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My book says that the speed of sound wave in a gas is always lesser than the r.m.s. speed of the gas at the same temperature. I understand that speed of sound is given by:

$$v_s=\sqrt{\frac{\gamma RT}{M}}$$

and r.m.s. speed is given by:

$$v_r=\sqrt{\frac{3RT}{M}}$$

Then my book says, "since $\gamma$ is always lesser than three, $v_s$ is always lesser than $v_r$". I am unable to understand why is this so? It would be of great help if someone can explain the reason for this fact. Also, why can't $v_s$ be greater than or equal to $v_r$? Is there any intuitive reason behind this? What happens if $v_s$ is greater than $v_r$? Sound waves are just compressions and rarefactions and I don't find anything wrong in having a larger wave speed than the r.m.s. speed as still gas molecules have even larger speeds than the r.m.s. speed.

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  • $\begingroup$ I found this on the sidebar which is related to that of mine. But that doesn't seem to clarify my queries. $\endgroup$
    – user268404
    Commented Jun 25, 2020 at 5:13
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    $\begingroup$ What''s the rms speed? $\endgroup$
    – Gert
    Commented Jun 25, 2020 at 5:29
  • $\begingroup$ @Gert Root mean square speed. $\endgroup$
    – user268404
    Commented Jun 25, 2020 at 5:32
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    $\begingroup$ Isn't that a common abbreviation? Or do I need to mention it in the question? $\endgroup$
    – user268404
    Commented Jun 25, 2020 at 5:32
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    $\begingroup$ NEVER assume an acronym is known by all. It NEVER is. $\endgroup$
    – Gert
    Commented Jun 25, 2020 at 12:33

3 Answers 3

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this is what I think ..sound wave is something that travels with the use of vibrating/moving molecules in a certain substance;in gas when it is related to the question. gas molecules are molecules that move/vibrate constantly and continuously at different directions and dimensions at different speed (overall at rms).but when a sound wave is origined thereby, using these moving molecules, the wave gets transmitted. So imagine the wave is transmitted to +x direction using a molecule that is already going towards the -x direction which will cause a deacceleration for the molecule. that will eventually make its speed lower..likewise even for other directions this would remain same..not for all the molecules but as a resultant it would be so..which makes the speed of soundwave in a gas be relatively less compared to the speed/rms of gas itself at a certain temperature.. This is my opinion .may not be the exact answer..hope this helps though...also note that is am a student still learning so might be wrong as well..yet it's open to all decided to answer using what my hypothesis was regarding this😌

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  • $\begingroup$ books.google.lk/… $\endgroup$
    – Tia DS
    Commented Jun 25, 2020 at 16:30
  • $\begingroup$ The above is the URL to a web article in which I found my hypothesis is true and correct (found now) page 556 $\endgroup$
    – Tia DS
    Commented Jun 25, 2020 at 16:32
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Let's have a closer look at the heat capacity ratio, $\gamma$. It may be predicted by considering the degrees of freedom, $f$ of the ideal gas that the sound is moving through:

$$\gamma = 1 + \frac{2}{f}$$

For a monotonic gas (e.g., helium) there are three degrees of freedom for the three directions in which the molecules can translate, and $\gamma = \frac{5}{3} = 1.\overline{66}$, which is obviously less than three. In fact, this is the highest possible value for $\gamma$, which answers your question.

To convince yourself, consider a larger molecule like carbon-dioxide. It has the same three translational degrees of freedom, but also an additional three rotational degrees of freedom, and so the equation predicts $\gamma = \frac{8}{6} = 1.\overline{33}$. More complex gas molecules will have an even lower adiabatic index.

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  • $\begingroup$ I think user268404 knows that $\gamma <3$ and he would like to get an intuitive explanation why $v_s$ is less than the rms velocity $v_r$. $\endgroup$
    – hyportnex
    Commented Aug 1, 2023 at 0:48
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At a microscopic level a sound wave is a local disturbance propagated in a gas medium by particle collisions. Thus, you cannot really expect to have this disturbance propagate faster than the average speed of particles in that gas.

A possible interpretation as to why exactly the speed of sound should be less than the rms speed of particles (all this goes in line with the ideal gas kinetic theory) is that the disturbance goes locally in a single direction (the wave direction), so only one third of the local kinetic energy (one degree of freedom) of the particles may be mobilised to contribute to the disturbance propagation (hence the $1/\sqrt{3}$ factor. The $\gamma$ factor comes in as an additional complexity because the process is not isothermal.

Maybe a slighlty more interesting insight given by the microscopic perspective on wave propagation is that the mean free path of a particle should always by much smaller than the length of the disturbance (the wavelength). This, in effect, puts a bound on the maximum frequency at which a wave can propagate at a given temperature.

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