energies,
$$
E_{\rm ac}=\frac12\rho_0|\mathbf u'|^2+\frac{p'^2}{2\rho_0c_s^2},\qquad\langle I\rangle=\frac{p_{\rm rms}^2}{\rho_0c_s}.
$$
\b[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.
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