= Critical speed of a polytropic flow
{title2=$c_{\rm cr}$}
For a steady <isentropic flow> with $p=K\rho^\gamma$, $K>0$ and $\gamma>1$, write the <Bernoulli function> as
$$
C_B=\frac{u^2}{2}+\frac{c_s^2}{\gamma-1}+\Phi,\qquad c_s^2=\gamma K\rho^{\gamma-1}.
$$
At a fixed position, hold $C_B$, $K$ and the external potential $\Phi$ fixed and assume $H=C_B-\Phi>0$. The positive scalar <mass flux> is $j=\rho u$ for $u>0$. Differentiating the Bernoulli constraint gives $d\rho/du=-\rho u/c_s^2$, hence
$$
\frac1j\frac{dj}{du}=\frac1u-\frac{u}{c_s^2}.
$$
The maximum occurs at $u=c_s=c_{\rm cr}$. Substituting this equality back into the energy constraint gives
$$
c_{\rm cr}^2=\frac{2(\gamma-1)}{\gamma+1}(C_B-\Phi).
$$
Both the small-speed endpoint and the zero-enthalpy endpoint have vanishing $j$, so this interior stationary point is the maximum. The familiar formula with $C_B$ alone assumes a zero potential at the chosen section, or defines $C_B$ to exclude that fixed contribution.
For a <perfect gas> with no potential change, the result is equivalent to $T_{\rm cr}=2T_{\rm stag}/(\gamma+1)$. The local maximum explains <choked flow>; a complete <nozzle> solution must also connect the prescribed upstream state to the controlling section.<Mass Flow Choking by NASA Glenn Research Center>[Equations (10)–(19), fixed-total-state flux maximum at Mach one and the throat interpretation]{ref}
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