Others have already answered the question: water _is_ compressible, and yet this is mostly unrelated to pressure.
But let me give you a very simple proof that water can be compressed: it conducts sound as everyone can test in a bathtub or lake. Sound is produced by small differences in pressure that propagate as a wave. Anything that cannot be compressed cannot conduct sound. And indeed, even other types of compression such as pushing on a stick or pumping water into one end of a pipe are moving through the system at its speed of sound!
However, there are no proper substances that truly are incompressible; as more abstract examples, vacuum and black holes could maybe be used, both of which indeed do not compress nor conduct sound.
So realistically water CAN be compressed, just not well. It’s so little they basically just say it can’t be compressed because it’s so small its not worth mentioning. So how does water pressure work? It works because water CAN’T be compressed really! If water was compressed like air then it wouldn’t come out of the pipe as easily as you’d need to shove much more water in it to push it out.
The pressure of something coming out of lets say like a tube isn’t because it can be compressed. It’s because it is being pushed through. You’d just have to push more to get the same amount of pressure if it was air vs water because the air could be compressed.
You are mixing up compression and pressure they are two separate things.
Let’s say you want to increase the pressure of the water to 400 PSI you compress it, it’s volume won’t go down that much.
To do the same thing with air, the volume will go down a significant amount, you’ll have small space of 400 psi air.
Industry actually uses that to their advantage you can fit a heck of a lot of compressed gas in a compressed gas cylinder.
You don’t have to compress something to put it under pressure. Compression is what happens when you increase the pressure of something, pressure isn’t the result of something being compressed.
If what you are compressing is very squashy, like gas, then the pressure doesn’t change much as you change the volume a little bit. But with a fluid that is largely incompressible, the pressure will vary wildly under very small changes in volume. If you were to take a large steel tank, fill it with water under high pressure, seal it and remove the pump, only a tiny amount of water would flow out of the tank when you opened it, and most of that flow would come from the stretch in the steel tank, not the water’s compression. Fill a tank with air under pressure in the same way, and lots of air will flow out.
With the pressurised water systems like domestic water, the water isn’t under pressure because of compressed water – the pressure is either provided by gravity, by having a high water tower that provides pressure from gravity; or in smaller systems, a sealed tank is half filled with water and half with air, and the high-pressure but squashy air pushes down on the water, putting it under pressure. When you turn on the tap, water flows all the way from the water tower or the pressurised tank to your tap.
Lets say you have two syringes, one full of air, one full of water. You seal them so nothing gets out.
If you push on the air plunger, you can move it in, and if you move it in halfway, the pressure is about double what it started at(for more info on that ‘about’ search for adiabatic vs isothermal compression.)
If you push on the water plunger, the pressure rises in proportion to how hard you push, without changing in volume.
Pressure is just how much force there is pushing against an area, compression is about how much stuff you shoved into a given volume.
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