Normal Shock — Downstream Mach Number Calculator
Solve the downstream Mach number M₂ behind a normal shock, going either from upstream Mach M₁ or backwards from M₂ to M₁ — with step-by-step math, a live shock schematic, growth-curve charts, and the full companion shock-property set.
M₁ ↑ — M₂ falls toward a floor
As the incoming flow gets faster, M₂ drops quickly at first and then flattens out. The floor — approached but never reached — is √((γ−1)/(2γ)), about 0.378 for air.
M₁ → 1⁺ — the shock disappears
As M₁ approaches 1 from above, M₂ approaches 1 from below. The shock degenerates into a Mach wave and the flow remains essentially unchanged across it.
γ sets the floor
The limiting value √((γ−1)/(2γ)) ranges from ≈0.154 (γ → 1) to ≈0.448 (γ = 1.67). A monatomic gas therefore leaves a faster subsonic flow behind a very strong shock than a diatomic one does.
Independent of pressure and temperature
M₂ depends only on M₁ and γ — never on p₁, T₁ or ρ₁ individually. This is a direct consequence of the normal-shock jump relations being expressible in terms of Mach number alone.
How to Use This Calculator
The normal-shock Mach relation links two quantities: the upstream Mach number M₁ and the downstream Mach number M₂, with γ as a gas property. Pick which one you want the tool to solve, then fill in the other.
1. Choose what to solve for
Pick M₂ (downstream Mach from upstream Mach) or M₁ (upstream Mach from a known M₂) — the tool inverts the relation automatically.
2. Set the gas with γ
Use a preset (air 1.40, CO₂ 1.29, steam 1.33, helium/argon 1.67) or type any value between 1.05 and 1.67.
3. Read the result and steps
The result card shows the solved Mach number plus engineering (×10ⁿ) notation. The step-by-step panel shows exactly how the number was reached.
4. Check the schematic and curve
The live shock schematic shows the downstream arrows shortening as M₂ drops, and the chart plots M₂ against M₁ for three different gases.
5. Switch to Advanced for depth
Advanced mode adds the limiting-M₂ curve, a sensitivity check on a 1% input change, and the full set of companion shock metrics (p₂/p₁, ρ₂/ρ₁, T₂/T₁, p₀₂/p₀₁).
6. Quick lookup, or copy
Use the M₁ → downstream properties converter further down the page to see every post-shock quantity at once, or "Copy result" to grab the solved value.
What This Tool Offers
Built around the normal-shock downstream Mach relation M₂² = (1 + ((γ−1)/2)M₁²) / (γM₁² − (γ−1)/2), a core result of compressible-flow gas dynamics.
Solve either direction
Go from upstream Mach number to downstream Mach number, or invert the relation to recover M₁ from a measured M₂.
Any gas via γ
Presets and a continuous γ slider covering diatomic, polyatomic and monatomic gases, so the tool is not locked to air.
Live shock schematic
A duct diagram that updates with your inputs, showing supersonic inflow, the shock itself, and the shortening subsonic outflow arrows behind it.
Growth-curve charts
M₂ vs. M₁ for three gases, plus an advanced curve of the limiting M₂ floor √((γ−1)/(2γ)) as a function of γ.
Full companion metrics
Advanced mode reports p₂/p₁, ρ₂/ρ₁, T₂/T₁, p₀₂/p₀₁ and the stagnation-pressure loss, all from the same M₁ and γ.
Reference tables
A full M₁ → M₂ lookup table for air and a γ reference table for common gases, ready for quick cross-checking.
Reference Tables & Conversions
An M₁ → downstream properties converter, plus normal-shock lookup tables and a γ reference for common gases.
