Is sound really adiabatic because it is a fast process?

Question

In many books I have consumed so far there is the statement that sound is adiabatic because heat transfer does not have nearly enough time to reach isothermal equilibrium. Doesn't this contradict meteorological processes, which are very very slow as compared to sound and yet adiabatic to a very good approximation?

Isn't it the other way around, that sound becomes isothermal at very high frequencies due to the increased temperature gradient due to the short wavelengths? I found this in another textbook and it appears more logical to me.

What is the truth now? Are so many textbooks really wrong or do I have a misinterpretation?

Answer 1

Sound propagation is indeed better modeled as an isothermal process at higher frequencies, for exactly the reason you note: With a shorter wavelength, the hotter and colder regions are closer and more readily exchange heat. The topic is discussed in Wu's "Are sound waves isothermal or adiabatic?".

More broadly, to consider whether an adiabatic assumption is reasonable without the answer depending on the circumstances or the units used, you might consider using a dimensionless number such as the Fourier number $\mathrm{Fo}=\alpha t/L^2$, which incorporates a time scale $t$, length scale $L$, and material thermal diffusivity $\alpha$ (which itself incorporates the thermal conductivity, density, and specific heat capacity).

If the Fourier number is much greater than one, enough time has passed (for that length scale and those material properties) for most of the material to be at nearly the same temperature—good justification for the isothermal idealization.

If the Fourier number is much less than one, so little time has passed that most of the material doesn't thermally "know" what's happening at the perimeter—good justification for the adiabatic idealization.