Machmeter

A Machmeter is an aircraft pitot-static system flight instrument that
shows the ratio of the true airspeed to the speed of sound, 
a dimensionless quantity called Mach number. This is shown on a Machmeter as a decimal fraction.
An aircraft flying at the speed of sound is flying
at a Mach number of one, expressed as "Mach 1".
Use.
As an aircraft in transonic flight approaches the speed of sound, 
it first reaches its critical mach number, where air flowing
over low-pressure areas of its surface locally reaches the
speed of sound, forming shock waves. The indicated airspeed
for this condition changes with ambient temperature, 
which in turn changes with altitude.
Therefore, indicated airspeed is not entirely adequate to
warn the pilot of the impending problems. Mach number is
more useful, and most high-speed aircraft are limited to a maximum operating Mach number, also known as MMO.
For example, if the MMO is Mach 0.83, then at where the speed of sound under standard conditions is , the true airspeed at MMO is . The speed of sound increases with air temperature, so at Mach 0.83 at where the air is much warmer than at , the true airspeed at MMO would be .
Operation.
Modern electronic Machmeters use information from an air data computer system which makes calculations using inputs from a pitot-static system. Some older mechanical Machmeters use an altitude aneroid and an airspeed capsule which together convert pitot-static pressure into Mach number. The Machmeter suffers from instrument and position errors.
Calibration.
In subsonic flow the Mach meter can be calibrated according to:
where:
When a shock wave forms across the pitot tube the required formula is derived from the Rayleigh Supersonic Pitot equation, and is solved iteratively:
where:
Note that the inputs required are total pressure and static pressure. Air temperature input is not required.