M₁ → all downstream properties
Downstream Mach Number for Air (γ = 1.4)
Standard normal-shock table. The M₂ column is exactly what this calculator computes; the remaining columns are the companion downstream ratios.
| M₁ | M₂ | p₂/p₁ | ρ₂/ρ₁ | T₂/T₁ | p₀₂/p₀₁ |
|---|
Ratio of Specific Heats for Common Gases
| Gas | γ = c_p/c_v | Limiting M₂ = √((γ−1)/(2γ)) | Notes |
|---|---|---|---|
| Helium / Argon | 1.667 | 0.4479 | Monatomic — highest possible downstream Mach floor |
| Steam (H₂O) | 1.330 | 0.3522 | Polyatomic, moderate temperature |
| Methane (CH₄) | 1.310 | 0.3439 | Polyatomic |
| Carbon dioxide (CO₂) | 1.289 | 0.3349 | Polyatomic — lowest floor in this set |
| Air (standard) | 1.400 | 0.3780 | Diatomic, the standard reference value |
| Hot air / combustion gas | 1.30 – 1.33 | 0.3397 – 0.3522 | γ falls as temperature rises |
Downstream Mach Number Quick Lookup (Air, γ = 1.4)
| M₁ | M₂ | Shock strength |
|---|---|---|
| 1.00 | 1.0000 | No shock (Mach wave) |
| 1.20 | 0.8422 | Weak |
| 1.50 | 0.7011 | Weak–moderate |
| 2.00 | 0.5774 | Moderate |
| 3.00 | 0.4752 | Strong |
| 5.00 | 0.4152 | Very strong |
| ∞ | 0.3780 | Limit — approached but never reached |
What Does M₂² = (1 + ((γ−1)/2)M₁²) / (γM₁² − (γ−1)/2) Measure?
A normal shock is a compression discontinuity standing perpendicular to a supersonic flow. Across it the flow slows abruptly from supersonic (M₁ > 1) to subsonic (M₂ < 1), while static pressure, temperature, density and entropy all rise. The downstream Mach number M₂ is the headline number for the flow state behind the shock — it tells you how much kinetic energy the shock removed, and it is fixed entirely by the incoming Mach number and the gas.
Normal Shock Downstream Mach Formula
The defining relationship is:
Rearranged to solve for the upstream Mach number given a known M₂:
The companion post-shock relations used for the advanced metrics are:
ρ₂/ρ₁ = (γ + 1) M₁² / ( 2 + (γ − 1) M₁² )
T₂/T₁ = (p₂/p₁) · (ρ₁/ρ₂)
p₀₂/p₀₁ = [ (γ+1)M₁² / (2 + (γ−1)M₁²) ]γ/(γ−1) · [ (γ+1) / (2γM₁² − (γ−1)) ]1/(γ−1)
Normal Shock Variables
| Symbol | Meaning | Notes |
|---|---|---|
| M₁ | Upstream Mach number | Must be ≥ 1 for a normal shock to exist |
| M₂ | Downstream Mach number | Always subsonic (M₂ < 1) when M₁ > 1 |
| γ | Ratio of specific heats c_p/c_v | 1.40 for air, ≈1.67 for monatomic gases |
| a | (γ − 1)/2 | Shorthand used in the formula |
| p₂/p₁ | Static pressure ratio | Always ≥ 1 — normal shocks only compress |
| p₀₂/p₀₁ | Stagnation (total) pressure ratio | Always < 1 — measures shock irreversibility |
How Upstream Mach Number Affects M₂
The relation is such that M₂ falls rapidly as M₁ rises from 1, then flattens as it approaches the floor √((γ−1)/(2γ)). Going from M₁ = 1.5 to M₁ = 3.0 in air drops M₂ from 0.7011 to 0.4752 — a change of only 0.23, even though the upstream Mach number doubled. In the limit M₁ → ∞, M₂ approaches 0.378 for air but never crosses it. That floor is the absolute minimum downstream Mach number a normal shock can produce in a given gas.
