# Why can't supersonic flow work its way upstream?

I am reading this textbook today. And it described supersonic flow as follows:

In a supersonic flow, because the local flow velocity is greater than the speed of sound, disturbances created at some point in the flow cannot work their way upstream (in contrast to subsonic flow)

And I am confused, I don't understand why the flow cannot work their way upstream. I don't really understand what upstream or downstream mean. Can someone explain to me please?

I don't really understand what upstream or downstream mean. Can someone explain to me please?

For an observer sitting in a flying airplane, it looks as if a stream of air is rushing towards and then past him. Therefore, upstream means "in the direction of flight" and downstream means the opposite direction, the perceived direction of travel of the air.

why do fluid particles interact at the speed of sound?

In fluids (matter in the liquid or gaseous state) the molecules are not packed together in a rigid structure (that would be the case for solid matter) but have some space for moving around. The average speed of this movement is equivalent to the temperature of the fluid (more precisely: The square root of the temperature, measured from absolute zero). The average length of undisturbed motion is very small, though, because molecules are constantly bouncing into each other. A local pressure change (say, by an approaching airplane) will cause more bouncing in one direction, because some molecules will be hit and pushed away by this airplane. Now this disturbance will propagate into the fluid at the speed of their movement.

Propagation of disturbances. The circles symbolize the distance a disturbance has travelled, so bigger circles show disturbances which emanated earlier from the airplane.

At rest ("Stillstand"), the disturbances (which we perceive as sound) will propagate equally in all directions. For Mach 0.8, this is no longer true because the source of sound moves while the sound waves propagate away from it. Still, sound manages to move in all directions, so the pressure changes caused by the approaching airplane are "announced" to air upstream of the airplane.

At Mach 1.4, however, the airplane moves faster than the disturbance. It will hit air which is completely unaware of its coming. This is described by "disturbances created at some point in the flow cannot work their way upstream" in your text. The sound caused by the airplane will not reach the air upstream (= ahead) of it before itself does so.

If you want to know what consequences this has, read this answer.

• Very good. I would add one (minor) addition or disagreement. What you say also applies identically to solids as well as fluids. Imagine a steel bar flying past you faster than the speed of sound (in steel). Disturbances in that steel bar would only move downstream, not upstream, ( relative to your position), for exactly the same reasons. Commented May 15, 2022 at 12:27
• Thank you very much! The three examples really cleared up my confusions. Commented May 15, 2022 at 13:26
• @CharlesBretana Agreed, there is also a speed of sound in solids. Commented May 15, 2022 at 16:31
• One more scenario where understanding comes from understanding frame translation. Commented May 15, 2022 at 21:47
• Speed of sound in water is around 1800m/s, in aluminium around 6000m/s off the top of my head.
– Frog
Commented Aug 25, 2022 at 10:03

Because the speed of sound is actually the mean velocity of the fluid particles, that's why it is dependent on temperature. The temperature is really just another way to measure the average kinetic energy (aka velocity) of all the particles in the fluid. A disturbance moves due to the particles interacting with one another. Clearly, if the average free-stream velocity of a body in a fluid is faster than the average velocity of the fluid particles, any disturbance or change in the fluid cannot propagate upstream. The stream is moving faster in the opposite (downstream) direction than the disturbance can move in the upstream direction.

• Oh so if I understand correctly, fluid particles interact at the speed of sound, but if the free stream velocity is supersonic, all these particles would get pushed toward the downstream direction. Is that correct? And just one more question, why do fluid particles interact at the speed of sound? Sorry if it's a really dumb question. Commented May 15, 2022 at 3:46
• @Yihong Zhu it’s not so much that the particles interact at the speed of sound, but the speed of sound is the speed at which particles interact. Hopefully it’s obvious when stated that way.
– Frog
Commented May 15, 2022 at 10:02
• @Yihong, Particles interact at the speed of sound because sound is really nothing more than movement of the fluid particles. Sound moves through the interaction of fluid particles, so it of course moves at the speed of the particles. Commented May 15, 2022 at 12:16
• When something, like our vocal chords, or a speaker cone, pushes on a fluid, (actually, on anything), the particles it pushes directly on move a tiny bit, and push on the particles next to them, and so on,.. and so on... Commented May 15, 2022 at 12:18
• Not my downvote but "[...] is actually the mean velocity of the fluid particles" is not quite right. It's related, yes, but not an actual 1:1 relation. If it's up for debate, cite your source(s) as I have checked mine. Also the dependence on temperature is a square root dependence so it's not a strong dependence. Thanks.
– user14897
Commented May 15, 2022 at 12:30

Unfortunately air and waves moving in it are invisible. But luckily in nature waves exist that can be actually easily seen. These are the waves generated by perturbations at the interface between a liquid and a gas: the waves generated by a duck swimming in a lake 😄

When the duck makes some movement at zero speed (maybe it is just moving the neck or stretching the wings), it generates waves on the surface of the water which spread away with a certain constant speed. By a physical and mathematical point of view, exactly the same happens when a body moves in the air: it generates waves that spread away with the speed of sound. This correspond to the left part of this picture, taken from another answer, that I repost here:

Now, when the duck starts to swim, the waves tend to get closer in front of it and rarefy on the back. The duck "moves toward" the wave front. This correspond to the picture in the middle.

The nicest part is anyway when the duck overcome the "sound barrier" and swim faster than the wave's speed: the waves that it generates are now slower than itself and therefore they lag behind it i.e. they

cannot work their way upstream.

Downstream, they coalesce forming the typical Mach cone.

Beware of the duck approaching at supersonic speed. Source: https://i.pinimg.com/originals/94/d5/7e/94d57e09290b005273aa26313d355443.jpg

If you don't want to wait for some duck, just play with a twig 😉

• Actually, the angle of the wake on water is independent of the speed of the object which causes it and is $\text{arctan}(1/\sqrt{8}$) ≈ 19.47° to either side. Commented Oct 26, 2022 at 21:56
• @PeterKämpf: the duck appreciate this precision🦆 Anyway never spoke about angles in the answer or any relationship with the actual speed... Plus, at zero speed the waves look concentric and not forming any angle, so there it seems to be a relationship with speed. Commented Oct 26, 2022 at 23:08