# How does a Mach Meter determine the speed of sound at a given altitude?

By my understanding, the Mach Number at a given altitude is calculated by dividing IAS by the speed of sound at that altitude. So how is this speed of sound calculated to display the Mach Number on the Mach Meter? Does the Mach Meter share the same pitot tube used to calculate airspeed?

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Most modern jets use an Air Data Computer (ADC) to calculate (among other things) Mach Number.

Air Data Computer

An ADC is simply a computer which accepts measurements of atmospheric data to calculate various flight related data.

A typical ADC may be connected to$^1$:

Inputs

• Static System Pressure
• Pitot Pressure
• Total Air Temperature (TAT)

Outputs (Calculated)

• Pressure Altitude
• Baro-Corrected Altitude
• Vertical Speed
• Mach Number
• Total Air Temperature
• Calibrated Airspeed
• True Airspeed
• Digitized Pressure Altitude (Gillham)
• Altitude Hold
• Airspeed Hold
• Mach Hold
• Flight Control Gain Scheduling.

Each of the inputs and outputs may be analog or digital depending on the design of the system, and are used for many purposes throughout the airplane. Each output is a purely calculated value based on the various input measurements and data stored within the unit.

To answer your question about the pitot source for the Mach Meter: Yes, they use the same pitot and static sources as the airspeed indicator.

In the case of mechanical instruments, they are both connected directly to the pitot static system.

In the case of an ADC, the pitot static system is connected directly to the ADC and then electrical signals communicate the airspeed and mach number to the electric airspeed indicator and mach meter (or EFIS), which no longer require actual pitot static connections.

# The Math

A simplified example for the Mach Number calculation$^2$ would be based on the pressure inputs:

$$Mach~number=5((PT/PS+1)^{0.2857}–1)^\frac12$$

Where:

$PT$ = Total Pressure
$PS$ = Static Pressure

The actual calculation makes corrections to the pressure data to compensate for installation errors and nonlinear sensor readings.

Note that it doesn't actually calculate the (local) speed of sound (LSS) in order to determine the current mach number, but with the TAT input and the calculated mach number, it could calculate it by calculating the outside air temperature (OAT/SAT) first:

$$SAT=\frac{TAT}{1+0.2\times{Mach}^2}$$

$$LSS=38.945\sqrt{SAT}$$

For example, let's say that the TAT is -36C (237.16K) and we are flying Mach 0.80:

$$SAT=\frac{237.16}{1+0.2\times0.8^2}=\frac{237.16}{1.128}=210.25°K=-63°C$$

$$LSS=38.945\sqrt{210.25}=38.945\times14.5=564.70knots$$

Again, these are simplified formulas because the actual ones would consider sensor error, etc.

$^1$ List of inputs and outputs obtained from Air Data Computers.
$^2$ Formula from TAT Sensor Operation and Equations.

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Thank you for the good answer! Is it possible for the Mach meter to determine speed of sound from the TAT alone, without any input from the static system? – shortstheory Feb 19 '14 at 5:59
@shortstheory Unfortunately, no but not for the most obvious reason. You need to know SAT in order to calculate the speed of sound, but in order to calculate SAT you need to know your airspeed in order to calculate the ram rise (difference between TAT and SAT) caused by flying at high speed. For all practical purposes, the ADC needs all three inputs in order to make any meaningful calculations.... – Lnafziger Feb 19 '14 at 6:04
@shortstheory I added an example calculation that could be used to determine the speed of sound. – Lnafziger Feb 19 '14 at 7:00
Thanks again! But, I have another question. Mach 0.8 at such LSS means an IAS of nearly 450 knots. That's much higher than the 250 KIAS/Mach 0.86 of my 747 in FS2004 at 40000 feet ;) Where does the difference arise from? – shortstheory Feb 20 '14 at 9:03
@shortstheory Ahhh, it's only a slight difference in terminology, but Mach 0.8 at this LSS is a TAS (True Airspeed) of nearly 450 knots, not an IAS (Indicated Airspeed). Because of the lower air density at high altitudes, the airspeed indicator reads lower than it would at the same speed down at a lower altitude. An ADC will normally calculate TAS as well (see the list of outputs above). You probably have a TAS readout somewhere in the cockpit of the 747 that you can use to verify this. :) – Lnafziger Feb 20 '14 at 9:09

An (analog) machmeter looks something like this:

So it's more like an more complex version of the airspeed indicator, in this case correcting for the altitude in the process. That being said, I found this extract apparently from an FAA publication:

Some older mechanical Machmeters not driven from an air data computer use an altitude aneroid inside the instrument that converts pitot-static pressure into Mach number. These systems assume that the temperature at any altitude is standard; therefore, the indicated Mach number is inaccurate whenever the temperature deviates from standard. These systems are called indicated Machmeters. Modern electronic Machmeters use information from an air data computer system to correct for temperature errors. These systems display true Mach number.

Most systems today use more detailed data from sensors to give a correct value through a variety of (complex) calculations.

A little more discussion is available on PPruNe.

Side note: Speed of sound ($a$) itself is solely determined by temperature (that being said, you are able to determine it from pressure, as pressure is a function of temperature) hence the problem with the analog system above.

For air:

$$a=\sqrt{R{\gamma}T}~m/s$$

Where:

$R=287$ Specific Gas constant [dimensionless]

$\gamma=1.4$ Specific heat ratio [dimensionless]

$T=$ Absolute temperature [K]

Remember that you're reading off indicated airspeed [IAS] in knots in the cockpit, which is not the same as True airspeed [TAS] converted to m/s, in case you're trying to work out your mach speed manually ($M=\frac{TAS}{a}$)

For use without knowledge of airspeed & temperature, Wikipedia gives the following formula for subsonic flows:

$$M=\sqrt{5((\frac{P_T}{P}+1)^\frac27-1)}$$

Where:

$P_T=$ Total Pressure

$P=$ Static Pressure

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The gas constant of dry air (R_d) is not dimensionless, it has units of J kg-1 K-1. You are correct on the other units so that RgammaT have units of m2 s-2, and the correct units m/s when the square root is taken. – casey Feb 18 '14 at 19:55

A Machmeter does not determine the speed of sound. It doesn't even need to:

$$Mach~Number=\frac{P_T-P_S}{P_S}$$

Mach number is simply the ratio between total pressure minus static pressure, divided by the static pressure.

Here is why:

$$Mach~Number=\frac{TAS}{LSS}$$

The Mach number is true airspeed versus the local speed of sound

$$TAS=IAS\sqrt{T}\div\sqrt{P}\div16.97$$

Converting indicated speed to true speed, we need to multiply with the square root of absolute temperature (in °K)

$$LSS=38.94\sqrt{T}$$

Also the local speed of sound is directly proportional with the square root of absolute temperature (in °K)

If you divide $TAS=IAS\sqrt{T}\div\sqrt{P}\div16.97$ by $LSS=38.94\sqrt{T}$, the $\sqrt{T}$ will cancel eachother out.

$$Mach~Number=\frac{IAS}{\sqrt{P}x}$$

IAS we already have, it's dynamic pressure minus static pressure, and $P$ is just static pressure, or, like I said in the beginning:

$$Mach~Number=\frac{P_T-P_S}{P_S}$$

See, no thermometer... only dynamic and static pressure.

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