Oblique Shock Calculator
θ→β, β→θ, or M from two measured angles — both roots and the detachment limit, at any γ.
Results
What this is computing
A wedge in supersonic flow gets no warning it is coming, so the flow turns all at once, across a single straight wave. Everything here follows from one fact: an oblique shock is a normal shock in disguise — split the velocity across the wave, and the tangential component crosses unchanged while the normal component alone shocks. Turn the wall the other way instead and there is no shock at all: the flow expands through a fan.
β is measured from the upstream flow, θ to the wedge surface. A shock needs Mn1 ≥ 1, so β never falls below the Mach angle μ = arcsin(1/M₁). The θ–β–M relation is explicit given β; given θ it is a cubic in tan β.
Two roots, and what chooses between them
For every θ below the maximum, two wave angles turn the flow by exactly that much — a shallow weak shock and a steep strong one, both satisfying every conservation law. The wedge does not choose between them: the downstream pressure does, and in free external flow it is always the weak one.
Detachment
Push θ up and the two roots converge, meeting at θmax; past it there is no attached solution and the shock stands off ahead of the body as a curved bow shock. For air θmax is 34.07° at M₁ = 3 and rises towards 45.58° as M₁ → ∞, so a wedge blunter than about 45.6° has no attached shock at any speed. A cone of the same half-angle stays attached far longer — 40.69° against 22.97° at M₁ = 2.
The weak solution is not always supersonic
M₂ = 1 occurs at a slightly smaller wave angle than θmax does, leaving a narrow band of weak solutions with subsonic flow behind them — 0.27° of deflection wide at M₁ = 2, and 0.065° at M₁ = 3. That is why the diagram draws the maximum-deflection locus and the M₂ = 1 locus as two separate curves.