# what is the actual meaning, role and aim of $p_0$, $V_{a_0}$ and $V_c$ for Mach-meter?

Roskam, book "Airplane Aerodynamichs and Performance" at subparagraph: "2.5.2 HIGH-SPEED AIRSPEED INDICATORS (COMPRESSIBLE FLOW)" uses the following three equations:

a) $\left(\frac{\gamma}{\gamma-1}\right)\frac{p}{\rho}+\frac{1}{2}V^2=\left(\frac{\gamma}{\gamma-1}\right)\frac{p_t}{\rho_t}$

b) $\frac{p}{\rho^\gamma}=\frac{p_t}{\rho^\gamma_t}$

c) $V_a^2=\gamma\frac{p}{\rho}$ ,

and, after some steps, it reachs $M^2=\frac{2}{\gamma-1}\left[\left(\frac{p_t-p}{p}+1\right)^\frac{\gamma-1}{\gamma}-1\right]$ (called eq.n (2.44) )

At this point the book says: "...From this equation it can be seen that when $(p_t-p)/p$ is measured, the Mach number can be calculated. An instrument which measures both $(p_t-p)$ and the static pressure, $p$, to indicate the Mach number through Eqn (2.44) is called a Mach meter. Note, that a conventional airspeed indicator will only measure $(p_t-p)$.

When, in the calibration of an airspeed indicator, $(p_t-p)/p$ is replaced by $(p_t-p)/p_0$ and $V_a$ is replaced by $V_{a_0}$ , the resulting airspeed is called the calibrated airspeed, $V_c$ ...."

After some steps it yelds an equation with $M^2$ depending by variables $\gamma$, $\delta$ * , $V_c$ , $V_{a_0}$ and it tells that Fig 2.6, obtained with this last equation, represents a plot of true Mach numbers versus calibrated airspeed for constant pressure altitudes.

My QUESTION is: what is the actual meaning, role and aim of $p_0$, $V_{a_0}$ and $V_c$ ? I suppose Roskam considers they play an experimental role to set the MachMeter before it will fitted to use, but I have to understand exactly how to use those variable. How to obtain them?

Note that equation with calibrated variables is not something realistic about the flight because for example we have $\frac{(p_t-p)}{p_0}$, so we have in the same formula $p$ and $p_0$, something is changed and something is not changed, so it's not represent a specific physical situation but only something artificial in order to calibrate the Mach-meter, I suppose. But ... can someone explain me the details? Maybe that Machmeter is measuring $V_c$ every moment during flight?

• $\delta$ is $p/p_0$
• Hi, is this still relevant to you? If so could you add the definition of p0 p_0, V_a0, and V_c given by Roskam? My assumption would be that it has something to do with the fact that in order to measure "the" static ambient pressure you need to make some assumptions with respect to the static pressure distribution across the hull of the aircraft (or probe). – rul30 Jan 14 '18 at 10:37