Editing Euler equations (fluid dynamics) (section)

====Ideal gas====
The sound speed in an ideal gas depends only on its temperature:
<math display="block">a_s (T) = \sqrt {\gamma \frac T m}.</math>

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|Deduction of the form valid for ideal gases
|In an ideal gas the isoentropic transformation is described by the [[Poisson's law]]:
<math display="block">d\left(p\rho^{-\gamma}\right)_s = 0</math>
where ''γ'' is the [[heat capacity ratio]], a constant for the material. By explicitating the differentials:

<math display="block">\rho^{-\gamma} (d p)_s + \gamma p \rho^{-\gamma-1} (d \rho)_s =0</math>

and by dividing for ''ρ''<sup>−''γ''</sup> d''ρ'':

<math display="block">\left({\partial p \over \partial \rho}\right)_s = \gamma p \rho</math>

Then by substitution in the general definitions for an ideal gas the isentropic compressibility is simply proportional to the pressure:

<math display="block">K_s (p) = \gamma p </math>

and the sound speed results ('''Newton–Laplace law'''):

<math display="block">a_s (\rho,p) = \sqrt {\gamma \frac p \rho} </math>

Notably, for an ideal gas the [[ideal gas law]] holds, that in mathematical form is simply:

<math display="block">p = n T </math>

where ''n'' is the [[number density]], and ''T'' is the [[absolute temperature]], provided it is measured in ''energetic units'' (i.e. in [[joules]]) through multiplication with the [[Boltzmann constant]]. Since the mass density is proportional to the number density through the average [[molecular mass]] ''m'' of the material:

<math display="block"> \rho = m n </math>

The ideal gas law can be recast into the formula:

<math display="block"> \frac p \rho = \frac T m </math>

By substituting this ratio in the Newton–Laplace law, the expression of the sound speed into an ideal gas as function of  temperature is finally achieved.

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Since the specific enthalpy in an ideal gas is proportional to its temperature:

<math display="block">h = c_p  T = \frac {\gamma}{\gamma-1} \frac T m, </math>

the sound speed in an ideal gas can also be made dependent only on its specific enthalpy:

<math display="block">a_s (h) = \sqrt {(\gamma -1) h} .</math>