Mach Number — Physica

Equivalent forms

a = \sqrt{\gamma_h R T}

M = \frac{v}{\sqrt{\gamma_h R T}}

A single ratio classifies all flows: subsonic, transonic, supersonic, hypersonic.

Symbol	Name	SI unit	Dimension	Range
$v$	Object Velocity Speed of object through fluid	m/s	L·T⁻¹	0 – 3000
$a$	Speed of Sound Local speed of sound in the medium	m/s	L·T⁻¹	1 – 1500
$M$	Mach Numberoutput Velocity ratio (dimensionless)	—	1	0 – 10

Unit systems

Symbol	SI	CGS	Imperial
$v$	m/s	cm/s	ft/s
$a$	m/s	cm/s	ft/s
$M$	—	—	—

Where it holds

Defined for any flow, but its physical meaning matters most when M ≳ 0.3 (compressibility effects). Speed of sound depends on local temperature (and composition); at sea-level standard atmosphere

a \approx 340\,\mathrm{m/s}

.

Dimensional analysis

[v]/[a $] = (L\cdot T^{-1})/(L\cdot T^{-1}) = 1 \checkmark$ (dimensionless)

Discovery

Ernst Mach · 1887

Ernst Mach photographed shock waves around supersonic bullets in his 1887 experiments, and the ratio v/a was named in his honor by Jakob Ackeret in 1929 during early supersonic-flight research.

Try this

What does it mean for a jet to break the sound barrier?

An aircraft flies at 680 m/s through air where the speed of sound is 340 m/s. What is its Mach number?

Research status: stable

Real-world applications

Aircraft and missile design
Rocket nozzles and shock-diamonds
Supersonic wind tunnels
Re-entry vehicle thermal protection

Common misconceptions

Mach 1 is NOT a fixed speed — it changes with temperature (lower altitudes = warmer = higher 'Mach 1' in absolute m/s)
Breaking the sound barrier isn't a 'wall' — it's a sharp rise in drag at transonic speeds $(M \approx 0.8-1.2)$
Mach number applies in liquids too, but compressibility usually keeps M ≪ 1

Experimental verification

Mach's 1887 schlieren photographs of bullets show conical shock waves with half-angle

\mu = \arcsin (1/M)

, confirming the geometric meaning of Mach number. Modern wind tunnels and CFD reproduce shock angles and drag-divergence Mach numbers to within 1%.

Derivation

From dimensional analysis, compressibility effects in a gas depend on how fast the body moves relative to the speed at which pressure information propagates — the speed of sound,

a = \sqrt{\gamma _h\cdot R\cdot T}

for an ideal gas.

The dimensionless ratio

M = v

/a falls naturally out of the linearized compressible Euler equations, where

(1-M^{2})

multiplies the streamwise term in the small-disturbance equation, changing its character from elliptic (M<1) to hyperbolic (M>1).

Limiting cases

M < 0.3⟶ IncompressibleDensity changes are negligible; standard fluid dynamics applies.

M \approx 1

⟶ Transonic shockMixed sub-/supersonic regions form, shocks appear on the body; drag spikes (the 'sound barrier').

M > 5⟶ HypersonicAir heats so much that chemistry (dissociation, ionization) matters.

What if…

What if you fly at

M = 0.85

?

You're in the transonic regime — local pockets of supersonic flow form over the wings, causing shock-induced drag rise. Most jetliners cruise just below this.

What if temperature drops

20 ^{\circ}C

?

a drops about 3%, so the same airspeed corresponds to a slightly higher Mach number. Critical for high-altitude jets.

What if you go hypersonic (M > 5)?

Stagnation temperatures soar (thousands of K), air dissociates, and conventional aerodynamics gives way to reactive gas dynamics.

1

Supersonic jet

Given ·

v:: 680
a:: 340

Find · M

Steps

$M = v$ /a
$M = 680/340$
$M = 2$ (Mach 2, twice the speed of sound)

Result ·

M = 680/340 = 2

2

Concorde at cruise

Given ·

v:: 605
a:: 295

Find · M

Steps

At cruise altitude, lower temperature $\to a \approx 295\,\mathrm{m/s}$
$M = 605/295 \approx 2.05$
Concorde cruised at Mach 2.04

Result ·

M = 605/295 \approx 2.05

Reynolds Number→Bernoulli's Equation→