Downstream Mach Calculation Examples
a = (γ − 1)/2 = 0.2
Numerator = 1 + 0.2 × 4 = 1.8
Denominator = 1.4 × 4 − 0.2 = 5.4
M₂² = 1.8 / 5.4 = 0.3333
M₂ = √0.3333 = 0.5774
a = (γ − 1)/2 = 0.2
Numerator = 1 + 0.2 × 0.3333 = 1.0667
Denominator = 1.4 × 0.3333 − 0.2 = 0.2667
M₁² = 1.0667 / 0.2667 = 4.000
M₁ = √4 = 2.000
Frequently Asked Questions
A normal shock is an irreversible compression that converts ordered kinetic energy into random thermal motion. The conservation equations force the downstream flow to be subsonic whenever the upstream flow is supersonic — the second law of thermodynamics only permits a supersonic-to-subsonic transition across a normal shock, never the reverse. So M₂ < 1 always for M₁ > 1.
As M₁ → ∞, M₂ approaches √((γ−1)/(2γ)). For air (γ = 1.4) that is √(1/7) ≈ 0.3780. For helium (γ = 1.667) it is √(0.2) ≈ 0.4479. This floor is never reached by any finite M₁, but it is the theoretical limit for the gas.
No. M₂ depends only on M₁ and γ. The upstream pressure, temperature and density cancel out when the conservation equations are combined, so the downstream Mach number is fixed by the incoming Mach number alone. Only the absolute values of pressure, temperature and density require knowing the upstream state.
Solving for M₁ from M₂ requires the denominator γM₂² − (γ−1)/2 to be positive, which means M₂ > √((γ−1)/(2γ)). Below that value no finite M₁ can produce the given M₂ — the equation has no physical solution because that M₂ is smaller than the shock can ever produce in that gas.
The formula gives M₂ = 1, but a shock of zero strength is not a shock at all — it degenerates into a Mach wave (an infinitesimally weak pressure disturbance). Real normal shocks require M₁ > 1 to have any physical meaning.
Not directly, but the same relation applies to the normal component of the flow across an oblique shock. If you know the shock angle β, you would use M₁ sin β in place of M₁ in this formula to get the downstream normal component, then reconstruct the full downstream Mach number from that. This page covers only the normal-shock case where β = 90°.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| M2 | Downstream Mach number | |
| γ | Ratio of specific heats | |
| M1 | Upstream Mach number |
What it means
This relation gives the Mach number downstream of a normal shock, which is always subsonic. It is derived from the normal shock relations. The downstream Mach number depends only on the upstream Mach number and the specific heat ratio. For M₁ approaching infinity, M₂ tends to √((γ−1)/(2γ)), which for γ=1.4 is about 0.378. This relation is used in supersonic diffuser design and in evaluating shock losses. It helps in determining the recovery pressure and the flow conditions after a shock. Understanding this formula is essential for compressible flow analysis and for designing efficient intake systems.
Worked example
Normal Shock Downstream Mach – Two Examples
Real‑World| Parameter | Value |
|---|---|
| γ | 1.4 |
| M₁ | 2.0 |
| Parameter | Value |
|---|---|
| M₁ | 3.0 |
Common mistakes
- Normal shock downstream Mach number: M₂² = (1 + ((γ−1)/2)·M₁²) / (γ·M₁² − (γ−1)/2).
- M₁ > 1 ⇒ M₂ < 1 (shock decelerates flow).
- γ: Specific heat ratio.
- Valid for normal shock.
- Check: for M₁→∞, M₂→√((γ−1)/(2γ)).
Applications
The normal shock downstream Mach number, M₂² = (1 + ((γ−1)/2)M₁²)/(γM₁² − (γ−1)/2), gives the Mach number after a normal shock. This is crucial for designing supersonic inlets, as the downstream flow must be subsonic for the compressor. Engineers use it to ensure that the shock is positioned correctly and that the flow entering the engine is within acceptable limits. The shock also causes total pressure loss, which affects engine performance. By understanding the downstream Mach number, aerospace engineers can predict shock losses and design efficient supersonic propulsion systems.
- Supersonic inlet shock positioning and stability
- Engine performance impact of shock losses
- Design of shock‑trapping and bleed systems
- Supersonic combustion (scramjet) inlet design
- Supersonic vehicle aerodynamics (e.g., fighter aircraft)
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