True, Equivalent, and Calibrated Airspeeds 

The energy equation for adiabatic flow of an ideal gas is 

_{} 

where v is the flow velocity and h is the specific enthalpy. This implies that if a flow with free stream velocity v_{0} and enthalpy h_{0} is brought to rest adiabatically, and if h_{1} is the enthalpy of the resting gas, then 

_{} 

Also, for an ideal gas we have dh/dT = c_{p}, where the specific heat c_{p} is essentially constant for relatively small changes in temperature, so this equation can be written as 

_{} 

Now, the speed of sound in an ideal gas is given by _{} where g = c_{p}/c_{v} is the ratio of specific heats and R is the gas constant, so we can express any given temperature T in terms of the speed of sound at that temperature. With these substitutions, the preceding equation becomes 

_{} 

Also, for an ideal we have R = c_{p}  c_{v}, so we can make this substitution, divide through by a_{0}^{2}, and rearrange terms to give 

_{} 

The ratio v_{0}/a_{0} is, by definition, the Mach number M of the free stream, and since temperature is proportional to the square of the speed of sound, the ratio (a_{1}/a_{0})^{2} equals the ratio of temperatures T_{1}/T_{0}. Hence we have 

_{} 

Furthermore, using the ideal gas equation p = rRT we can substitute for the temperatures to give 

_{} 

As discussed in The Speed of Sound, for an adiabatic quasistatic (i.e., isentropic) process, the ratio p/r^{g} is constant. Thus if we rewrite the above equation as 

_{} 

the first factor on the left side is unity, and we arrive at the formula for the isentropic pressure rise of an ideal fluid being brought to a stop from a Mach number M 

_{} 

Solving this equation for M and then multiplying through by the freestream speed of sound _{}, we have the formula for the true airspeed v_{true} = v_{0} as a function of the freestream static pressure p_{0}, the total (stagnation) pressure p_{1}, and the freestream static temperature T_{0} 

_{} 

If the pressure ratio p_{1}/p_{0} is known but the freestream static temperature T_{0} is not (as is sometimes the case with primitive instrumentation), we can agree by convention to simply use for T_{0} the standard sea level atmospheric temperature 

T_{standard} = 518.67 R 

When this is done, the result is called equivalent airspeed. The factor of _{} in the above equation is then simply taken to be the speed of sound at standard sea level temperature, so the equation for equivalent airspeed is 

_{} 

where 
a_{standard} = 1116 ft/sec = 661.47 knots 

Of course, the square root in this expression is simply the true Mach number, so equivalent airspeed can also be written as 

_{} 

Thus, noting that a_{0}M is the true airspeed, and letting q denote the ratio T_{0}/T_{standard}, we have 

_{} 

Sometimes we lack not only the static temperature, but the pressure ratio as well. With certain kinds of primitive instrumentation systems we can measure only the difference p_{1}_{ }_{ }p_{0} between the total and static pressures. (This difference is sometimes called the impact pressure, which is the same as the dynamic pressure 1rv^{2} for incompressible flow, but not for compressible flow.) Even if we can rewrite the ratio p_{1}/p_{0} as (p_{1}p_{0})/p_{0} + 1, we would still need to know the static pressure p_{0} (in addition to the impact pressure) in order to compute equivalent airspeed. However, lacking the static pressure, we can agree by convention to simply use the standard sea level static pressure in place of p_{0}. When this is done, we get the following formula for calibrated airspeed as a function only of the impact pressure 

_{} 

where 
p_{standard} = 14.696 psia 

Given the definition of calibrated airspeed, we sometimes need to compute it based on the actual measured values of the static pressure p_{0} and freestream Mach number M. (Of course, knowing p_{0} and M, we could compute equivalent airspeed, but convention may still force us to deal with calibrated airspeed.) For convenience, we will set g = 1.4, which is the value for atmospheric air. Then p_{1} is given in terms of p_{0} and Mach by the relation 

_{} 

Substituting this into the calibrated airspeed equation gives 

_{} 

where d denotes the pressure ratio p_{0}/p_{standard}. For values of M less than _{} this can be expanded into a convergent power series in M, the first few terms of which are 

_{} 

In terms of the true airspeed this is 

_{} 

For small Mach numbers (much less than 1) the zeroth order term of the expansion is often accurate enough for practical purposes, but for Mach numbers approaching 1 it is necessary to take account of the higherorder terms  or else simply use the exact analytical expression. 
