The continuous <transonic branch> passes from $dj/du>0$ to $dj/du<0$, so $j(u)$ has a maximum where its derivative vanishes. At this <critical speed of a polytropic flow>,
$$
\boxed{u=c_s=c_{\rm cr}}.
$$
The maximum mass flux is therefore attained at the <sonic point>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 314 / 2 / b / Solution

{title2=$c_{\rm cr}$}

For a steady polytropic streamline with Bernoulli function $C_B$, the critical speed is the sound speed at which the scalar mass flux $\rho u$ is maximal. It satisfies
$$
c_{\rm cr}^2=\frac{2(\gamma-1)}{\gamma+1}C_B.
$$


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