Nozzle area-velocity relation ($\frac{dA}{A}=(M^2-1)\frac{du}{u}$)

Steady mass conservation, inviscid momentum balance and $dp=c_s^{2}d\rho$ give $d\ln A=(M^2-1)d\ln u$, where $M=u/c_s$. A smooth sonic crossing needs an area extremum; a minimum-area throat supports an accelerating subsonic-to-supersonic branch. A throat is a necessary geometric condition, not a sufficient specification of operating pressures.