Example data table
| M1 | M2 | p2/p1 | T2/T1 | ρ2/ρ1 | p0,2/p0,1 |
|---|---|---|---|---|---|
| 1.50 | 0.7011 | 2.4583 | 1.3202 | 1.8621 | 0.9298 |
| 2.00 | 0.5774 | 4.5000 | 1.6875 | 2.6667 | 0.7209 |
| 3.00 | 0.4752 | 10.3333 | 2.6790 | 3.8571 | 0.3283 |
| 5.00 | 0.4152 | 29.0000 | 5.8000 | 5.0000 | 0.0617 |
Formula used
This calculator applies classic normal shock relations for a calorically perfect gas:
- M2² = [1 + (γ−1)/2 · M1²] / [γ · M1² − (γ−1)/2]
- p2/p1 = 1 + 2γ/(γ+1) · (M1² − 1)
- ρ2/ρ1 = ( (γ+1)·M1² ) / ( (γ−1)·M1² + 2 )
- T2/T1 = (p2/p1) / (ρ2/ρ1)
- p0,2/p0,1 = (p2/p1) · [(1+(γ−1)/2·M2²)^(γ/(γ−1))] / [(1+(γ−1)/2·M1²)^(γ/(γ−1))]
With temperature and gas constant, speed of sound is computed as a = √(γRT), and velocity as V = Ma. Density uses the ideal-gas relation ρ = p/(RT).
How to use this calculator
- Enter an upstream Mach number of 1.0 or higher.
- Set γ and R for your working gas or mixture.
- Provide upstream temperature and static pressure with units.
- Click Calculate to view results above the form.
- Download CSV or PDF to document your calculation.
Normal shock purpose and scope
Normal shocks convert supersonic flow to subsonic flow across a thin pressure jump. This calculator models that jump directly using one-dimensional, steady relations for a perfect gas. Enter upstream Mach number, heat-capacity ratio, gas constant, temperature, and static pressure. The output is useful for inlet lips, diffuser throats, shock tubes, and nozzle testing when the shock is approximately normal to the flow.
Key state ratios you should track
The most important indicators are the downstream Mach number and the ratios p2/p1, ρ2/ρ1, and T2/T1. For air with γ=1.4 at M1=2.0, the calculator predicts M2≈0.577, p2/p1≈4.50, ρ2/ρ1≈2.67, and T2/T1≈1.69. It also reports total pressure recovery p0,2/p0,1, which captures irreversible losses and strongly affects propulsion and compressor matching.
Velocity and speed of sound implications
With temperature and gas properties, the tool computes speed of sound a=√(γRT) and velocity V=Ma on both sides. Using air at T1=288 K gives a1≈340 m/s; with M1=2, V1≈681 m/s. After the shock, T2 rises and M2 drops, so the flow slows markedly. These values help estimate residence time, aerodynamic heating, and the change in dynamic pressure across the discontinuity.
Design interpretation for ducts and inlets
A normal shock raises static pressure but reduces total pressure, so it is rarely "free" compression. Stronger shocks occur at higher M1: for air at M1=3.0, p2/p1≈10.33 and M2≈0.475, while total pressure recovery can fall below about 0.33. In practical intakes, designers try to weaken a normal shock by using multiple oblique shocks or variable geometry to maintain higher recovery.
Quality checks and practical tips
Validate results by checking trends: p2/p1 and T2/T1 must increase with M1, while M2 must remain below 1. Compare a few points against the example table to confirm unit handling. Remember the limits: real-gas chemistry, vibrational excitation, boundary layers, and shock-boundary interaction are not included. For high-enthalpy flows, use this output as a baseline before CFD or tunnel correlation.
FAQs
What upstream Mach numbers are valid?
Use M1 at or above 1.0. Values slightly above one create weak shocks with small changes. Very high Mach numbers may violate perfect-gas assumptions, so treat results as first-order estimates.
Why does total pressure decrease across a shock?
A shock is irreversible, so entropy increases as kinetic energy converts to internal energy. Static pressure rises, but stagnation pressure drops, reducing recoverable work for compressors, turbines, and expansions.
Which gamma and R should I use?
Pick properties for your working gas and temperature range. Dry air near room temperature typically uses γ≈1.4 and R≈287 J/(kg·K). For other gases, consult property tables or verified supplier data.
Can this handle oblique shocks?
This calculator is for normal shocks. For an oblique shock, compute the normal Mach component Mn1 = M1·sin(β), apply these relations to Mn1, then recombine with the unchanged tangential component to find downstream M2.
Do units affect the ratios?
No. Ratios like p2/p1 and T2/T1 are dimensionless. Units only affect absolute outputs such as pressure, temperature, density, speed of sound, and velocity. Inputs are converted internally for consistent calculations.
How should I validate results?
Confirm p2/p1 and ρ2/ρ1 increase with M1 while M2 stays below one. Cross-check one or two cases with a trusted reference table, and verify γ, R, and unit selections before exporting.