Oblique Shock Calculator
Solve the oblique shock relations for a supersonic flow deflected by a wedge or ramp. Enter the upstream Mach number and deflection angle to find the shock wave angle, downstream Mach number, and the pressure, temperature and density jumps across the shock.
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How does this calculator work?
An oblique shock forms when supersonic flow meets a wedge of angle θ. The shock angle β is found numerically from the θ-β-M relation: tan θ = 2cot β (M₁²sin²β − 1)/(M₁²(γ+cos2β)+2). Post-shock conditions follow from the normal Mach component Mn₁ = M₁ sin β using standard normal-shock formulas. A = 1.4 for air; deflection above the detachment limit produces no attached shock.
Formula
How this is calculated
When a supersonic flow meets a wedge or ramp it cannot adjust subsonically, so a shock wave forms at an angle β (the shock angle) to the upstream flow. The flow is deflected by the wedge half-angle θ. The θ-β-M relation — a transcendental equation — links these angles to the upstream Mach number M₁ and the ratio of specific heats γ (1.4 for diatomic gases like air at standard conditions). This calculator solves that equation numerically using bisection to find the weak-shock solution (the physically preferred one for attached shocks).
Once the shock angle β is known, the component of the upstream Mach number normal to the shock is Mn₁ = M₁ sin β. This normal component drives the thermodynamic changes across the shock, just as in a normal shock: the pressure ratio p₂/p₁ = 1 + 2γ(Mn₁² − 1)/(γ+1), the density ratio ρ₂/ρ₁ = (γ+1)Mn₁² / ((γ−1)Mn₁²+2), and the temperature ratio T₂/T₁ = (p₂/p₁)/(ρ₂/ρ₁). The downstream Mach number M₂ is recovered from the post-shock normal component Mn₂ and the geometry: M₂ = Mn₂ / sin(β − θ).
A key limitation: for each Mach number there is a maximum deflection angle beyond which no attached oblique shock exists — the shock detaches and a curved bow shock forms. The calculator returns no result when the requested deflection exceeds this detachment limit. The results assume a perfect (calorically ideal) gas with constant γ and no real-gas or viscous effects.
Frequently asked questions
For most M₁ and θ combinations there are two mathematical solutions for β: a weak shock (smaller β, M₂ > 1 usually) and a strong shock (larger β, M₂ < 1). In practice attached shocks on wedges are almost always weak shocks. This calculator returns the weak-shock solution.
If the deflection angle θ exceeds the maximum value for a given M₁, no attached oblique shock solution exists. A detached curved bow shock forms ahead of the body. The calculator returns no result in this case — reduce θ or increase M₁.
For the weak-shock solution the flow downstream of an oblique shock is typically (but not always) still supersonic, because the shock is weaker than a normal shock. This is why oblique shocks are more efficient for decelerating supersonic flows than normal shocks — the total pressure loss is smaller.
Also known as
TG we-Calculate Editorial Team. (2026). Oblique Shock Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/oblique-shock-calculator
TG we-Calculate Editorial Team. "Oblique Shock Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/oblique-shock-calculator.
TG we-Calculate Editorial Team, "Oblique Shock Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/oblique-shock-calculator
@misc{wecalculate_oblique_shock_calculator, title = {Oblique Shock Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/oblique-shock-calculator}}, year = {2026}, note = {TG we-Calculate} }
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