Oblique Shock Calculator
A Mach 2 flow turned by a 10° wedge forms an attached oblique shock at β = 39.3139 °, leaving the flow at Mach 1.64 with 1.71 times the pressure.
Air 1.4; hot combustion gases about 1.3; helium 1.667.
39.3139318448189 °
Full answer, then cut to fewer decimals (not rounded):
- 39.3139318448189 °
- 39.313931844818 °
- 39.31393184481 °
- 39.3139318448 °
- 39.313931844 °
- 39.31393184 °
- 39.3139318 °
- 39.313931 °
- 39.31393 °
- 39.3139 °
- 39.313 °
- 39.31 °
- 39.3 °
- Downstream Mach number M₂
- 1.640522229
- Pressure ratio p₂/p₁
- 1.706578604
- Temperature ratio T₂/T₁
- 1.170151284
- Density ratio ρ₂/ρ₁
- 1.458425613
- Stagnation pressure ratio p₀₂/p₀₁
- 0.9846440225
- Normal Mach component M₁ sin β
- 1.267138036
- Strong-shock solution β
- 83.70008038 °
- Maximum deflection θ_max at this Mach
- 22.97353173 °
Step-by-step solution
- Solve tan θ = 2 cot β (M₁² sin²β − 1) ÷ (M₁²(γ + cos 2β) + 2) for β → 39.31393184° (weak)
- Mₙ₁ = M₁ sin β = 1.267138036; normal-shock relations give Mₙ₂ and the ratios
- M₂ = Mₙ₂ ÷ sin(β − θ) = 1.640522229
Oblique Shock Formula
tan θ = 2 cot β (M₁² sin²β − 1) / (M₁²(γ + cos 2β) + 2)
normal component M₁ sin β obeys the normal-shock relations; M₂ = Mₙ₂ / sin(β − θ)
Worked Example
Mach 2 flow turned by a 10° wedge:
- Solve tan θ = 2 cot β (M₁² sin²β − 1) ÷ (M₁²(γ + cos 2β) + 2) for β → 39.31393184° (weak)
- Mₙ₁ = M₁ sin β = 1.267138036; normal-shock relations give Mₙ₂ and the ratios
- M₂ = Mₙ₂ ÷ sin(β − θ) = 1.640522229
Pressure ratio p₂/p₁: 1.706578604 · Temperature ratio T₂/T₁: 1.170151284 · Density ratio ρ₂/ρ₁: 1.458425613 · Stagnation pressure ratio p₀₂/p₀₁: 0.9846440225 · Normal Mach component M₁ sin β: 1.267138036 · Strong-shock solution β: 83.70008038 ° · Maximum deflection θ_max at this Mach: 22.97353173 °
GATE tip
A supersonic diamond airfoil sees an oblique shock at the leading edge, a Prandtl–Meyer expansion at the shoulder and a shock at the trailing edge; the pressure difference gives wave drag.
More Compressible Flow Calculators
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Isentropic Flow Calculator
T₀/T = 1 + (γ − 1)M²/2, p₀/p = (T₀/T)^(γ/(γ−1))
A/A* = (1/M)[(2/(γ+1))(1 + (γ−1)M²/2)]^((γ+1)/(2(γ−1)))
Example: Air (γ = 1.4) at Mach 2: Pressure ratio p/p₀ = 0.127805.
-
Normal Shock Calculator
M₂² = (1 + (γ−1)M₁²/2) / (γM₁² − (γ−1)/2)
p₂/p₁ = 1 + 2γ(M₁² − 1)/(γ + 1); ρ₂/ρ₁ = (γ+1)M₁²/((γ−1)M₁² + 2)
Example: A normal shock in air at M₁ = 2: Downstream Mach number M₂ = 0.57735.
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Prandtl-Meyer Expansion Calculator
ν(M) = √((γ+1)/(γ−1)) arctan √((γ−1)(M²−1)/(γ+1)) − arctan √(M² − 1)
expansion by θ: ν(M₂) = ν(M₁) + θ, isentropic
Example: Mach 2 air expanding round a 10° corner: Downstream Mach number M₂ = 2.38489.
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Nozzle Flow Calculator
ṁ = A* p₀ √(γ/(RT₀)) (2/(γ+1))^((γ+1)/(2(γ−1)))
exit Mach from Aₑ/A* (supersonic branch); vₑ = Mₑ√(γRTₑ); F = ṁvₑ + (pₑ − pₐ)Aₑ
Example: A rocket nozzle with 7 MPa, 3000 K chamber, 100 cm² throat and expansion ratio 10, in vacuum: Mass flow rate ṁ = 44.3031 kg/s.
Frequently asked questions
Weak or strong solution?
Two shock angles satisfy the relation; in external flows the weak one almost always occurs. The strong one needs a high downstream pressure.
What happens above θ_max?
No attached oblique shock can turn the flow, so a detached bow shock stands ahead of the body.
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Oblique Shock Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/oblique-shock-calculator/
IEEE
Scientific Converters, "Oblique Shock Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/oblique-shock-calculator/