= Solution
For a small longitudinal disturbance, let the <equilibrium> pressure and <mass density> be $p_0,\rho_0$, their perturbations be $p',\rho'$, and the fluid <velocity> be $v$. Linearizing <conservation of mass> and the inviscid momentum equation in one dimension gives
$$
\partial_t\rho'=-\rho_0\partial_xv,\qquad
\rho_0\partial_tv=-\partial_xp'.
$$
When compression is adiabatic, the <equation of state> gives $p'=c_s^2\rho'$ with $c_s^2=(\partial p/\partial\rho)_s$. Differentiating and eliminating $v$ yields
$$
\partial_t^2\rho'=c_s^2\partial_x^2\rho',\qquad
\boxed{c_s=\sqrt{\gamma p_0/\rho_0}=\sqrt{\gamma RT/M_{\rm mol}}}
$$
for an <ideal gas> with heat-capacity ratio $\gamma=C_p/C_v$ and molar <mass> $M_{\rm mol}$. Compression raises <temperature> as well as density, explaining the factor $\gamma$ compared with the isothermal value $\sqrt{p_0/\rho_0}$.
A rightgoing plane <sound wave> has pressure and particle <velocity> in phase, $p'=\rho_0c_sv$, and mean <intensity> $\langle p'v\rangle=\widehat p^{\,2}/(2\rho_0c_s)=\rho_0c_s\widehat v^{\,2}/2$ for peak amplitudes. Particle <displacement> is a quarter-cycle out of phase with particle <velocity>: particles oscillate locally while the compression pattern transports <energy>. This continuum linear description requires <wavelength> much larger than the <mean free path>, small pressure perturbations and negligible viscous attenuation. Adiabatic propagation requires heat exchange across the compressed region to be slow on the oscillation time; strong thermal coupling would instead favour an isothermal response.
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