The speed of sound is defined as the speed of propagation of small pertubations in density and pressure in a fluid in a compressible fluid. For the derivation of the sound speed it is also assumed that the fluid is homogeneous, i.e. the average value of density ${\rho }_0$ and pressure $p_0$ are spatially (and temporarily ?) constant. Then we can split the local value of density and pressure in a spatially constant average value and a small spatially an temporarily varying pertubation ${\rho }^{\prime }$ and $p^{\prime }$ respectively

	$\rho$	$={\rho }_0+{\rho }^{\prime }(x,t),{\rho }^{\prime }\ll {\rho }_0$		(B.1)
	$p$	$=p_0+p^{\prime }(x,t),p^{\prime }\ll p_0$		(B.2)

For an ideal fluid a temporal change in density or pressure has to adiabatic, i.e. with constant entropy. The derivation of the equation of motion for a small pertubation in density and pressure then leads us to a wave equation with a propagation speed

$c=\sqrt{{\left({\displaystyle \frac{\partial p_0}{\partial {\rho }_0}}\right)}_S}$

(B.3)

which is called the adiabatic speed of sound.

Further on for an adiabatic change of state it applies

	$p{V}^{\gamma }$	$=\text{const.}=C\mathit{\hspace{1em}}|\cdot {\displaystyle \frac{1}{M^{\gamma }}}$		(B.4)
	$p{\left({\displaystyle \frac{V}{M}}\right)}^{\gamma }$	$={\displaystyle \frac{C}{M^{\gamma }}}$		(B.5)
	$p{\left({\displaystyle \frac{1}{\rho }}\right)}^{\gamma }$	$=C^{*}$		(B.6)
	$⟹p$	$=C^{*}{\rho }^{\gamma }⟺{\displaystyle \frac{p}{\rho }}=C^{*}{\rho }^{\gamma -1}$		(B.7)

Using this we get

${\displaystyle \frac{\partial p_0}{\partial {\rho }_0}}=\gamma C^{*}{\rho }_0^{\gamma -1}=\gamma {\displaystyle \frac{p_0}{{\rho }_0}}.$

(B.8)

With this result the adiabatic speed of sound for an ideal, compressible, homogeneous fluid yields

$c=\sqrt{\gamma {\displaystyle \frac{p_0}{{\rho }_0}}}$

(B.9)

Using equation (A.4) we can compute the sound speed for an ideal gas directly from the internal energy

$c=\sqrt{\gamma (\gamma -1)u_0}$

(B.10)