Normal Shock Pressure Ratio Calculator
Solve the static pressure ratio across a normal shock — going either from upstream Mach number to p₂/p₁, or backwards from a measured pressure ratio to M₁ — with step-by-step math, a live shock schematic, growth charts, and full downstream property recovery.
M₁ ↑ — quadratic pressure jump
Because the relation is quadratic in M₁, doubling the upstream Mach number far more than doubles the pressure jump. At M₁ = 2 the ratio is 4.5; at M₁ = 4 it is already 18.5 for air.
The shock is always compressive
For any M₁ > 1 the term (M₁² − 1) is positive, so p₂/p₁ > 1 always. A normal shock converts supersonic kinetic energy into static pressure, temperature and entropy — it can never expand the flow.
Total pressure always drops
Static pressure rises, but stagnation pressure p₀₂ is always lower than p₀₁ — the shock is irreversible. At M₁ = 2 air loses about 28% of its total pressure; at M₁ = 5 it loses over 90%.
γ sets the slope
The coefficient 2γ/(γ+1) ranges from 1.00 (γ → 1) to 1.25 (γ = 5/3). For air (γ = 1.4) it is 1.1667, which is why the air curve sits between the monatomic and diatomic extremes.
How to Use This Calculator
The normal-shock pressure relation links two quantities: the upstream Mach number M₁ and the static pressure ratio p₂/p₁, 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 p₂/p₁ (pressure ratio from Mach number) or M₁ (upstream Mach number from a measured pressure ratio) — the tool inverts the formula 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 value plus engineering (×10ⁿ) notation. The step-by-step panel shows exactly how the number was reached.
4. Check the schematic and charts
The live shock schematic shows the wave thickening as the shock strengthens, and the chart plots p₂/p₁ against M₁ for three different gases.
5. Switch to Advanced for depth
Advanced mode adds the stagnation-pressure recovery curve, a sensitivity check on a 1% input change, and the full set of downstream shock metrics (M₂, ρ₂/ρ₁, T₂/T₁, p₀₂/p₀₁).
6. Convert to real pressures, or copy
Use the upstream-to-downstream static pressure converter further down the page to turn the ratio into an absolute pressure, or "Copy result" to grab the solved value.
What This Tool Offers
Built around the normal-shock pressure relation p₂/p₁ = 1 + (2γ/(γ+1))(M₁² − 1), one of the core results of compressible-flow gas dynamics.
Solve either direction
Go from upstream Mach number to pressure ratio, or invert the relation to recover M₁ from a measured p₂/p₁.
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 subsonic outflow with M₂.
Growth-curve charts
p₂/p₁ vs. M₁ for three gases, plus an advanced total-pressure-recovery curve showing how irreversible the shock becomes.
Full downstream metrics
Advanced mode reports M₂, ρ₂/ρ₁, T₂/T₁, p₀₂/p₀₁ and the percentage stagnation-pressure loss, all from the same M₁.
Absolute pressure converter
Feed in an upstream static pressure and get the downstream static pressure in the same units — no manual multiplication needed.
Reference Tables & Conversions
An upstream-to-downstream pressure converter, plus normal-shock property tables and a γ reference for common gases.
Upstream → downstream static pressure
Normal Shock Properties for Air (γ = 1.4)
Standard normal-shock table. The pressure ratio column is exactly what this calculator computes; the remaining columns are the companion downstream metrics.
| M₁ | p₂/p₁ | ρ₂/ρ₁ | T₂/T₁ | p₀₂/p₀₁ | M₂ |
|---|
Ratio of Specific Heats for Common Gases
| Gas | γ = c_p/c_v | 2γ/(γ+1) | Notes |
|---|---|---|---|
| Helium / Argon | 1.667 | 1.250 | Monatomic — strongest pressure jump per unit M₁ |
| Steam (H₂O) | 1.330 | 1.142 | Polyatomic, moderate temperature |
| Methane (CH₄) | 1.310 | 1.134 | Polyatomic |
| Carbon dioxide (CO₂) | 1.289 | 1.126 | Polyatomic — weakest jump |
| Air (standard) | 1.400 | 1.167 | Diatomic, the standard reference value |
| Hot air / combustion gas | 1.30 – 1.33 | 1.13 – 1.14 | γ falls as temperature rises |
Pressure Ratio Quick Lookup (Air, γ = 1.4)
| M₁ | p₂/p₁ | Shock strength |
|---|---|---|
| 1.00 | 1.000 | No shock (Mach wave) |
| 1.20 | 1.513 | Weak |
| 1.50 | 2.458 | Weak–moderate |
| 2.00 | 4.500 | Moderate |
| 3.00 | 10.333 | Strong |
| 5.00 | 29.000 | Very strong |
What Does p₂/p₁ = 1 + (2γ/(γ+1))(M₁² − 1) Measure?
A normal shock is a compression wave standing perpendicular to a supersonic flow. Across it the flow abruptly slows from supersonic (M₁ > 1) to subsonic (M₂ < 1), while static pressure, temperature, density and entropy all jump upward. The pressure ratio p₂/p₁ is the headline number: it tells you how much the shock compresses the gas, and it is driven almost entirely by how fast the incoming flow is.
