# What changes in a normal shock?

Normal shock relations for a perfect gas: from the upstream Mach number (or M₂, or p₂/p₁) and γ, the downstream Mach number and the pressure, density, temperature, velocity and total pressure ratios.

- Page: https://www.acalculator.org/physics/normal-shock-calculator
- JSON spec: https://www.acalculator.org/physics/normal-shock-calculator.json
- Version: 6ebec366d8b2

## Default answer

Example with the default inputs (I know Upstream Mach M₁, Upstream Mach number (M₁) 2, Ratio of specific heats (γ) 1.4): For M₁ = 2 and γ = 1.4, the flow leaves the shock at M₂ = 0.57735 with p₂/p₁ = 4.5.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| by | I know | The upstream Mach number M₁, the downstream Mach number M₂, or the static pressure ratio p₂/p₁. |
| M1 | Upstream Mach number (M₁) | The Mach number of the flow before the shock: 1 or more. |
| M2 | Downstream Mach number (M₂) | The Mach number of the flow after the shock: above 0 and at most 1. |
| p | Static pressure ratio (p₂/p₁) | The static pressure after the shock divided by the pressure before it: 1 or more. |
| g | Ratio of specific heats (γ) | γ = cp ÷ cv of the gas: 1.4 for air, 1.67 for helium or argon, 1.3 for steam or carbon dioxide. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Mach number found | M₂ when M₁ is typed; otherwise M₁, to 6 significant figures. |
| M1 | Upstream Mach number (M₁) | The Mach number before the shock. |
| M2 | Downstream Mach number (M₂) | The Mach number after the shock: below 1 for M₁ above 1. |
| p | Static pressure ratio (p₂/p₁) | The static pressure after the shock over the pressure before it. |
| rho | Density ratio (ρ₂/ρ₁) | The density after the shock over the density before it. |
| T | Static temperature ratio (T₂/T₁) | The static temperature after the shock over the temperature before it. |
| u | Velocity ratio (u₂/u₁) | The flow speed after the shock over the speed before it: ρ₁/ρ₂. |
| pt | Total pressure ratio (p₀₂/p₀₁) | The stagnation pressure after the shock over the stagnation pressure before it. |
| pitot | Pitot pressure ratio (p₀₂/p₁) | The stagnation pressure behind the shock over the static pressure before it. |

## Method

M₂² = [(γ − 1)M₁² + 2] ÷ [2γM₁² − (γ − 1)]; p₂/p₁ = [2γM₁² − (γ − 1)] ÷ (γ + 1); ρ₂/ρ₁ = (γ + 1)M₁² ÷ [(γ − 1)M₁² + 2]; T₂/T₁ = (p₂/p₁) ÷ (ρ₂/ρ₁); p₀₂/p₀₁ = (ρ₂/ρ₁)^(γ/(γ − 1)) × (p₁/p₂)^(1/(γ − 1)).

## Assumptions

- A perfect gas with constant γ, in steady one-dimensional flow with no heat added and no friction (NASA).
- The shock is normal: at a right angle to the flow.
- Typed numbers are read as exact decimals; M₂ is a root rounded once; the total pressure ratios are 64-bit floats.

## Worked examples

1. by = m1, M1 = 2, g = 1.4 gives M2 = 0.57735, p = 4.5, rho = 2.66667, T = 1.6875, u = 0.375, pt = 0.720874, pitot = 5.64044, result = M₂ = 0.57735. Source: NASA Glenn Research Center, Normal Shock Wave Equations (M₁² = [(γ − 1)M² + 2] ÷ [2γM² − (γ − 1)]; T₁/T₀ = [2γM² − (γ − 1)][(γ − 1)M² + 2] ÷ [(γ + 1)²M²]; p₁/p₀ = [2γM² − (γ − 1)] ÷ (γ + 1); r₁/r₀ = (γ + 1)M² ÷ [(γ − 1)M² + 2]; pt₁/pt₀ = {(γ + 1)M² ÷ [(γ − 1)M² + 2]}^[γ/(γ − 1)] × {(γ + 1) ÷ [2γM² − (γ − 1)]}^[1/(γ − 1)]; Tt₁/Tt₀ = 1), https://www.grc.nasa.gov/www/k-12/airplane/normal.html (retrieved 2026-10-19); NASA Glenn Research Center, Isentropic Flow Equations (Eq. 6: p/pt = [1 + M²(γ − 1)/2]^−[γ/(γ − 1)]), https://www.grc.nasa.gov/www/k-12/airplane/isentrop.html (retrieved 2026-10-19).
2. by = m1, M1 = 3, g = 1.4 gives M2 = 0.475191, p = 10.3333, rho = 3.85714, T = 2.67901, pt = 0.328344, pitot = 12.061. Source: NASA Glenn Research Center, Normal Shock Wave Equations (M₁² = [(γ − 1)M² + 2] ÷ [2γM² − (γ − 1)]; T₁/T₀ = [2γM² − (γ − 1)][(γ − 1)M² + 2] ÷ [(γ + 1)²M²]; p₁/p₀ = [2γM² − (γ − 1)] ÷ (γ + 1); r₁/r₀ = (γ + 1)M² ÷ [(γ − 1)M² + 2]; pt₁/pt₀ = {(γ + 1)M² ÷ [(γ − 1)M² + 2]}^[γ/(γ − 1)] × {(γ + 1) ÷ [2γM² − (γ − 1)]}^[1/(γ − 1)]; Tt₁/Tt₀ = 1), https://www.grc.nasa.gov/www/k-12/airplane/normal.html (retrieved 2026-10-19); NASA Glenn Research Center, Isentropic Flow Equations (Eq. 6: p/pt = [1 + M²(γ − 1)/2]^−[γ/(γ − 1)]), https://www.grc.nasa.gov/www/k-12/airplane/isentrop.html (retrieved 2026-10-19).
3. by = m1, M1 = 1, g = 1.4 gives M2 = 1, p = 1, rho = 1, T = 1, pt = 1, pitot = 1.89293. Source: NASA Glenn Research Center, Normal Shock Wave Equations (M₁² = [(γ − 1)M² + 2] ÷ [2γM² − (γ − 1)]; T₁/T₀ = [2γM² − (γ − 1)][(γ − 1)M² + 2] ÷ [(γ + 1)²M²]; p₁/p₀ = [2γM² − (γ − 1)] ÷ (γ + 1); r₁/r₀ = (γ + 1)M² ÷ [(γ − 1)M² + 2]; pt₁/pt₀ = {(γ + 1)M² ÷ [(γ − 1)M² + 2]}^[γ/(γ − 1)] × {(γ + 1) ÷ [2γM² − (γ − 1)]}^[1/(γ − 1)]; Tt₁/Tt₀ = 1), https://www.grc.nasa.gov/www/k-12/airplane/normal.html (retrieved 2026-10-19); NASA Glenn Research Center, Isentropic Flow Equations (Eq. 6: p/pt = [1 + M²(γ − 1)/2]^−[γ/(γ − 1)]), https://www.grc.nasa.gov/www/k-12/airplane/isentrop.html (retrieved 2026-10-19).
4. by = m2, M2 = 0.5, g = 1.4 gives M1 = 2.64575, p = 8, rho = 3.5, pt = 0.443112, result = M₁ = 2.64575. Source: NASA Glenn Research Center, Normal Shock Wave Equations (M₁² = [(γ − 1)M² + 2] ÷ [2γM² − (γ − 1)]; T₁/T₀ = [2γM² − (γ − 1)][(γ − 1)M² + 2] ÷ [(γ + 1)²M²]; p₁/p₀ = [2γM² − (γ − 1)] ÷ (γ + 1); r₁/r₀ = (γ + 1)M² ÷ [(γ − 1)M² + 2]; pt₁/pt₀ = {(γ + 1)M² ÷ [(γ − 1)M² + 2]}^[γ/(γ − 1)] × {(γ + 1) ÷ [2γM² − (γ − 1)]}^[1/(γ − 1)]; Tt₁/Tt₀ = 1), https://www.grc.nasa.gov/www/k-12/airplane/normal.html (retrieved 2026-10-19).
5. by = p, p = 4.5, g = 1.4 gives M1 = 2, M2 = 0.57735, result = M₁ = 2. Source: NASA Glenn Research Center, Normal Shock Wave Equations (M₁² = [(γ − 1)M² + 2] ÷ [2γM² − (γ − 1)]; T₁/T₀ = [2γM² − (γ − 1)][(γ − 1)M² + 2] ÷ [(γ + 1)²M²]; p₁/p₀ = [2γM² − (γ − 1)] ÷ (γ + 1); r₁/r₀ = (γ + 1)M² ÷ [(γ − 1)M² + 2]; pt₁/pt₀ = {(γ + 1)M² ÷ [(γ − 1)M² + 2]}^[γ/(γ − 1)] × {(γ + 1) ÷ [2γM² − (γ − 1)]}^[1/(γ − 1)]; Tt₁/Tt₀ = 1), https://www.grc.nasa.gov/www/k-12/airplane/normal.html (retrieved 2026-10-19).
6. by = m1, M1 = 1.5, g = 1.3 gives M2 = 0.694249, p = 2.41304, T = 1.24732, pt = 0.9261. Source: NASA Glenn Research Center, Normal Shock Wave Equations (M₁² = [(γ − 1)M² + 2] ÷ [2γM² − (γ − 1)]; T₁/T₀ = [2γM² − (γ − 1)][(γ − 1)M² + 2] ÷ [(γ + 1)²M²]; p₁/p₀ = [2γM² − (γ − 1)] ÷ (γ + 1); r₁/r₀ = (γ + 1)M² ÷ [(γ − 1)M² + 2]; pt₁/pt₀ = {(γ + 1)M² ÷ [(γ − 1)M² + 2]}^[γ/(γ − 1)] × {(γ + 1) ÷ [2γM² − (γ − 1)]}^[1/(γ − 1)]; Tt₁/Tt₀ = 1), https://www.grc.nasa.gov/www/k-12/airplane/normal.html (retrieved 2026-10-19).

