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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

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up vote 4 down vote accepted

Fluids are complicated systems described by non-linear differential equations that can't be reasonably treated in a full generality (certainly not analytically). Just consider the kinds of waves that propagate in the sea -- deep or shallow water, solitons, tsunami and many others (this is not to say that these are sound waves; but as an illustration of complicated wave behavior it should suffice). So, to proceed one often employs some approximation.

By far the most popular one (with many applications) rests on the linearization of the problem around an equilibrium solution where one replaces complicated non-linear equations with second-order wave equation that describes propagation of the perturbation in the system. Now, this places some consistency conditions on how big those perturbation can be so that higher-order effects can be ignored. Depending on the precise form of the equations, this might require that the temperature or some other parameter stays constant (otherwise the induced heat transport effects might destroy the linearization, for example)

Roughly, isothermal processes are slow (so that there is enough time for the transfer of the heat with the environment which keeps the temperature constant) whereas in the adiabatic case the wave propagates so fast that the environment can't catch up with it and so no heat is exchanged (but the temperature can change). I think for the most familiar types of materials where the speed of sound is quite big one uses the adiabatic approximation (certainly for the propagation of sound in the air). I guess for less standard materials (as encountered in astrophysics) you might need isothermal approximation too but it's hard to say more than this without knowing what system you have in mind precisely.

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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.

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