Solve the oblique-shock θ–β–M relation for shock angle and downstream conditions.
IDEAL-GAS MODEL: weak-shock solution of the θ–β–M relation for a calorically perfect gas. If the deflection exceeds the maximum for the Mach number, no attached oblique shock exists (the shock detaches) and the tool reports invalid. Real shocks involve viscosity, heat transfer and, at high Mach, dissociation — this is a textbook estimate.
Shock angle β (°)
How to use this tool
Type the supersonic Mach number ahead of the shock. It must be above 1.
Type the wedge or ramp angle θ in degrees — how much the flow is turned by the shock.
Use 1.4 for air. The tool solves the θ–β–M relation for the weak-shock solution.
The tool reports β and the pressure, density and temperature ratios and the downstream Mach number. If no solution exists, the shock is detached.
Formula & worked example
Mach 2 flow deflected 15° (γ = 1.4): β ≈ 45.34°, p2/p1 ≈ 2.195, ρ2/ρ1 ≈ 1.729, T2/T1 ≈ 1.269, M2 ≈ 1.446 — still supersonic, as expected for a weak shock.
Limitations — read this before you rely on it
Perfect-gas, inviscid, steady two-dimensional model returning the weak-shock solution only (the strong-shock solution, with subsonic downstream flow, is not reported). At hypersonic Mach numbers, real-gas effects (vibration, dissociation) change the ratios significantly.
Frequently asked questions
A shock wave inclined to the flow, formed when supersonic flow is deflected by a wedge, ramp or cone. Unlike a normal shock, it turns the flow while compressing it less severely.
For a given Mach number and deflection below the maximum, the θ–β–M relation gives a weak shock (small β, supersonic downstream) and a strong shock (large β, subsonic downstream). The weak solution is the one normally observed on a wedge.
No attached straight shock can turn the flow that much, so the shock detaches and stands off ahead of the body as a curved bow shock.
Only the velocity component normal to the shock is strongly slowed; the tangential component is unchanged, so the total downstream Mach number often stays above 1.
A normal shock (β = 90°) is the strongest case for a given Mach number, with the largest pressure rise and total-pressure loss. Oblique shocks compress more gently, which is why supersonic inlets use them.