Isentropic Nozzle – Area–Mach Relation Calculator
Solve the compressible-flow area–Mach relation for the area ratio A/A*, or invert it numerically for Mach number on the subsonic or supersonic branch — with a full step-by-step solution and bonus isentropic property ratios.
Mach ↑ (subsonic, 0<M<1)
As M rises toward 1, A/A* falls toward 1 — the channel converges. This is the converging half of a converging–diverging nozzle.
Mach ↑ (supersonic, M>1)
Past the throat, A/A* rises again with M — the channel must diverge to keep accelerating supersonic flow (opposite of subsonic intuition).
γ ↑
A larger specific-heat ratio steepens the relation, so the same Mach number maps to a somewhat different area ratio — monatomic gases (γ≈1.667) diverge less than air for a given supersonic M.
Throat (M = 1)
A/A* always equals exactly 1 at the throat, independent of γ — this is the unique minimum-area, sonic point of the duct.
How to Use This Calculator
The area–Mach relation links three quantities: Mach number M, area ratio A/A*, and the gas property γ. Pick which one you want the tool to solve, then fill in the other two.
1. Set the gas property γ
Use a preset pill (air, combustion gas, monatomic) or type a custom value. γ must stay above 1.
2. Choose what to solve for
Pick "A/A* from Mach" if you know M and want the area ratio, or "M (sub)"/"M (super)" if you know A/A* and want the Mach number on the converging or diverging branch.
3. Enter the known value
Type into the number box or drag the slider for whichever field is still editable — the greyed-out field is the one being solved.
4. Read the result and steps
The result card shows the solved value plus engineering (×10ⁿ) notation. The "step-by-step solution" panel below it shows exactly how the number was reached.
5. Switch to Advanced for more depth
Advanced mode adds sensitivity to a ±1% change in the known input, plus the companion T/T₀, p/p₀ and ρ/ρ₀ isentropic ratios at that Mach number.
6. Copy or reset
Use "Copy result" to grab the solved value, or "Reset" to return to the default worked example (M = 2, γ = 1.4, air).
What This Tool Offers
Built specifically around the isentropic area–Mach relation used to size converging–diverging (de Laval) nozzles and supersonic wind-tunnel sections.
Solve either direction
Go from Mach number to area ratio directly, or from area ratio back to Mach number — the relation isn't algebraically invertible, so the tool handles the inversion for you.
Dual-branch Mach solving
Every area ratio above 1 has two valid Mach numbers. Choose the subsonic (converging) or supersonic (diverging) branch explicitly, so you always get the physically correct root.
Adjustable gas properties
Quick presets for air, hot combustion gas, and monatomic gases, or enter any custom γ for a specific propellant or working fluid.
Transparent step-by-step math
Every calculation shows the intermediate term, exponent, and numerical-solver steps — not just the final number — so results can be checked by hand.
Companion isentropic ratios
Advanced mode also reports T/T₀, p/p₀, and ρ/ρ₀ at the solved Mach number — the properties nozzle designers usually need right alongside the area ratio.
Built-in sensitivity check
See how much the result moves for a ±1% nudge in the known input or in γ, useful for quick tolerance and design-margin checks.
What Is the Area–Mach Relation?
The area–Mach relation is a core result of one-dimensional isentropic compressible-flow theory. It ties the local cross-sectional area of a duct, A, to the area at the point where the flow is exactly sonic (M = 1), called the throat area A*. For any given Mach number and specific-heat ratio γ, there is exactly one value of A/A* — but for any given A/A* greater than 1, there are two possible Mach numbers: one subsonic, one supersonic. This is precisely why a rocket nozzle or supersonic wind tunnel must first converge to a throat and then diverge again to keep accelerating the flow past Mach 1.
Area–Mach Relation Formula
The relationship, in its squared form, is:
Taking the positive square root gives the form the calculator solves directly for A/A* given M:
Because M appears both outside and inside a nonlinear power term, there is no closed-form formula for M in terms of A/A* — the calculator finds it with a numerical bisection search on whichever branch you select, then checks the residual by substituting the solved M back into the formula above.
Area–Mach Relation Variables
| Symbol | Meaning | Typical range |
|---|---|---|
| M | Local flow Mach number | 0.01 – 5 (subsonic < 1, supersonic > 1) |
| A/A* | Local area divided by throat (sonic) area | ≥ 1, minimum of 1 at the throat |
| γ | Ratio of specific heats, c_p/c_v, of the gas | 1.33 – 1.667 for common gases |
How Mach Number Affects Area Ratio
The relation is not monotonic over its full range: on the subsonic branch, A/A* decreases as M increases toward 1; on the supersonic branch, A/A* increases as M increases past 1. Both branches meet at exactly A/A* = 1 when M = 1, which is why the throat is the single narrowest point of a converging–diverging nozzle.
