In an ideal gas, the speed of sound $v_s$ is related to the r.m.s. molecular speed $v_m$ by
$$\frac{v_s}{v_m}=\sqrt{\frac{\gamma}{3}} \qquad ,$$
where $\gamma=C_p/C_v=7/5$ for a diatomic gas. I understand how to prove this relation from first principles. However, it seems mysterious to me that it pops out like this in the end.
I suppose it's inevitable for dimensional reasons that there will be some relation of this form, since if we want to describe a sample of an ideal gas, a sufficient set of unitful parameters is $m$ and $kT$, and there is a unique way of combining these to give units of velocity. (In a solid or liquid, we have other parameters such as the Young's modulus.) However, this type of dimensional argument doesn't prove that the ratio of the speeds is of order unity.
Is there any straightforward physical plausibility argument for the fact that this ratio of speeds is constant, and for the fact that the ratio is of order unity? I guess it's implausible to have $v_m \gg v_s$, since it seems like then a sound wave would sort of get scrambled and lose its identity because of molecular motion, and this scrambling would take place in much less than a period.
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