# What's the difference between different speeds of sound?

In astrophysics, I often come across the speed of sound. I understand that, broadly, it represents the speed at which perturbations travel through a medium. But there's more than one speed of sound. The most common seem to be isothermal and adiabatic, which are defined as $c_s^2=(dp/d\rho)_T$ and $c_s^2=(dp/d\rho)_S$, respectively.

My question is, when do these different speeds apply? When do perturbations travel at the adiabatic sound speed, and when the isothermal? Are there any other useful sound speeds?

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A related question I asked –  Bernhard Heijstek Jul 19 '11 at 15:58
I saw that. Even the first answer doesn't specify which sound speed is meant. I believe part of the confusion is because, for an ideal gas, the isothermal and adiabatic sound speeds are the same? –  Warrick Jul 20 '11 at 8:33

The isothermal sound speed applies when the cooling timescale is very fast compared to the propagation speed of the wave. Often in astrophysics the temperature of the gas is set by thermal balance between heating sources and radiative cooling and the timescale to get into balance is short compared to the sound wave travel timescale. One example is spiral density waves traveling in a galactic disk where an effective sound speed would be that of the $10^4$K gas and about that of the velocity dispersion of the molecular clouds.