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For a steady isentropic flow with $p=K{\rho }^{\gamma }$, $K>0$ and $\gamma >1$, write the Bernoulli function asAt a fixed position, hold $C_B$, $K$ and the external potential $\Phi$ fixed and assume $H=C_B-\Phi >0$. The positive scalar mass flux is $j=\rho u$ for $u>0$. Differentiating the Bernoulli constraint gives $d\rho /du=-\rho u/c_s^2$, henceThe maximum occurs at $u=c_s=c_{\mathrm{cr}}$. Substituting this equality back into the energy constraint givesBoth the small-speed endpoint and the zero-enthalpy endpoint have vanishing $j$, so this interior stationary point is the maximum. The familiar formula with $C_B$ alone assumes a zero potential at the chosen section, or defines $C_B$ to exclude that fixed contribution.

For a perfect gas with no potential change, the result is equivalent to $T_{\mathrm{cr}}=2T_{\mathrm{stag}}/(\gamma +1)$. The local maximum explains choked flow; a complete nozzle solution must also connect the prescribed upstream state to the controlling section.^[ref]

Treat the gas as a smooth quasi-one-dimensional flow through a rigid passage, neglecting axial body forces and heat input. It is the confining medium that is incompressible; the flowing gas must remain compressible. With area $\mathcal{A}(x)$ and axial speed $u$, continuity and inviscid momentum balance giveCombining their logarithmic differentials yields the nozzle area-velocity relationA smooth crossing of $\mathcal{M}=1$ requires an area extremum. The given radius is positive and has derivative $r^{\prime }(x)=16r_{0}x\exp [-16x^2/L^2]/L^2$, which vanishes only at $x=0$ within the slab and changes from negative to positive there. Thus this station is the unique throat, with $r_s=r_0/2$ and ${\mathcal{A}}_s=\pi r_0^2/4$. The specified subsonic inlet and supersonic outlet select the smooth accelerating branch, soA throat by itself does not guarantee transonic flow for arbitrary reservoir and back pressures; that branch is specified here. Before the throat a converging section accelerates subsonic flow; beyond it a diverging section accelerates supersonic flow.^[ref]

For the temperature task, evaluate the mass flux and sound speed at the throat using $p=K{\rho }^{\gamma }$:The ideal-gas relation then givesThis expresses the sonic temperature in terms of the entropy constant, geometry, mass flux and composition; no numerical value of $K$ or $\dot{M}$ is provided. Equivalently $c_s^{\gamma +1}=\gamma K(\dot{M}/{\mathcal{A}}_s{)}^{\gamma -1}$ and $T_s=\mu c_s^2/(\gamma R^{\ast })$. The slab length drops out because the sonic thermodynamic state depends on the minimum area, not on the axial distance over which a lossless smooth nozzle contracts. As a check, conserved stagnation enthalpy gives $T_s=2T_{\mathrm{stag}}/(\gamma +1)$.^[ref]

The result assumes cross-sectional thermodynamic uniformity and negligible transverse kinetic energy. Exact multidimensional flow through a sharply varying passage can have a curved sonic surface; the radius profile alone does not solve that boundary-value problem. The original numerical plot below computes the standard quasi-one-dimensional branch, rather than reproducing the supplied passage sketch.