Transonic small disturbance equation

Remarks

The equation is of mixed type: it is elliptic where $1 - M_\infty^2 - (\gamma + 1) M_\infty^2 \varphi_x / U > 0$, which corresponds to locally subsonic flow, and hyperbolic where this quantity is negative. Example: if $M_\infty < 1$ and $\varphi_x / U$ is negligible next to $1 - M_\infty^2$, the equation reduces to the linear equation $(1 - M_\infty^2)\varphi_{xx} + \varphi_{yy} = 0$ of the Prandtl–Glauert transformation. Non-example: that linear equation is not a transonic small disturbance equation, because it lacks the term $\varphi_x \varphi_{xx}$, and so it cannot change type or describe a sonic line. Its linear analogue of mixed type is the Tricomi equation $y\, u_{xx} + u_{yy} = 0$. Solutions with shocks are understood as weak solutions of the conservation form $\partial_x\bigl((1 - M_\infty^2)\varphi_x - \frac{\gamma + 1}{2U} M_\infty^2 \varphi_x^2\bigr) + \partial_y \varphi_y = 0$.