 Plane-parallel isothermal sonic transition at a potential maximum

In a nonzero steady one-dimensional flow with an isothermal equation of state with $p=c_s^2\rho$ and constant cross section, the continuity equation gives $\rho u=\text{constant}$. Eliminating density from momentum gives $(u-c_s^2/u)u^{\prime }=-{\Phi }^{\prime }$. With Mach number $\mathcal{M}=|u|/c_s$, integration yields ${\mathcal{M}}^2/2-\ln \mathcal{M}+\Phi /c_s^2=C$. The function $f(\mathcal{M})={\mathcal{M}}^2/2-\ln \mathcal{M}$ has minimum $1/2$ at $\mathcal{M}=1$, so a smooth transonic crossing at $x_{\ast }$ requires a local maximum of $\Phi$ and $C=1/2+\Phi (x_{\ast })/c_s^2$. At a nondegenerate maximum, $f=1/2+(\mathcal{M}-1{)}^2+\cdots$ gives ${\mathcal{M}}^{\prime }(x_{\ast })=\pm \sqrt{-{\Phi }^{\prime \prime }(x_{\ast })/(2c_s^2)}$.