= Plane-parallel isothermal sonic transition at a potential maximum
{title2=$C=\tfrac12+\Phi_{\max}/c_s^2$}
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 $(u-c_s^2/u)u'=-\Phi'$. 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 $x_*$ requires a local maximum of $\Phi$ and $C=1/2+\Phi(x_*)/c_s^2$. At a nondegenerate maximum, $f=1/2+(\mathcal M-1)^2+\cdots$ gives $\mathcal M'(x_*)=\pm\sqrt{-\Phi''(x_*)/(2c_s^2)}$.
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