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Aerospace
Downstream Mach number of a normal shock
Mach number behind a steady normal shock for upstream Mach number M1 > 1 and heat-capacity ratio γ.
- Ref
- AER-38
- Rev
- v1
- Field
- Aerospace
- Result
- M2
Across a normal shock the Mach number drops from M1 > 1 to a subsonic M2, given by M2² = [1 + (γ−1)M1²/2]/[γM1² − (γ−1)/2]. At M1 = 2 it is 0.577, and as M1 → ∞ it tends to √((γ−1)/(2γ)) = 0.378 for air. At M1 = 1 there is no shock and M2 = 1. For an oblique shock apply the formula to the Mach number normal to the shock.
Mach number behind a steady normal shock for upstream Mach number M1 > 1 and heat-capacity ratio γ. The result is always subsonic and falls toward a limit of about 0.378 for air at very high M1. For an oblique shock apply it to the normal component only.
Calculator
- Temperature ratio T₂/T₁1.687
- Pressure ratio p₂/p₁4.5
- Limit of M₂ as M₁ → ∞, √((γ − 1)/(2γ))0.378
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The flow slows from Mach 2 to Mach 0.5774, and its temperature rises by a factor of 1.687.
- However strong the shock, M₂ cannot fall below 0.378, so the downstream flow stays subsonic but never stagnates.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| M1 Upstream Mach number | 1.01 to 6 | 0.4042 | 0.9901 | 0.586 | +10% | −5.25 % |
| γ Heat-capacity ratio | 1.1 to 1.67 | 0.5252 | 0.6073 | 0.08206 | +10% | +2.93 % |
Sweep of Upstream Mach number
| Upstream Mach number | Downstream Mach number |
|---|---|
| 1.01 | 0.9901 |
| 1.634 | 0.6585 |
| 2.258 | 0.5396 |
| 2.881 | 0.4826 |
| 3.505 | 0.451 |
| 4.129 | 0.4316 |
| 4.753 | 0.419 |
| 5.376 | 0.4104 |
| 6 | 0.4042 |
Sweep one variable
Variables and how each one contributes
Downstream Mach number
What the equation gives you. Everything else feeds it.
Upstream Mach number
The upstream Mach number, above 1. A stronger shock (higher M1) gives a lower downstream Mach number. Raising it by 10 % (to 2.2) moves the result by −5.25 %, from 0.5774 to 0.5471.
Heat-capacity ratio
The heat-capacity ratio cp/cv of the gas, about 1.4 for air. A lower γ (hotter, more complex gas) changes the compressible-flow exponents and the sizes of the shock and expansion effects. Raising it by 10 % (to 1.54) moves the result by +2.93 %, from 0.5774 to 0.5943.
Worked example with the values above
- Numerator: 1 + (γ − 1)M1²/2 = 1.8.
- Denominator: γM1² − (γ − 1)/2 = 5.4.
- M2 = √(1.8 / 5.4) = 0.57735.
Assumptions and limits
- A steady normal shock in a calorically perfect gas, with the same γ on both sides and no heat addition.
- Upstream Mach number above 1; the downstream flow is subsonic.
Common mistakes
- Using the upstream Mach number on the downstream side.
- Using total pressure where static pressure is needed, or the reverse.
- Applying normal-shock formulas to the full Mach number of an oblique shock.
Statistics and uncertainty
Normal-shock relations are exact for a perfect gas. At high Mach numbers, dissociation and real-gas effects change the pressure and temperature ratios by several per cent, and weak shocks are close to isentropic, so small pressure losses are sensitive to inputs.
Where it is used
- Estimating shock losses in inlets, nozzles and supersonic tunnels.
- Interpreting pitot readings behind a bow shock.
Maths you need
- Rankine–Hugoniot relations: conservation of mass, momentum and energy across a shock
- Stagnation pressure loss: a measure of the irreversibility of the shock
- Entropy: a measure of irreversibility