Area-Mach Relation Calculator

Compute A/A* for isentropic nozzle flow and the alternate Mach number — every area ratio A/A* > 1 has one subsonic and one supersonic solution.

Compressible Flow · Nozzle

Gas presets — click to fill γ

Inputs

Mach number M (dimensionless)

Specific heat ratio γ (dimensionless)

c_p/c_v — use gas presets above

Supply throat area A* — computes the local cross-section area A = A* × (A/A*)

Theory

Area-Mach number relation (isentropic, quasi-1D):

A/A* = (1/M) × [(2/(γ+1)) × (1 + (γ−1)/2 × M²)]^{(γ+1)/(2(γ−1))}

T/T₀ = [1 + (γ−1)/2 × M²]⁻¹

P/P₀ = (T/T₀)^γ/(γ−1)

Where:

A = local cross-sectional area of the duct [m²]
A* = throat area where M = 1 — minimum area in the nozzle [m²]
M = local Mach number
T₀, P₀ = stagnation (total) temperature and pressure — constant throughout isentropic flow
A/A* has a minimum value of 1.0 at M = 1; it increases for both M < 1 and M > 1

Area-Mach table — air (γ = 1.4):

M	A/A*	T/T₀	P/P₀	ρ/ρ₀
0.2	2.964	0.9921	0.9725	0.9803
0.4	1.590	0.9690	0.8956	0.9243
0.6	1.188	0.9328	0.7840	0.8405
0.8	1.038	0.8865	0.6560	0.7400
1.0	1.000	0.8333	0.5283	0.6339
1.5	1.176	0.6897	0.2724	0.3950
2.0	1.688	0.5556	0.1278	0.2300
2.5	2.637	0.4444	0.0585	0.1317
3.0	4.235	0.3571	0.0272	0.0762
4.0	10.72	0.2381	0.0066	0.0277
5.0	25.00	0.1667	0.0019	0.0113

A converging-diverging (Laval) nozzle accelerates subsonic flow through the throat (M = 1) to supersonic exit. The diverging section must match A/A* at the design Mach number. If back pressure is too high, a normal shock stands inside the diverging section and the exit is subsonic.