I don't know what you mean exactly by "the principle".
The simpliest principle is that a capped pipe floats vertically in the water. That's extremely easy to demonstrate.
But making a small pipelauncher with a small rocket that demonstrates
how a full scale pipelauncher would function is to say it mildly, is very difficult. I have a swimming pool, 20' from my computer, but I don't know how to make a pipelaucher which would work in that pool. It would be easier if my pool was 20' deep. I have made a pool toy- using piece 8" diameter air duct which is 6' tall- that was fairly amusing. I have constructed a larger pipe from smaller pipe- 10' by about 8" diameter. (though I haven't made one from pipe and spacer type construction- it's a newer idea).
Let me give example of very simple small pipelauncher- though it's too small to be "fully" functional:
Model size:
Steel pipe 14" diameter .156" weighs 23.07 lbs per foot. And it has
about cu' volume per foot length.
20' total displacement: 20 cu' times 62 equals 1240 [saltwater: 64.08
=1281.6 lbs]
20' pipe weighs 461.4 lbs.
Cap: 1/8" plate steel 14" in diameter- about 5 lb
Take 2" sq piece of plate steel 1/4 to 1/2" thick
weld to middle of cap. Next drill through this piece and cap with large
drill, which you will tap so it has threads.
Weld cap to top of pipe.
Screw in pipe which has some kind of valve (could be ball valve).
Above the valve you will need fittings to attach to whatever you going
power this pipelauncher with (more about this later).
Depth of waterlevel to float weight of pipe: 7.2' at psi 3.2 psi.
So with 7.8' of water in pipe, the top of pipe floats 5' above water.
With this amount of water inside the pipe, the pipelauncher will float
vertical, and person could stand on top of the pipelauncher- and as long
as person didn't weigh over 300 lbs, the top of pipelauncher would stay
vertical and above the water.

This is pretty easy to make. The problem is the length (& weight) of
this thing makes it hard to move around. And *really* it's length is too
short to function as a pipelauncher. But if you can buy this size of pipe,
you can build the rest of it in couple of hours. Now, besides the hassle of moving this thing around and of course storing it, the next item is the cost, the pipe itself probably cost around $500- the rest of stuff is pretty cheap.
Now if instead you used an aluminum pipe, it would instead cost about $900. It would weigh less the half the weight- so two people without too much struggling could lift it. And of course it could lift more weight- two people could stand on it:)

Now as far as powering this. I figure you would need about a cup (8ozs) of liquid air. You want the liquid air to fall on the water [inside the pipe]. The water's heat might provide enough energy to turn the liquid into gas. This should blast pipelauncher out of the water, but the pipelauncher is too short, so it will be unstable- unless you can bleed enough air out of it as it lands (doubtful).
Another way to power it would be with compressed air, unless you are working with extremely high pressure (like those CO2 container used for air guns or diving tanks) you will needs a fairly large diameter pipe feeding through the top.