Is there any way that someone can give me more of a conceptual explanation for the fact that the Mach number downstream of a normal shock must be less than or equal to 1? I understand the mathematical reasons and how to show this is indeed true, but not exactly why. I understand why it must be less than the Mach number upstream of the shock, but what is the reason it HAS to be less than or equal to one?


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The short answer is that sound waves travel at the speed of sound. Since the sound speed increases downstream of a shock and the flow speed decreases, this results in a Mach number less than one.

The physical reason is that without a piston, the flow of a fluid cannot exceed its communication speed due to dissipative effects. You can see my response here that explains this in more detail.

I will answer in more detail later, but my 8 month old daughter is demanding attention.

Suppose one were to create a region of homogeneous fluid flow with a uniform bulk flow speed. The bulk flow speed could be larger than the speed of sound in that medium and nothing would be wrong with that. There would be no shock waves or violations of laws, unless one inserted an obstacle into the medium. The obstacle would then act as a piston or driver, that would obstruct and deflect the flow. If the bulk flow speed exceeds the speed of sound, then a shock wave would form if sufficient energy dissipation existed. I explain the need for energy dissipation here and wrote a response that explains energy dissipation (i.e., irreversibility) in more detail here.

You can also think in terms of traffic jams. Imagine there is a highway/road where all the cars are moving with the same speed. Now you insert a bunch of crazy drivers that try to move at twice that speed. The crazy drivers would immediately run into the slower drivers and, ignoring damage done to the cars, would start to exchange energy/momentum with the slower cars (analogous to a binary particle collisions). There would be a pile up of cars at the interaction region (analogous to a shock ramp). Assume that we continually insert crazy drivers. Eventually, the average speed of traffic on the road would increase slightly at the interaction region and behind it (i.e., direction from which the crazy drivers are coming, called the downstream region). However, a larger increase would occur in the range of speeds of all the different drivers from all the collisions (analogous to increasing the heat/temperature). This interaction region would move up the road faster than the slower drivers, thus expanding the downstream region and advancing the ramp region up the road.

If we were to ride in a helicopter and follow the ramp region (i.e., like transforming into the shock rest frame), it would appear as if the slower drivers were moving towards the ramp faster than their speed relative to the road. The "mess" downstream would appear to be moving away from the ramp region at a slightly slower rate due to the increased collision rate and scattering of cars.

The point is, sound waves propagate at the sound speed because that is roughly the average thermal speed of the molecules in a gas (though in space, this gets more complicated as I discussed here). To move faster than the average speed of the particles is to incur more collisions with those slower particles, thus lowering the net speed (on average) of this hypothetical faster particle. So downstream of a shock, the Mach number is generally less than one because the forces involved act to keep the flow from moving faster than the average.

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