# Fuel efficiency of supersonic planes

I stumbled upon an interesting plot; in particular, the dependence of wave drag on the Mach number: It is curious to see that the drag coefficient drops so abruptly in the supersonic regime, but I am even more curious if the total drag force acting on the airplane also drops, i.e. if the plane starts massively accelerating in the supersonic regime.

I did some quick analysis of the problem. The drag force is defined as $$F_{\text{drag}} = C(v) v^2 ,$$ where $$C(v) = A C_D(v) \rho(v)/2$$ is the newly introduced drag constant with the drag coefficient $$C_D(v)$$, flight medium density $$\rho(v)$$, and cross-sectional area $$A$$.

Assuming these relations, one can taylor expand the equation for the drag force to find that

$$\Delta F_{\text{drag}} = C'(v)v^2 \Delta v + 2Cv \Delta v.$$

As $$C'(v)$$ is clearly negative in the supersonic regime, this means that the total drag force drops if $$C'(v) < -\frac{2C(v)}{v}.$$

It seems that this criterion could indeed be satisfied in air, as the right hand term in the inequality is a fairly small number.

Is there anyone that could elaborate further on what actually happens to the drag force as the plane breaks the sound barrier? Do the plane engines lower their power, and if they didn't, would they be under too much heat load?

• Concorde used afterburners to accelerate from M0.9 to M1.7. It then turned off the burners and cruised without them. – BowlOfRed May 13 at 22:19
• @BowlOfRed Has Concorde done that to be able to break the sound barrier at all, or only to minimise the time spent in the transonic regime? – Akerai May 13 at 22:27
• Post this on the aviation stack exchange and ask Peter Kaempf for an explanation! He understands this stuff. – niels nielsen May 14 at 0:41
• The piston (the plane here) is driving a nonlinearly steepening sound wave and creating a substantial wake with a cross-section much larger than the piston during this transition. This is why condensation cones arise while transonic, i.e., the wake's reduced pressure changes the dew point/temperature ratio and water vapor condenses. After passing the sonic point, the effective cross section of the piston reduces greatly and flow patterns change to accommodate a supersonic piston (e.g., see discussion at physics.stackexchange.com/a/457894/59023 for some related details). – honeste_vivere May 14 at 14:54
• @honeste_vivere, I think you answered something that isn't my question :) I understand the process, I am just curious as to the magnitudes of the different effects here. – Akerai May 14 at 15:08