Originally Posted by Rivet gun
The difference between CAS and EAS is not a function of density. It is (as CabinMaster's formula shows) a function of static pressure and Mach number. The delta in the formula is pressure ratio, not density ratio.
Sorry, I was thinking of the difference between TAS and EAS.
Let me quote at length from John D. Anderson Jr's Introduction to Flight, one of the most ubiquitous textbooks on intro aerodynamics, with some change of symbolism from standard math to ASCII,
Originally Posted by Anderson, pp175-7
(Eqn 4.64) p0 = p + q This relation holds for incompressible flow only........
(Eqn 4.66) TAS = sqrt ( 2(p0 - p) / rho ). However, the measurement of atmospheric air density directly at the airplane's location is difficult. Therefore for practical reasons, the airspeed indicators on low-speed airplanes are calibrated by using the standard sea-level value of rho_s in [the equation]. This gives a value of velocity called the equivalent airspeed: (Eqn 4.67) EAS = sqrt( 2(p0 - p) / rho_s. The equivalent airspeed EAS differs slightly from TAS, the difference being the factor sqrt(rho / rho_s).
In the second quote, Anderson expressly introduces compressibility, talking about an F-16 flying at 300m/s at an altitude of 7km, a transonic speed.
Originally Posted by Anderson, pp189-90
EAS was introduced [in a previous section] for low-speed flight, where the flow is assumed to be incompressible. However, the concept of EAS has a broader meaning..... [F-16 example] Consider an airplane flying at some true airspeed at some altitude. Its equivalent airspeed at this condition is defined as the velocity at which it would have to fly at standard sea level to experience the same dynamic pressure. The equation for equivalent airspeed is straighforward ... It is EAS = TAS.sqrt(rho/rho_s).
That couldn't be clearer. Anderson does not introduce the notion of "calibrated airspeed", neither in this book nor in his Fundamentals of Aerodynamics.
Here are a couple of quotations from Shevell, Fundamentals of Flight, 1st edition.
Originally Posted by Shevell, p81-2
The speed read on a perfect airspeed indicator with zero instrument error using a static source which records true ambient air pressure is called the calibrated airspeed, V_CAL. (Eqn 6.12) V_CAL .. = sqrt( 2(p_T - p0) / rho_s, where rho_s = sea-level standard density. When compressibility effects are negligible, the calibrated airspeed is identical to another defined airspeed known as the equivalent airspeed, V_E. V_CAL = V_E = V_0.sqrt(rho/rho_s) = V_0.sqrt(sigma), where sigma is the density ratio (= rho/rho_s). Note that V_0 is true airspeed....
p_T is the total pressure, as measured by a pitot tube. p is freestream pressure. p_T is p multiplied by an algebraic function of Mach number.
Originally Posted by Shevell, p102
(Eqn 7.25) V_E = sqrt( (2/rho_s).(p_T - p).(1/(1+M^2/4+M^4/40+M^6/1600+....))). Eqn 7.25 differs from equation 6.12 for incompressible flow in the Mach number terms. The series in M is a Mach number correction applied to the scale of all airspeed indicators at sea level. At other altitudes, there are different values of Mach number corresponding to each value of V_E. The speed read by a perfect airspeed system (i.e., zero instrument and static error) with the Mach number correction based on a standard-day sea-level relationship between V_E and M is the calibrated airspeed, V_CAL. At sea level the Mach number correction is exact and V_CAL = V_E. At other altitudes, V_CAL = V_E + delta(V_C), where delta(V_C) is the difference between the true Mach number correction at the flight altitude and values based on sea level.
Shevell is somewhat misleading in that he implicitly suggests that Mach number versus AS correction is dependent on altitude, whereas of course it is dependent solely on temperature.
PBL