Are sound waves adiabatic or isothermal?

Question

I am doing a presentation on sound waves and I need to know if they are adiabatic or isothermal. I know that they can generate heat, but is the amount of heat created so small that it can still be considered adiabatic?

Answer 1

Sound waves are approximately adiabatic, and the sound speed is determined by the adiabatic compressibility. The reason is that long wave length disturbances are approximately solutions to the Euler equation (which conserves entropy), and the Navier-Stokes terms (which generate entropy) are small corrections. Sound is not isothermal, the pressure disturbances induce temperature oscillations.

The amount of heat generated is proportional to the amplitude squared, the frequency squared, and the dissipative coefficients (shear viscosity, bulk viscosity, and thermal conductivity). The sound absorption coefficient (the inverse sound absorption length) is $$ \gamma = (\langle\dot E\rangle/\langle E\rangle)/(2c) $$ with $$ \gamma = \frac{\omega^2}{2\rho c^3} \left[\frac{4}{3}\eta+\zeta + \kappa\left(\frac{1}{c_v}-\frac{1}{c_p}\right) \right] $$ where $\eta$ is shear viscosity, $\zeta$ bulk visosity, and $\kappa$ thermal conductivity.

Under typical conditions the sound absorption length is quite long, and not that much heat is produced. A more efficient mechanism for producing heat is sound absorption by a solid body. This is because the surface of the solid is isothermal, so large temperature gradients can occur at the surface boundary layer.

Answer 2

You can, under typical situations assume that the transmission of sound through air is isentropic, and therefore isothermal and adiabatic. But that's just an approximation. The heat generated and transferred outside any chosen control volume is minutely small, but nevertheless there.

But considering very high sound pressures this assumption may not lead to predictions that are as accurate at low sound pressure. Examples to consider might be the sound canons that have been used by cruise ships to thwart pirates!