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...but if we're going to dismiss inconvenient constraints, "Mach I without a vacuum" seems like a good place to start.

OK, let's discuss Mach 1 without a vacuum. A condition that I had not through, but which actually makes surprising sense as a design limit with the system that I am suggesting.

The Chapman–Jouguet condition says that a supersonic shock wave travels at the speed of sound in the air behind it. The ideal gas law says that this speed is almost entirely determined by temperature, not pressure. If you were to go anywhere above Mach 1, the temperature behind the shock wave gets hotter than room temperature very, very quickly. At Mach 1 exactly you should be able to maintain a thin (~0.2 micrometers is quoted by Wikipedia as the measured value) bow shock.

That shock can trap a huge pressure differential behind it, which could allow an open container to maintain normal air pressure while moving through a much lower density of air. If you are moving below that speed, the pressure differential that you maintain is much less than if you speed it up. Therefore for efficiency you'd want the vehicles to be moving at Mach 1 relative to the gas. Assuming that the gas is dragged along, this would let you actually go slightly faster than Mach 1. But if we assume that we have a lot of tunnel and occasional objects, the air would move much closer to the speed of the tunnel than the objects moving through, and therefore the top speed would be approximately Mach 1.

The exact relationship between the air pressure of the tube and the air pressure behind that flap is very complex. But with the flap design that I suggested, at pressures above 1 atmosphere there would be leakage and an opportunity to remove air. If the pressure fell below 1 atmosphere, that opportunity would disappear. Therefore the flap system that I suggested would do a good job of letting air leak out of the tunnel at about the necessary rate to maintain 1 atmosphere inside of the open containers as the speed of the whole system increases and air leaves the tunnel. (It should not be hard to design ways to bring air into the tunnel...)

This reinforces the point that the tunnel need not be evacuated, and explains why Elon said it would run at the speed that he predicted.

To mince words, Musk used the word "tunnels" (and not "tubes"), perhaps intentionally?? Tunnels are know for leaking water (and air), while tubes (or pipelines) are implied to not leak.

At any rate, the subject of Evacuated Tube Transport Technologies (ET3) tm was disclosed in US patent 5,595,543 in 1999. For the record, Coanda's system was pneumatic (and took alot of power to move the air). Search for "evacuated tube transport" on youtube for an overview. I have not noted any claims for 'hyperloop', that are not exceeded by ET3 (tm). see www.et3.com and www.et3.net

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