Enter upstream conditions. For a physical normal shock, M₁ must be greater than 1.
Assumptions: steady, one-dimensional flow; a normal shock; calorically perfect gas with constant γ; no shaft work; negligible potential energy changes.
Static pressure ratio: P₂/P₁ = 1 + (2γ/(γ+1)) (M₁² − 1)
Density ratio: ρ₂/ρ₁ = ((γ+1) M₁²) / ((γ−1) M₁² + 2)
Temperature ratio: T₂/T₁ = (P₂/P₁) / (ρ₂/ρ₁)
Downstream Mach: M₂² = (1 + (γ−1)M₁²/2) / (γM₁² − (γ−1)/2)
Total pressure ratio: P₀₂/P₀₁ = (P₂/P₁) · [ (1+(γ−1)M₂²/2)^(γ/(γ−1)) / (1+(γ−1)M₁²/2)^(γ/(γ−1)) ]
Velocity ratio: V₂/V₁ = ρ₁/ρ₂ (from continuity)
If upstream P₁ and T₁ are provided, the calculator also computes P₂, T₂, speed of sound (a = √(γRT)), and velocities (V = Ma).
- Enter M₁ (must be greater than 1) and γ.
- Optionally enter P₁ and T₁ to get absolute downstream values.
- Leave ρ₁ blank to let the ideal gas law estimate it.
- Click Calculate to display results above the form.
- Use Download CSV or Download PDF to export.
For oblique shocks, use the normal component of Mach number or a dedicated oblique shock tool.
Sample outputs for air (γ = 1.4). Values are rounded.
| M₁ | γ | M₂ | P₂/P₁ | ρ₂/ρ₁ | T₂/T₁ | P₀₂/P₀₁ |
|---|---|---|---|---|---|---|
| 1.5 | 1.4 | 0.701 | 2.458 | 1.862 | 1.320 | 0.930 |
| 2.0 | 1.4 | 0.577 | 4.500 | 2.667 | 1.687 | 0.721 |
| 3.0 | 1.4 | 0.475 | 10.333 | 3.857 | 2.679 | 0.328 |
1) What normal shock relations describe
A normal shock is a thin, nearly discontinuous region in supersonic flow where properties change rapidly. Across the shock, the flow decelerates from supersonic to subsonic, while static pressure, density, and temperature rise. This calculator uses standard perfect‑gas normal shock relations to connect upstream Mach number M₁ to downstream conditions M₂ and key ratios.
2) Inputs that control the solution
Two parameters dominate the shock jump: the upstream Mach number M₁ and the specific heat ratio γ. For air near room temperature, γ ≈ 1.4 is widely used. Higher M₁ increases the strength of the shock and amplifies P₂/P₁ and T₂/T₁. Changing γ shifts the compressibility behavior and modifies the downstream Mach number and ratios.
3) Typical data trends with Mach number
For γ = 1.4, a modest shock at M₁ = 1.5 produces about P₂/P₁ ≈ 2.46 and M₂ ≈ 0.70. At M₁ = 2.0, the pressure ratio rises to about 4.50 and M₂ ≈ 0.58. At M₁ = 3.0, P₂/P₁ ≈ 10.33 and M₂ ≈ 0.48, reflecting a much stronger compression and heating.
4) Stagnation pressure loss as a performance metric
Although total temperature remains constant for an adiabatic shock, total (stagnation) pressure drops due to irreversible entropy production. The calculator reports P₀₂/P₀₁ and a percentage loss, which is crucial for inlet design, diffusers, and propulsion systems. For γ = 1.4, P₀₂/P₀₁ decreases rapidly as M₁ increases, indicating growing losses.
5) When to enter upstream pressure and temperature
If you provide P₁ and T₁, the calculator converts ratios into absolute downstream values P₂ and T₂. This helps when preparing test‑cell reports, validating CFD boundary conditions, or estimating post‑shock loads. If density ρ₁ is not entered, it can be estimated from the ideal gas law using the supplied gas constant R.
6) Velocity and speed of sound outputs
Given temperature, the speed of sound is computed by a = √(γRT). Velocities follow from V = Ma. Across a normal shock, the velocity ratio is linked to the density ratio by continuity: V₂/V₁ = ρ₁/ρ₂. These values are helpful for estimating dynamic pressure changes and downstream residence times.
7) Entropy change and irreversibility
Normal shocks increase entropy. With the entropy option enabled, the tool reports Δs and Δs/R using Δs = cp ln(T₂/T₁) − R ln(P₂/P₁). A larger Δs corresponds to a lower P₀₂/P₀₁ and higher aerodynamic losses, especially at high M₁.
8) Practical engineering use cases
Engine inlets, supersonic wind tunnels, and shock tubes often rely on normal shock relations for quick sizing and sanity checks. Use this calculator to build a consistent set of ratios for documentation, to compare designs at different M₁, or to cross‑verify hand calculations. For oblique shocks, apply the relations to the normal Mach component or use a dedicated oblique shock method.
1) Why must M₁ be greater than 1?
A physical normal shock in a perfect gas requires supersonic upstream flow. If M₁ ≤ 1, the discontinuity does not form as a normal shock, and the standard relations are not applicable.
2) What γ value should I use for air?
For dry air near room temperature, γ is commonly taken as 1.4. At very high temperatures, γ can decrease, so use a value consistent with your temperature range and gas model.
3) Does total temperature change across a normal shock?
For an adiabatic normal shock without external work, the stagnation (total) temperature remains essentially constant. Static temperature increases because kinetic energy is converted into internal energy.
4) Why does total pressure drop across the shock?
A shock is irreversible and generates entropy. That irreversibility reduces stagnation pressure even when total temperature stays constant. The reported P₀₂/P₀₁ captures this loss directly.
5) Can I compute absolute downstream pressure and temperature?
Yes. Enter P₁ and T₁ with units, and the calculator multiplies by the computed ratios to produce P₂ and T₂. If T₁ is missing, velocity and sound‑speed outputs remain unavailable.
6) What if I do not know upstream density ρ₁?
Leave ρ₁ blank and provide P₁ and T₁. The tool estimates ρ₁ using the ideal gas law ρ₁ = P₁/(RT₁). If P₁ or T₁ is missing, ρ₂ cannot be computed absolutely.
7) Is this valid for oblique shocks?
Not directly. For an oblique shock, use the normal component of the upstream Mach number (Mₙ₁) in the normal shock relations, then transform results back to the flow direction if needed.