One nozzle, six personalities. Lower the back pressure pb and watch the same convergent–divergent nozzle morph: gentle venturi → choked throat → a shock trapped inside → overexpanded (shock diamonds!) → perfectly expanded → underexpanded. The white curve is p/p₀ along the nozzle — the story in one line. 🎯 challenges you to hit design condition like a propulsion engineer.
(A/A*)² = (1/M²)·[2/(γ+1)·(1 + ½(γ−1)M²)]^{(γ+1)/(γ−1)} — every area ratio has TWO answers, one subsonic and one supersonic. The back pressure picks which one you get.p/p₀ = (1 + ½(γ−1)M²)^{−γ/(γ−1)}.ṁ = p₀A*/√T₀ · √(γ/R)·(2/(γ+1))^{(γ+1)/(2(γ−1))} — lowering pb further can't pull more through.