# Mach Number

## Introduction to the Mach Number

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**The Mach Number** is a dimensionless value useful for analyzing fluid flow dynamics problems where compressibility is a significant factor.

The Mach Number can be expressed as

M = v / c (1)

where

M = Mach number

v = fluid flow speed (m/s, ft/s)

c = speed of sound (m/s, ft/s)

Alternatively the Mach Number can be expressed with the density and the bulk modulus for elasticity as

M = v (ρ / E)^{1/2}(2)

where

ρ = density of fluid (kg/m^{3}, lb/ft^{3})

E = bulk modulus elasticity (N/m^{2 }(Pa), lb_{f}/in^{2}(psi))

The bulk modulus elasticity has the dimension pressure and is commonly used to characterize the fluid compressibility.

The square of the Mach number is the Cauchy Number.

M^{2}= C (3)

where

C = Cauchy Number

### Subsonic and Supersonic speed

- If the mach number is
*< 1*, the flow speed is lower than the speed of sound - and the speed is**subsonic**. - If the mach number is
*~ 1*, the flow speed is approximately like the speed of sound - and the speed is**transonic**. - If the mach number is
*> 1*, the flow speed is higher than the speed of sound - and the speed is**supersonic**. - If the mach number is
*>> 1*, the flow speed is much higher than the speed of sound - and the speed is**hypersonic**.

### Example - Calculating an Aircraft Mach Number

An aircraft flies at speed *500 mph *at an altitude of *35000 ft*. The surrounding temperature is *-60 ^{o}F*.

The speed of sound at this altitude and temperature can be calculated

c = [kR T]^{1/2}

= [ 1.4 (1716 ft lb/slug^{o}R) ((-60^{o}F) + (460^{o}R)) ]^{1/2}

= 980 ft/s

where

k= 1.4

R = 1716 (ft lb/slug^{o}R)

The speed of the aircraft can be calculated as

v = (500 miles/hr) (5280 ft/miles) / (3600 sec/hr)

= 733 ft/sec

The Mach Number can be calculated as

M = (733 ft/s) / (980 ft/s)

= 0.75 - the aircraft is flying at subsonic speed

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