How γ Affects Area Ratio
γ changes both the multiplier (2/(γ+1)) and the exponent (γ+1)/(γ‑1) in the formula. For a fixed supersonic Mach number, a lower γ (more complex combustion-gas molecules) generally requires a smaller area ratio than a higher γ (simpler, monatomic gas) to reach the same Mach number — one reason nozzle contours differ between rocket propellant combinations.
Area–Mach Calculation Examples
A/A* = (1/2.0) × [(2/2.4) × (1 + 0.2×2.0²)]^(2.4/0.8)
A/A* = 0.5 × [0.8333 × 1.8]^3 = 0.5 × 1.5³
A/A* = 1.6875
Bisection search on M ∈ (0, 1) for f(M) = 1.6875
M converges to 0.3722 (the "other" root of the same area ratio)
Frequently Asked Questions
The area–Mach function decreases from infinity to 1 as M rises from 0 to 1, then increases again from 1 to infinity as M rises past 1. Any A/A* greater than 1 therefore intersects the curve twice — once on each branch — which is exactly why you must specify subsonic or supersonic when solving for M.
M appears both as a simple factor and raised to a power inside a bracketed term, so the equation is transcendental in M. There is no general closed-form inverse, so the tool uses a numerical bisection method on the chosen branch and reports the converged, residual-checked value.
Nothing physical — 1 is the mathematical minimum of the relation, reached only at M = 1. The calculator flags any entered area ratio below 1 as invalid.
Yes. It assumes steady, one-dimensional, isentropic (reversible and adiabatic) flow of a calorically perfect gas — the standard idealization used for preliminary nozzle and diffuser sizing before losses are added.
They are the companion isentropic relations at the same Mach number and γ: T/T₀ = 1/(1+((γ‑1)/2)M²), with p/p₀ and ρ/ρ₀ derived from it by raising to γ/(γ‑1) and 1/(γ‑1) respectively — useful for finding local static conditions once the area ratio is known.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| A/A* | Area ratio to throat area | |
| M | Local Mach number | |
| γ | Ratio of specific heats |
What it means
The area‑Mach number relation is a key equation in compressible flow, linking the local area (A) to the Mach number (M) for isentropic flow in a nozzle. A* is the throat area where M=1. This relation shows that for a given Mach number, there is a corresponding area ratio. It is used to design convergent‑divergent nozzles to accelerate flow to supersonic speeds. The equation is derived from continuity and the isentropic relations. It also applies to diffusers and wind tunnels. Understanding this relation is essential for nozzle design, propulsion, and for analysing supersonic flows. It shows that to achieve supersonic speeds, the flow must pass through a throat and then expand.
Worked example
Area‑Mach Number Relation – Two Examples
Real‑World| Parameter | Value |
|---|---|
| γ | 1.4 |
| M | 0.5 |
| Parameter | Value |
|---|---|
| M | 2.0 |
Common mistakes
- Area‑Mach number relation (isentropic nozzle): (A/A*)² = (1/M²) · ((2/(γ+1))·(1+((γ−1)/2)·M²))^((γ+1)/(γ−1)).
- A*: Throat area where M=1.
- For M=1, A/A* = 1.
- For a given area ratio, there are two solutions: subsonic and supersonic M.
- Valid for isentropic, adiabatic flow.
Applications
The area‑Mach number relation for isentropic nozzles, (A/A*)² = (1/M²)·((2/(γ+1))·(1+((γ−1)/2)·M²))^((γ+1)/(γ−1)), describes the relationship between the local area and Mach number in a variable‑area duct. It is fundamental for designing converging‑diverging nozzles for supersonic flow. Engineers use it to size the throat area, to determine the exit Mach number for a given area ratio, and to design rocket nozzles and wind tunnels. The relation also applies to supersonic inlets and diffusers. By using this equation, aerospace engineers can design nozzles that accelerate exhaust to supersonic speeds for maximum thrust.
- Design of convergent‑divergent nozzles for rockets and jets
- Supersonic wind tunnel design (nozzle contouring)
- Design of supersonic inlets and diffusers
- Optimisation of nozzle expansion ratio for altitude
- Analysis of compressible flow in ducts
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