# Why is this equation for TAS only valid for subsonic flow?

I came across the equation for calculating True airspeed (TAS) on Wikipedia:

$$\text{TAS} = a_0 \sqrt{\frac{5 T}{T_0} \left[ \left( \frac{q_c}{P} + 1 \right)^{2 / 7} - 1 \right]}$$

But this equation is apparently valid for only subsonic flows.

Why is that? Does this have something to do with pressure?

That equation works for subsonic flow only because you have replaced the Mach number of the original TAS formula:

$$\text{TAS} = a_0 M \sqrt{\frac{T}{T_0}}$$

with the formula of Mach number for subsonic flow:

$$M = \sqrt{\frac{2}{\gamma - 1} \left[ \left( \frac{q_c}{p} + 1 \right)^\frac{\gamma - 1}{\gamma} - 1 \right]}$$

which exploit the simple form of Bernoulli's formula, working only for incompressible flows, i.e., subsonic flows. Check the Mach number and the Bernoulli formula pages for further details.

• Actually I have not gone through the derivation of this equation and how it exploits the basic form of Bernoulli's equation. The formula was just mentioned under high speed flow and yet it was mentioned (for subsonic flows). I'll look into this. Ty Mar 31 at 3:55

Probably because Supersonic flow is totally different than subsonic flow. Subsonic, the movement of one molecule, (just from it's kinetic motion due to temperature), is faster than the flow velocity, so the effects are felt by other molecules downstream before the flow reaches them. This is in fact the speed of oscillations in the fluid, i.e., the speed of sound. The molecules in front of the flow therefore, move away from the ones pushing them before the flow reaches them. The flow is incompressible.
In supersonic flow, otoh, the flow is faster than the speed of the molecules, so the molecules don't feel the effects of the flow until the flow itself reaches it. The flow is, therefore, compressible, since the molecules "bunch-up" in front of the flow. This bunching up is in fact the shock wave created by the Supersonic flow.

• Here we read that sub and supersonic flows are different, and see a comparison of their characteristics. But the explanation of how the above invalidates the equation is missing Mar 30 at 14:27
• Exactly..I am aware about those characteristics too. But that doesn't answer my question. Thanks anyways. Mar 30 at 18:46
• Hey, Why is the formula for Chemical reactions different than for Nuclear Reactions? They are different, so the formulas are different. You didn't ask why either the subsonic formula is as it is, (how it is derived from the physics), or why the supersonic formula is as it is, (how it is derived from the physics), you just asked why they're different. Mar 31 at 1:39
• I did not ask why they are different. I asked why the formula is no longer valid in the other case. Never have I asked why are the two flows different and respective characteristics. Anyway the question has been answered. Thanks Mar 31 at 3:49
• Well, you're welcome, but the answer you selected doesn't answer that question either. Simply showing the other equation doesn't answer the why. The only thing that answers the why as you seem to mean it is to describe the derivation of the two equations, which would be based on the physics - i.e., the characteristics of the two different flows. Apr 1 at 14:36