Sonic point of a scalar flux ($f\prime (s_{\ast })=0$)

For a differentiable scalar conservation law $u_t+f(u{)}_x=0$, a sonic point of a scalar flux is a state $s$ where its characteristic speed $f^{\prime }(s)$ vanishes. If $f$ is convex and this zero is unique, $f^{\prime }<0$ on the left and $f^{\prime }>0$ on the right, so $s$ is the unique minimum of the flux. The definition refers to a vanishing transport speed in a scalar conservation law. In compressible fluid mechanics, a sonic point instead refers to flow at the local sound speed.