Calibrated Airspeed from Impact Pressure

The pitot tube feels the total pressure of the air meeting it. Subtract the static pressure and what is left is the impact pressure $q_c$. Calibrated airspeed turns that pressure into a speed by asking: at what speed would standard sea-level air give the same impact pressure, with the compressibility of air counted? It is the airspeed an indicator would show if it had no instrument error.

The constants are the standard sea-level state. $a_0=340.294$ m/s (661.4788 kn) is the speed of sound, from $\sqrt{\gamma R T_0}$ with $\gamma=1.4$ and $T_0=288.15$ K, and $p_0=101\,325$ Pa is the pressure, both from the U.S. Standard Atmosphere, 1976. The air is treated as a perfect gas with a ratio of specific heats of 1.4, and the exponent 2/7 and the 5 inside the root come from the isentropic relation for that gas. At low speed the formula collapses to the familiar $\sqrt{2q_c/\rho_0}$ with $\rho_0=1.225$ kg/m3: 100 Pa gives 12.78 m/s (24.84 kn), and compressibility lowers that by 0.02 percent. At half the speed of sound the full formula gives 170.15 m/s from 18 868 Pa, where the simple formula would put the speed 3 percent too high.

NACA Report 1135 supplies the published check. Its Table I lists the static pressure over the total pressure as 0.8430 at Mach 0.50, 0.6560 at Mach 0.80 and 0.5283 at Mach 1.00. At the standard sea-level state those correspond to impact pressures of 18 870.7 Pa, 53 133.8 Pa and 90 469.4 Pa, and the page returns 330.74 kn, 529.18 kn and 661.4788 kn, which are 0.5, 0.8 and 1 times $a_0$. The table's four digits limit the match to about 2e-4. Exactly at Mach 1 the impact pressure is 90 476.05 Pa. For a simple case, 5 000 Pa gives 89.573 m/s, or 174.116 kn.

The formula holds for subsonic speeds, below 340.294 m/s of calibrated airspeed. Above that a shock stands in front of the pitot tube and the Rayleigh pitot formula takes over. Calibrated airspeed is the same as the true airspeed only at standard sea level; at altitude the true airspeed is higher, and the equivalent-airspeed page has the density step. The report itself does not use the name calibrated airspeed; the formula is derived from its isentropic relation.