## FAQ

### What is a normal shock?

A shock wave at a right angle to a supersonic flow. Across it the flow drops from supersonic to subsonic in a very thin layer, and its pressure, density and temperature jump while its total (stagnation) pressure falls.

### How do I find the Mach number after a normal shock?

M₂² = [(γ − 1)M₁² + 2] ÷ [2γM₁² − (γ − 1)]. For air (γ = 1.4) at M₁ = 2: M₂² = (0.4 × 4 + 2) ÷ (2.8 × 4 − 0.4) = 3.6 ÷ 10.8 = 1/3, so M₂ = 0.57735.

### What is the pressure ratio across a normal shock?

p₂/p₁ = [2γM₁² − (γ − 1)] ÷ (γ + 1). At Mach 2 in air it is (11.2 − 0.4) ÷ 2.4 = 4.5; at Mach 3 it is 10.333.

### Why does the total pressure fall across a shock?

A shock is not reversible: some of the flow’s energy becomes heat, so entropy rises and the stagnation pressure drops, while the total temperature stays the same. At Mach 2 in air, p₀₂/p₀₁ = 0.72087: 28% of the total pressure is lost.

### What is the pitot pressure ratio?

A pitot tube in a supersonic flow reads the total pressure behind the normal shock in front of it: p₀₂/p₁ = (p₀₂/p₀₁) × (p₀₁/p₁), with p₀₁/p₁ = [1 + (γ − 1)M₁²/2]^(γ/(γ − 1)). At Mach 2 in air it is 5.6404.

### What γ should I use?

The ratio of specific heats of the gas: 1.4 for air and other two-atom gases, 5/3 (1.667) for helium and argon, and about 1.3 for steam and carbon dioxide.

## Sources

- NASA Glenn Research Center, Normal Shock Wave Equations (last updated May 13 2021): M₁² = [(γ − 1)M² + 2] ÷ [2γM² − (γ − 1)], the static temperature, static pressure, density and total pressure ratios, and Tt₁/Tt₀ = 1, for a compressible gas with no viscous effects in one-dimensional flow without heat addition, retrieved 2026-10-19. https://www.grc.nasa.gov/www/k-12/airplane/normal.html
- NASA Glenn Research Center, Isentropic Flow Equations (last updated May 13 2021): Eq. 6, p/pt = [1 + M²(γ − 1)/2]^−[γ/(γ − 1)], retrieved 2026-10-19. https://www.grc.nasa.gov/www/k-12/airplane/isentrop.html