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\end{equation}  then the adiabatic sound speed is  \begin{equation}  \left( c^{\rm{ad}}_{s}\right)^2 c^{\rm{ad}}_{s}  = \frac{dP}{d\rho} \sqrt{\frac{dP}{d\rho}}  = \gamma \frac{P}{\rho} \sqrt{\gamma \frac{P}{\rho}}  \end{equation}  $P = \frac{\left( c^{\rm{ad}}_{s}\right)^2}{\gamma} \rho$