A rocket nozzle is shaped for one outside pressure. Below that altitude the air squeezes the exhaust: shock waves cross inside the jet and leave a chain of bright diamonds. At the design altitude the jet leaves straight. Above it the gas spreads out, and in near vacuum the plume opens into a vast, faint fan. Everything on the stage, from the angle at the lip to the spacing of the diamonds, is computed from the engine and the air.
Changing the area ratio or chamber pressure keeps the engine's propellants and makes it your own engine. Sound starts only when you press Sound.
The nozzle. Gas expands from the chamber through the throat, where it reaches the speed of sound, to the exit. For an ideal gas with ratio of specific heats γ the exit Mach number follows from the area ratio alone, and the exit pressure from the Mach number: pe/pc = (1 + (γ−1)/2·Me²)−γ/(γ−1). These are the guide's relations (NASA Glenn), solved by bisection. Specific impulse is the thrust coefficient, including the pressure term ε(pe − pa)/pc, times the characteristic velocity c* of the propellants, divided by g₀; losses are ignored, so real engines run a few percent lower.
The air. Outside pressure comes from the US Standard Atmosphere 1976, layer by layer to 86 km, and an exponential tail above it. (The guide uses a single 7.2 km scale height; the layers matter for the design altitude.)
The plume. Once outside, the jet adjusts to the outside pressure. Its fully expanded Mach number comes from the chamber-to-outside pressure ratio and its fully expanded diameter from the same area–Mach relation. If the exit pressure is higher than outside, the gas turns outward around the lip by the Prandtl–Meyer angle ν(Mj) − ν(Me); if lower, an oblique shock turns it inward by the angle that raises its pressure to the outside value (θ–β–M relation). The boundary then overshoots and oscillates around the fully expanded diameter, with shock cells about 1.22·Dj·√(Mj² − 1) long (Prandtl's rule of thumb, as in the guide; Tam's analysis gives 1.31). In strongly under-expanded jets (exit pressure more than twice the outside) a normal shock, the Mach disk, stands at about 0.67·D*·√(pc/pa) from the exit, where D* is the throat diameter. That correlation (Crist, Sherman and Glass) was measured on sonic nozzles; for a bell nozzle it is an estimate.
Separation. When the exhaust leaves far below the outside pressure, the air pushes back along the wall and the flow separates inside the nozzle. The page uses Schmucker's criterion: separation where the wall pressure falls below pa·(1.88·M − 1)−0.64. Past it, the jet leaves the wall at the separation point, and thrust is computed as if the nozzle ended there. Between 60 % and 100 % of the limit the page says "at the edge": a one-dimensional estimate cannot decide, because the pressure on a bell's wall near the exit is higher than the average. The RS-25 at sea level sits in that band and in reality runs attached once started; vacuum engines such as the RL10B-2 separate at sea level.
Colour. The light is drawn, not computed. Kerosene burns rich and sooty: hot carbon glows yellow-orange, and leftover fuel burns again where the jet mixes with air. Hydrogen and methane burn clean; their plumes are almost transparent with bluish or pinkish diamonds where shocks reheat the gas. The afterburning glow fades as the air thins.
A study in light architecture: a restricted palette of black, white and one warm colour, structure made visible by a medium, flicker at the edges and stillness at the core. It recreates no specific work.