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The speed cannot increase unboundedly of course, so what happens?

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The approximation on the basis of which your statment was derived breaks down. The statement therefore becomes invalid. –  Arnold Neumaier May 16 '12 at 14:53
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@ArnoldNeumaier: You could say that about literally anything beyond low energy, low speed, non-quantum, non-relativistic physics. –  Colin K May 16 '12 at 15:20
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up vote 9 down vote accepted

It's a bit misleading to simply say the speed of sound is proportional to $\sqrt{T}$ because life is a bit more complicated than that. You've probably seen http://en.wikipedia.org/wiki/Speed_of_sound and this does indeed say in the introduction that the speed of sound is a function of the square root of the absolute temperature. However if you read on you'll find:

$$c = \sqrt{\frac{P}{\rho}}$$

but for an ideal gas the ratio P/$\rho$ is roughly proportional to temperature hence you get $c \propto \sqrt{T}$.

Imagine doing an experiment in the lab at atmospheric pressure where you raise the temperature and see what effect it has on the speed of sound. In this experiment the pressure, $P$, is constant at one atmosphere, so as you increase the temperature the density falls and the speed of sound does indeed increase. But because the density is falling the mean free path of the air molecules increases, and at some temperature the mean free path becomes comparable with the wavelength of sound. When this happens the air will no longer conduct sound so the speed of sound ceases to be physically meaningful.

You could do a different experiment where you put a known amount of gas into a container of constant volume and then increase the temperature. In this experiment the density is constant, so as you increase the temperature the pressure increases and the speed of sound increases. In this experiment the mean free path is roughly constant so you don't run into the problem with the first experiment. However as the temperature rises the gas molecules will dissociate and then ionise to form a plasma. The speed of sound is then given by http://en.wikipedia.org/wiki/Speed_of_sound#Speed_in_plasma. You can keep increasing the temperature and the speed will indeed carry on increasing until it runs into a relativistic limit.

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