Normal Shock Calculator
Behind a normal shock at Mach 2 in air the flow slows to M₂ = 0.57735, the static pressure rises 4.5 times and 28 % of the stagnation pressure is lost.
Air 1.4; hot combustion gases about 1.3; helium 1.667.
0.577350269189626
Full answer, then cut to fewer decimals (not rounded):
- 0.577350269189626
- 0.57735026918962
- 0.5773502691896
- 0.577350269189
- 0.57735026918
- 0.5773502691
- 0.577350269
- 0.57735026
- 0.5773502
- 0.577350
- 0.57735
- 0.5773
- 0.577
- 0.57
- 0.5
- Static pressure ratio p₂/p₁
- 4.5
- Density ratio ρ₂/ρ₁
- 2.666666667
- Temperature ratio T₂/T₁
- 1.6875
- Stagnation pressure ratio p₀₂/p₀₁
- 0.7208738615
- Entropy rise Δs/R = −ln(p₀₂/p₀₁)
- 0.3272911064
- Pitot pressure ratio p₀₂/p₁ (Rayleigh)
- 5.640440813
Step-by-step solution
- M₂² = (1 + (γ−1)M₁²/2) ÷ (γM₁² − (γ−1)/2) → M₂ = 0.5773502692
- p₂/p₁ = 1 + 2γ(M₁² − 1)/(γ + 1) = 4.5
- ρ₂/ρ₁ = (γ+1)M₁² ÷ ((γ−1)M₁² + 2) = 2.666666667; T₂/T₁ = 1.6875
Normal Shock Formula
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)
Worked Example
A normal shock in air at M₁ = 2:
- M₂² = (1 + (γ−1)M₁²/2) ÷ (γM₁² − (γ−1)/2) → M₂ = 0.5773502692
- p₂/p₁ = 1 + 2γ(M₁² − 1)/(γ + 1) = 4.5
- ρ₂/ρ₁ = (γ+1)M₁² ÷ ((γ−1)M₁² + 2) = 2.666666667; T₂/T₁ = 1.6875
Density ratio ρ₂/ρ₁: 2.666666667 · Temperature ratio T₂/T₁: 1.6875 · Stagnation pressure ratio p₀₂/p₀₁: 0.7208738615 · Entropy rise Δs/R = −ln(p₀₂/p₀₁): 0.3272911064 · Pitot pressure ratio p₀₂/p₁ (Rayleigh): 5.640440813
GATE tip
For a shock in a nozzle, use p₀₂/p₀₁ to find the new A* behind the shock: A₂*/A₁* = p₀₁/p₀₂. Then the exit Mach follows from the subsonic area relation.
More Compressible Flow Calculators
-
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.
-
Oblique Shock Calculator
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)
Example: Mach 2 flow turned by a 10° wedge: Shock wave angle β (weak) = 39.3139 °.
-
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.
-
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
Why is the flow always subsonic after a normal shock?
An expansion shock would decrease entropy, which the second law forbids; only compressive shocks from supersonic to subsonic are possible.
What stays constant across the shock?
Stagnation temperature (and enthalpy), mass flux and the total of momentum flux plus pressure. Stagnation pressure falls.
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). Normal Shock Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/normal-shock-calculator/
IEEE
Scientific Converters, "Normal Shock Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/normal-shock-calculator/