# Hard-sphere gases and the wave equation

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.

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You might be interested in the Eulerian limit for lattice (Boltzmann) gases. – Steve Huntsman Nov 9 '11 at 3:14

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.

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I don't think that is really appropriate for this site, but whatever. Right now I cannot find a single written discussion encompassing the whole derivation, but there are basically three steps: (1) derivation of the ideal gas law, (2) derivation of the equation for an ideal gas in an adiabatic process, and lastly (3) modeling longitudinal sound waves as the limiting process of a series of springs which expand and contract according to the adiabatic process.

(1) If you start by considering a gas as a collection of discrete particles, then set up the system using Hamiltonian mechanics, and assuming ergodicity, then we can use the equipartition theorem to derive the ideal gas law. You can find this derivation in any statistical mechanics book.

(2) To derive the equation for an ideal gas undergoing an adiabatic process, one considers the rotational and translational degrees of freedom of the gas molecule (vibrational modes are ignored). Notice that, in this derivation, we DO NOT assume that the gas is made up of spheres; the molecule structure is important. You can find this derivation in any thermodynamics book.

(3) Now we think of sound waves as a series of springs with a variable spring constant given by the adiabatic expansion and contraction of the air. You can find this derivation in a number of places. When I use Google, I find a number of sources.

The primary resources I used for the above were "Thermal Physics" (Kittel, Kroemer), and "Waves" (Frank Crawford).

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I find the first sentence puzzling. Isn't one of the points of Mathoverflow to act as a resource for researchers who want/need something outside their specialization and think that specialists in other areas would know? – Yemon Choi Nov 9 '11 at 2:06
Sure, I guess I consider this a problem purely of physics. – Christopher A. Wong Nov 9 '11 at 2:17
Does "assuming ergodicity" indicate that the best available mathematical account remains heuristic and could admit a more rigorous treatment? – David Feldman Nov 10 '11 at 6:28
I'm not exactly an expert on this issue, but to my knowledge, the assumption that the trajectories of particles are ergodic is an axiom. But all this is developed with full mathematical rigor. So it's not entirely heuristic. – Christopher A. Wong Nov 10 '11 at 7:10

For a very clear explanation of the physics, though not much mathematical rigor, see Volume 1 of The Feynman Lectures on Physics, Lectures 39 and 47. In terms of Christopher Wong's breakdown of the problem, Lecture 39 has the ideal gas law and adiabatic expansion, and lecture 47 has the wave equation.

You might also try physics.stackexchange.com, if you don't like the answers you got here.

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