TITLE: Hard-sphere gases and the wave equation
QUESTION [5 upvotes]: I'm trying to bridge a basic gap in my own education:
Where can I find a written discussion concerning the derivation of the wave 
equation (for the propagation of sound, say), assuming nothing (much?) more than
having a gas consisting of colliding hard spheres?
NB  I plead total ignorance concerning whether such a derivation even
exists, so I may need help here just to ask the right question.

REPLY [5 votes]: Even if C. Wong's answer contains good advices, I wish to add my stone to the wall.
I doubt that you can find a simple mathematic explanation that uses only hard sphere model. Point 2) in Wong's answer is fundamental: the sound speed does depend on the number $D$ of freedom degrees of the molecules, through the adiabatic constant $\gamma=c_p/c_v$. For a monoatomic gas (Hydrogen, Argon, ...) $\gamma=\frac53$. For a diatomic gas (Oxygen, Nitrogen, ..., air in first approximation) $\gamma=\frac75$. More generally $\gamma=1+\frac2D$.
The second fundamental point is that in an ideal gas, the sound speed is proportional to the square root of the temperature. The latter quantifies the standard deviation of the molecule velocities around their mean velocity. Thus it seems difficult to do anything meaningfull without digging into the Boltzman equation. Mind that when $D\neq3$ (there are rotational degrees of freedom), the Boltzmann model is quite complicated, and not that much realistic.