How to compute the airspeed change based on the Altitude change

Question

The plot shows that: when I fly with fixed MACH, the true air speed decreases when the altitude increase; when I fly with fixed CAS, the true airspeed increases when altitude increase. So how could I compute compute how much TAS change I will get based on the altitude change when I fly with fixed MACH or CAS?

Answer 1

At the most basic level the Mach formula is: Mach = TAS/Speed of Sound

The speed of sound is a slightly non-linear function of temperature: 38.967854*SQRT(OAT+273.15) where OAT is expressed in degrees Celsius.

So:

1. TAS = Mach * 38.967854*sqrt(OAT+273.15)
2. Mach = TAS / 38.967854*sqrt(OAT+273.15)

PART II - CAS/TAS

This is about to get real... Mach <-> TAS is a bit easier because I’m mostly just accommodating temperature’s (OAT’s) affect on the speed of sound. You can get there directly by entering an observed OAT at your present altitude, or you can estimate what your OAT will be at various altitudes by using a surface observation and using the standard atmosphere (ISA) lapse rate which is -0.0019812°C per foot, or -1.9812°C per thousand feet. (Caution-For FAA written test purposes, you’re expected to use the approximation of -2°C per thousand feet.)

Converting CAS <-> TAS requires us to look at air density, which is a function of both the pressure and temperature of the air. We’ll ignore humidity to keep things “less complicated.” We’re either going to have to calculate rho (air density) or calculate density altitude. We already use density altitude in GA, and it’s a less complicated formula. So now we need:

1. Your pressure altitude in feet (what height above MSL your altimeter says you are, assuming the correct altimeter setting.) We’ll call that PA.
2. We’ll need to use the standard temperature lapse rate that I mentioned above, which is -0.0019812°C per foot. We don’t need to convert this to Kelvin since it’s a rate of change. We’ll call this LR.
3. We need to know the ISA Standard temperature for that PA. In physics, all temperature comparisons are done in Kelvin, so we have to express this in °K, which is °C + 273.15. This is going to be sea level standard temperature with the lapse rate applied: 15°C – (0.0019812 * PA) +273.15. We’ll call this STK, for Std Temp Kelvin.
4. We also need to convert your observed OAT from °C to °K, again by adding 273.15. We’ll call this OATK.
5. Density Altitude will be: DA = PA + (STK/LR)*(1-(STK/OATK)^0.2349690)

Only part way there. Now we’ll take our DA and plug it into this:

TAS = CAS / ( 1 - ( .0000068755856 * DA ) )^2.127940

Or

CAS = TAS * ( 1 - ( .0000068755856 * DA ) )^2.127940

If you put the DA formula and the Kelvin conversion all into one you end up with:

TAS = CAS / ( 1 - ( .0000068755856 * (PA + (  (15-(0.0019812 * PA) + 273.15)  / 0.0019812 )*( 1-( ( 15-(0.0019812 * PA) + 273.15)/( OAT + 273.15) )^0.2349690)) ) )^2.127940

Sample problem: PA = 10,000’; OAT = 3.5°C; CAS = 150 Kts. Should get a TAS of 177. I typed it up in Excel format so you should be able to paste this into Excel and it should work:

=150 / ( 1 - ( .0000068755856 * (10000 + (  (15-(0.0019812 * 10000) + 273.15)  / 0.0019812 )*( 1-( ( 15-(0.0019812 * 10000) + 273.15)/( 3.5 + 273.15) )^0.2349690)) ) )^2.127940

It turn out that TAS goes up by about 1.015x per thousand feet assuming the standard atmospheric lapse rate, accurate to less than 1Kt.