Supersonic flow

Remarks

Take steady plane flow with a uniform stream along the $x$-axis and small disturbances. The perturbation potential $\varphi$ solves $$(M_\infty^2 - 1)\varphi_{xx} - \varphi_{yy} = 0.$$ This equation is hyperbolic for $M_\infty > 1$, in contrast with the elliptic subsonic case of the Prandtl–Glauert transformation. Its characteristics are the lines $y = \pm x / \sqrt{M_\infty^2 - 1}$, which make the Mach angle $\mu = \arcsin(1/M_\infty)$ with the stream. Example: air at $M_\infty = 2$; then $\mu = 30^\circ$, and a small disturbance at a point affects only the wedge of half-angle $30^\circ$ behind it. Non-example: a flow at $M_\infty = 0.8$ is subsonic, and its disturbances spread in all directions. In usage, supersonic often means $1.2 < M_\infty < 5$, the range between transonic and hypersonic flow.