Normal Shock Pressure Ratio Formula
The defining relationship for the static pressure jump is:
Rearranged to solve for the upstream Mach number given a measured pressure ratio:
The companion downstream relations used for the advanced metrics are:
T₂/T₁ = (p₂/p₁) · (ρ₁/ρ₂)
M₂² = ( 1 + ((γ − 1)/2) M₁² ) / ( γ M₁² − (γ − 1)/2 )
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 |
| p₂/p₁ | Static pressure ratio across the shock | Always ≥ 1 — normal shocks only compress |
| γ | Ratio of specific heats c_p/c_v | 1.40 for air, ≈1.67 for monatomic gases |
| M₂ | Downstream Mach number | Always subsonic (M₂ < 1) when M₁ > 1 |
| p₀₂/p₀₁ | Stagnation (total) pressure ratio | Always < 1 — measures shock irreversibility |
How Upstream Mach Number Affects the Pressure Jump
The relation is quadratic in M₁, so the pressure jump grows far faster than the speed itself. Going from M₁ = 2 to M₁ = 4 in air takes the pressure ratio from 4.5 to 18.5 — a fourfold increase for a doubling of Mach number. That quadratic sensitivity is why a normal shock is such an effective (and lossy) compressor: it converts ordered kinetic energy into static pressure very efficiently in terms of geometry, but irreversibly in terms of available energy.
Normal Shock Calculation Examples
2γ/(γ+1) = 2 × 1.4 / 2.4 = 1.16667
M₁² − 1 = 4 − 1 = 3
p₂/p₁ = 1 + 1.16667 × 3 = 4.500
(γ+1)/(2γ) = 2.4 / 2.8 = 0.85714
M₁² = 1 + 0.85714 × (4.5 − 1) = 1 + 3 = 4
M₁ = √4 = 2.000
Frequently Asked Questions
A normal shock is a compression discontinuity. The second law of thermodynamics only permits it when the flow is supersonic ahead of the wave — a supersonic-to-subsonic transition. If M₁ < 1 the formula would return a pressure ratio below 1, which is physically impossible for a normal shock (that would be an expansion, and a steady expansion shock cannot exist).
Because M₁ ≥ 1 makes (M₁² − 1) non-negative and 2γ/(γ+1) is always positive, the whole second term is positive. So the ratio is at minimum exactly 1 (at M₁ = 1, where no real shock exists) and rises from there.
Yes, but only through γ, and only mildly. The coefficient 2γ/(γ+1) runs from about 1.126 for CO₂ to 1.250 for helium/argon — roughly a 10% spread. For the same M₁, a monatomic gas produces a slightly larger pressure jump than a polyatomic one.
Static pressure p is the pressure you would measure moving with the flow. Stagnation pressure p₀ is the pressure the flow would reach if brought to rest isentropically. Across a normal shock, static pressure rises (p₂/p₁ > 1) but stagnation pressure falls (p₀₂/p₀₁ < 1) — that loss is the shock's irreversibility, and it shows up as an entropy increase.
There is no strict threshold, but in practice shocks with M₁ below about 1.5 are called weak (p₂/p₁ ≈ 2.5 or less), M₁ from 1.5 to 3 is moderate, and above M₁ ≈ 3 the shock is described as strong — pressure ratios exceed 10 and stagnation pressure losses become severe.
Not directly. An oblique shock is solved by decomposing the flow into components normal and tangential to the wave and applying exactly this normal-shock relation to the normal component M₁ sin β. This page covers only the normal-shock case, where β = 90°.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| p2/p1 | Downstream-to-upstream pressure ratio | |
| γ | Ratio of specific heats | |
| M1 | Upstream Mach number |
What it means
The normal shock pressure ratio describes the jump in pressure across a shock wave perpendicular to the flow. It is derived from the conservation of mass, momentum, and energy across the shock. This ratio increases with upstream Mach number. Normal shocks occur in supersonic flows, e.g., at the intake of a supersonic aircraft or in a diverging nozzle. The pressure rise is accompanied by a drop in velocity, increase in temperature, and entropy rise. This relation is essential for designing supersonic inlets and for understanding shock‑wave interactions. It also appears in the design of shock tubes and in aerodynamic heating estimates.
Worked example
Normal Shock Pressure Ratio – Two Examples
Real‑World| Parameter | Value |
|---|---|
| γ | 1.4 |
| M₁ | 2.0 |
| Parameter | Value |
|---|---|
| M₁ | 3.0 |
Common mistakes
- Normal shock pressure ratio: p₂/p₁ = 1 + (2γ/(γ+1))·(M₁² − 1).
- M₁: Upstream Mach number (>1).
- γ: Specific heat ratio.
- Pressure rises across the shock.
- Valid for normal shock (1D).
Applications
The normal shock pressure ratio, p₂/p₁ = 1 + (2γ/(γ+1))·(M₁² − 1), gives the pressure rise across a normal shock wave. Normal shocks occur in supersonic flows, causing a sudden increase in pressure, temperature, and density, and a drop in Mach number to subsonic. This is essential in the design of supersonic intakes, where shocks are used to decelerate airflow for the compressor. Engineers use this formula to calculate shock losses, to design variable‑geometry inlets, and to evaluate the performance of supersonic aircraft. By understanding normal shock relations, aerospace engineers can manage the thermodynamic changes across shocks and optimise engine‑airframe integration.
- Supersonic intake and diffuser design (e.g., F‑15, Concorde)
- Shock wave location and pressure recovery estimation
- Design of variable‑geometry inlets
- Supersonic wind tunnel and test section design
- Analysis of detached shocks around blunt bodies
Related calculators · Aerodynamics
Recently updated formulas
Suggest an improvement
Found a mistake? Have a better example? Let us know.
You may send up to 3 suggestions per 24 hours.