# Is the mach number of an aircraft measured with respect to stagnation speed of sound?

I was reading chapter 8 of JD Anderson Aerodynamics when I came across something I find weird. I found out that there are different types of Mach number, based on what the speed of sound was taken as. Now my question is this, when we say "That plane is flying at Mach 1", is it with respect to the stagnation speed of sound, ie, (approximately 330 m/s), or is it with respect to the actual speed of sound?

The Mach number is defined with respect to the static speed of sound.

Note however that the Mach number is not measured by somehow measuring the TAS (True AirSpeed) and dividing it by the known speed of sound because this would require knowing the static speed of sound, which is temperature dependent. Instead, we derive the Mach number from the ram air pressure $$p_0$$ measured by the pitot tube and the static air pressure $$p$$ measured by the static port:

$$M^2 = \frac{2}{\gamma - 1} \left[ \left( \frac{p_0}{p} \right)^{(\gamma - 1) / \gamma} - 1 \right]$$

[...]

The velocity can be determined from $$V^2 = a^2 M^2 = \frac{a_0^2 M^2}{1 + \frac{\gamma - 1}{2} M^2}$$ but this requires knowing either the static speed of sound $$a$$, or the stagnation speed of sound $$a_o$$.

(MIT Lecture Notes on Subsonic Pitot Tubes)

You can see in the velocity equation that the Mach number $$M = V / a$$ is defined via the static speed of sound $$a$$ instead of the stagnation speed of sound $$a_0$$.

The above equations are only valid for subsonic flow. If you want to determine the Mach number in supersonic flow, you have to solve the following equation iteratively:

$$M \approx 0.88128485 \sqrt{\left( \frac{p_0}{p} + 1 \right) \left( 1 - \frac{1}{7 M^2} \right)^{2.5}}$$

(Wikipedia - Mach Number)

• Thank you for the answer! I had another doubt in the same. We know the temperature changes with altitude, and in turn the speed of sound too, right? Now the temperature measured by say a thermometer at rest at an altitude at 10km, is that the static temperature or stagnation temperature? It must be stagnation cause it is at rest right? Or am I wrong? – Betelgeuse Sep 16 '20 at 8:13
• Temperature changes with altitude and therefore also speed of sound $a \sim \sqrt{T}$. The temperature measured by the aircraft is called TAT (True Air Temperature), the probe is called a TAT Probe. The temperature measured at rest is usually called the SAT (Static Air Temperature) and cannot be measured directly (but calculated). – Bianfable Sep 16 '20 at 8:17