Oblique Shock (θ-β-M)

The engineering

Turn a supersonic stream through angle θ and it obliges with a shock slanted at β — solved here by iterating the θ-β-M relation on the weak branch, the one nature almost always picks. Only the velocity component normal to the shock takes the compression, so the flow downstream can stay supersonic: M₂ 1.64 behind a 10° wedge at M 2.

Ask for more turning than θmax (about 23° at M 2) and no attached solution exists — the shock detaches into a bow wave standing off the nose, and this card tells you so instead of inventing a number. That limit is why supersonic leading edges are sharp and hypersonic vehicles are blunt on purpose.

Where this math comes from

Ludwig Prandtl's Göttingen school turned shock waves from photographs into design equations; his student Theodor Meyer worked out the oblique-shock and expansion-fan theory in his 1908 dissertation — a document so complete that a century of supersonic inlets is essentially its appendix.

The relation resists algebra (it's cubic in sin²β), so the twentieth century solved it by lookup: NACA Report 1135 (1953) carried the charts every aerodynamicist thumbed until computers made the iteration free. Wedge-and-cone staged compression built on this math fed the SR-71's inlets, which recovered most of Mach 3.2 as pressure the engines could use.

1. 1904Ludwig PrandtlGöttingen program makes supersonic flow an engineering science.
2. 1908Theodor MeyerOblique-shock θ-β-M theory in his dissertation under Prandtl.
3. 1953NACA (Ames)Report 1135 charts the θ-β-M relation for working engineers.
4. 1964Lockheed Skunk WorksSR-71 inlets stage oblique shocks to fly Mach 3.2 efficiently.

See the full timeline of the math behind every calculator →

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