Why is the speed of sound wave in a gas always lesser than the r.m.s. speed at the same temperature? 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.
 A: 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
