4
$\begingroup$

In many questions and answers about high speed aerodynamics here it is said that as speed approaches the speed of sound, there is also a decrease in air density.

Can someone explain why?.

$\endgroup$
1
  • 3
    $\begingroup$ Can you add a link to at least one such question? $\endgroup$
    – Jim
    Aug 26, 2022 at 14:38

1 Answer 1

8
$\begingroup$

We start with the Laplace equation for the speed of sound: $$a = \sqrt{\frac{\gamma \cdot p}{\rho}}$$ and differentiate and square it: $$a^2 = \frac{\delta p}{\delta\rho}$$ Now we can write: $$\delta p = \delta\rho \cdot a^2$$ Next to be replaced is the pressure differential $\delta p$, using the equation for the conservation of momentum in its differentiated form: $$\delta p = -\rho \cdot v \cdot\delta v$$ $$\rightarrow\:-\frac{\delta v}{v} = \frac{\delta\rho}{\rho} \cdot \left(\frac{a}{v}\right)^2$$ which can be written as $$-\frac{\delta v}{v} \cdot Ma^2 = \frac{\delta\rho}{\rho}$$ Interpretation: At small Mach numbers, changes in speed cause negligible changes in density, but as Mach approaches unity, both are of similar magnitude. With $Ma>>1$, changes in density will become dominant. A speed increase is always coupled to a decrease in density.

$\endgroup$
6
  • $\begingroup$ wonderful derivation! -NN $\endgroup$ Aug 27, 2022 at 18:19
  • $\begingroup$ A little bit late to the discussion, but I've been trying to figure this out for about a week now. Why do small Mach numbers cause negligible changes in density? Asked differently, what makes this relationship non-linear? Thanks. $\endgroup$
    – Wyatt
    Jan 20 at 4:13
  • 2
    $\begingroup$ @Wyatt Short answer: Physics. Longer answer: An integration over speed is involved to arrive at energy, so energy-related things like dynamic pressure or Mach effects go with the square of it. Mach = 1 is significant because that is the speed at which pressure changes travel in air. If you find this still too simple, pick a good book on molecular dynamics. And read it. $\endgroup$ Jan 20 at 8:21
  • $\begingroup$ @PeterKämpf oh okay. So it would make sense for dynamic pressure to not scale linearly. Does static pressure also not scale linearly? For me personally it's hard to understand why below Mach 0.3 density doesn't change much. (Also, are you talking about the pressure around shockwaves here? Someone on a different answer said that this answer was talking about shockwaves, so I didn't know if this one was also.) $\endgroup$
    – Wyatt
    Jan 20 at 17:19
  • $\begingroup$ @Wyatt At low Mach there is too little energy in the kinematics of gasses to affect density much. Not enough force to pull the molecules apart, if that helps. Static pressure is static, so speed has no influence on it. Here it is density: The static pressure is what is needed to keep the column of air above it suspended. Since this column shrinks when you go up, so does static pressure. This answer is not about shockwaves. $\endgroup$ Jan 20 at 18:57

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .