# Adiabatic approximation in the derivation of the speed of sound

• B
The speed of sound in a gas at temperature T is given to be ## v=\sqrt{\frac{\gamma RT}{M}}##, where ##\gamma## is the adiabatic exponent, R is the gas constant and M is the molar mass of the gas. In deriving this expression, we assumed that the compression and expansion processes were so fast that there isn't enough time for heat transfer to take place, and therefore that the processes are approximately adiabatic (Laplace's correction to Newton's formula). We therefore used ##PV^\gamma = constant##. However the relation ##PV^\gamma = constant## is valid only for a quasi-static adiabatic process, and cannot be used for very fast processes. The quasi-static assumption is made in the derivation of adiabatic process relations. So how are we allowed to use it in the speed of sound derivation? The very reason we assumed the process was adiabatic is because it is very fast in the first place. So how is it consistent to assume it is quasi-static as well?

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Mister T
Gold Member
The process has to be slow enough that it's quasi-static, but fast enough that negligible heat is transferred.

The process has to be slow enough that it's quasi-static, but fast enough that negligible heat is transferred.
How do we know that this case satisfies that condition? Is there a practical limit below which a process can be reasonably assumed to be quasistatic? I have read that we can make the quasistatic approximation for expansion and compression processes, if the boundaries are moving much slower than the speed of sound in the gas. This makes perfect sense because then there would be sufficient time for the boundary disturbance to propagate through the rest of the gas and equalize for each small part of the process. However, I don't see how I can use that here, since the very derivation is for speed of sound in the gas.

Mister T
Gold Member
How do we know that this case satisfies that condition?
It's an assumption used in the derivation. If the result of the derivation matches what we observe then that gives us confidence that the assumption is justified.

• Terry Bing
FWIW, my text derives the basic relation (##c = \sqrt{\frac{\Delta p}{\Delta \rho}}##) using mass and momentum equations, then justifies the use of the isentropic relation (##\frac{p}{\rho ^k} = const##) by referring to experimental results which "indicate that the relationship between pressure and density across a sound wave is nearly isentropic."

So, rather than being an assumption in the derivation, it is an experimental finding that is applied to the general result.

• vanhees71
It's an assumption used in the derivation. If the result of the derivation matches what we observe then that gives us confidence that the assumption is justified.
Thanks. I get that it matches the observed speed of sound. I am just curious as to why? Was there a way we could have guessed beforehand that it would work?

vanhees71