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 v₀ and enthalpy h₀ is brought to rest adiabatically, and if h₁ 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 γ = 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 a₀², and re-arrange terms to give

The ratio v₀/a₀ 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 (a₁/a₀)² equals the ratio of temperatures T₁/T₀. Hence we have

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

As discussed in The Speed of Sound, for an adiabatic quasi-static (i.e., isentropic) process, the ratio p/ρ^γ is constant. Thus if we re-write 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 = v₀ as a function of the freestream static pressure p₀, the total (stagnation) pressure p₁, and the freestream static temperature T₀

If the pressure ratio p₁/p₀ is known but the freestream static temperature T₀ is not (as is sometimes the case with primitive instrumentation), we can agree by convention to simply use for T₀ 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 a₀M is the true airspeed, and letting θ denote the ratio T₀/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 p₁ – p₀ between the total and static pressures. (This difference is sometimes called the impact pressure, which is the same as the dynamic pressure (1/2)ρv² for incompressible flow, but not for compressible flow.) Even if we can re-write the ratio p₁/p₀ as (p₁–p₀)/p₀ + 1, we would still need to know the static pressure p₀ (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 p₀. 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 p₀ and freestream Mach number M. (Of course, knowing p₀ and M, we could compute equivalent airspeed, but convention may still force us to deal with calibrated airspeed.) For convenience, we will set γ = 1.4, which is the value for atmospheric air. Then p₁ is given in terms of p₀ and Mach by the relation

Substituting this into the calibrated airspeed equation gives

where δ denotes the pressure ratio p₀/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 higher-order terms – or else simply use the exact analytical expression.

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