Speed of Sound/Push in Supersonic Flight

Question

I've learned from things like the Veritasium Slink Drop and the old "break the speed of light by pushing on the end of a very long pole" thought experiment that objects have a "speed of push" which is commonly referred to as the speed of sound. My understanding of this being that information about movement can't propagate through a medium faster than the speed of sound in that medium.

So my question is: What happens to air when a supersonic jet flies through it? On the one hand it would seem that air can't move out of the way of the jet faster than the speed of sound in the air, however the jet obviously displaces the air at supersonic speeds anyway. So what's happening?

Answer 1

Here is a quote by Ernst Mach (from a public lecture):

"If the projectile moves faster than sound, the air ahead of it cannot recede from it quickly enough. The air is condensed and warmed, and thereupon, as all know, the velocity of sound is augmented until the head-wave travels forward as rapidly as the projectile itself, so that there is no need whatever of any additional augmentation of the velocity of propagation. If such a wave were left entirely to itself, it would increase in length and soon pass into an ordinary sound-wave, travelling with less velocity. "

What he's saying there is that the speed of sound ahead of a ballistic projectile actually increases due to the effect of compression -- which raises the temperature of the air in front of the projectile. The formula is of course: $$c_{ideal}=(\gamma R T)^{0.5}$$ The rest is a continuation of the quote for the sake of context (the public lecture was about the double report of ballistic projectiles from artillery):

"But the projectile is always behind it and so maintains it at its proper density and velocity. Even if the projectile penetrates a piece of cardboard or a board of wood, which catches and obstructs the head-wave, there will, immediately appear at the emerging apex a newly formed, not to say newly born, head-wave. We may observe on the cardboard the reflexion and diffraction of the head-wave, and by means of a flame its refraction, so that no doubt as to its nature can remain."

Answer 2

You are right that the air can't get out of the way in a continuous fashion. When you move an object through a fluid at a speed that exceeds the fluid's sound speed, then discontinuities may form known as shocks. The sonic boom associated with superonic jets is one consequence of the formation of these shocks.

Supersonic fluid motion isn't always associated with shocks, though. You can smoothly accelerate fluid to supersonic velocities in a setup such as a de Laval nozzle. But decelerating a supersonic flow often requires a shock. In the rest frame of a supersonic jet, upstream air is approaching supersonically, then shocks and decelerates.

So how does all this fit with the idea that information can't move in a fluid at a speed faster than sound? You have to clearly define what you mean by information carried by the fluid. One way is to think in terms of fluid pressure. If we again think of a nozzle in which fluid is accelerated supersonically, then pressure variations in the supersonic part of the flow won't affect the subsonic part of the flow. People sometimes say that the supersonic part of the flow is carried off "ballistically," and is detached (in terms of pressure) from the subsonic part.