Question How does pressure increase with depth in water?

[steinbil]

This might be a stupid question, but I'd rather expose my lacking understanding of physics than not learning...

I have a very simplistic understanding of the physics behind the increase in pressure as water depth increases. As I understand it, it is simply the weight of water on top you at any given depth - just like I think of atmospheric pressure as the weight of air above me at any certain place at a given time. However, given this very simplistic understanding, I'm struggling to understand why the contour of the "water container" does not affect this pressure.

Why doesn't the first shape lead to more pressure at the bottom than the second one?

And while you are mesmerized by my mspaint skills, here comes the real reason I'm wondering:
Why is the pressure at depth in a cave the same as it would be in open water?

 

[tursiops]

Because water is a fluid and pressure is transmitted equally in all directions. The only thing that matters is the distance vertically (the direction of gravity) to the surface, not whether there is some tortuous path to get to the surface. Does this help?

 

[Angelo Farina]

Your question is not trivial.
I teach applied physics at the university (at an Engineering faculty) and this question often arises among my students.
The first thing to understand is the Pascal principle, which states that pressure is a scalar quantity, not a vector.
When in a fluid there is some pressure, this will exert a force always perpendicular to any surface in contact with the fluid.
So, albeit the hydrostatic pressure is caused by gravitational force (which always act vertically, pointing down), the resulting pressure is not pointing down. If you have an horizontal plate underwater, the water pressure on the upper surface of the plate will press it down, but the pressure acting on the lower surface of the plate will press it upwards, so in the end the resulting force on the plate is zero.
And the same happens to any submerged body, wathever its shape.
Now let's move from the force acting on a submerged body to the force acting on the container full of water.
In this case the net result of all the forces acting on the surfaces in contact with water will NOT be zero, it will be equal to the weight of the water in the container.
And the local force acting in each portion of the wet surface depends on his depth (giving the magnitude) and orientation of the surface (as the force will always be perpendicular to that surface).
Now let's move to what happens inside the water itself.
In each point, you should think that the water above you is not a liquid, it is a stack of coins.
There are many stacks of coins, all close together, but not touching. The pressure between each coin and the one just below it is given by the weight of all the coins above, divided by the surface of a coin.
So, at a given depth, the pressure is constant everywhere.
But remember, the coin model is wrong, as it describes a force pointing down, whilst thanks to mr. Pascal pressure has no direction.
So, even if in some point you are "under the rocks", so you have less water above you, you still perceive the higer pressure propagating horizontally from points where the water column is not interrupted up to the water surface.
So the pressure reached in such points pressurizes all points at the same depth.
All this works only if water is not flowing, hence this is called hydrostatics, ensuring that pressure is given by:
p=rho×g×H
where rho is water density (1000 kg/m3), g is gravity's acceleration (9.81 m/s2, usually approximated to 10) and H is depth (in m).
So at 10m depth you get a pressure of 100000 Pa, that is 1 bar. This must be added to atmospheric pressure at the surface, which is another bar.
I hope that this makes the topic more clear.
But, as said, fluid dynamics is counter-intuitive, and these questions are perfectly reasonable and legit.

 

[steinbil]

Because water is a fluid and pressure is transmitted equally in all directions. The only thing that matters is the distance vertically (the direction of gravity) to the surface, not whether there is some tortuous path to get to the surface. Does this help?

That does help my understanding somewhat. It is not so much the path of water that confuses me. What I'm struggling to understand is that the actual amount of water above doesn't affect the force of pressure below - only the vertical distance.

@Angelo Farina
Thank you for the step by step explanation. I will look up the Pascal principle and read through your explanation again. By the way, what happens when there is flow? Is the water pressure at depth different in the spring caves of Florida than the caves of Mexico?

 

[rjack321]

By the way, what happens when there is flow? Is the water pressure at depth different in the spring caves of Florida than the caves of Mexico?

Any plausible diving type flow has no effect on pressure. In hydroelectric turbines, orifices with high velocity flow and such, that's another matter in fluid dynamics that you dont need to be concerned with.

 

[tursiops]

the actual amount of water above doesn't affect the force of pressure below - only the vertical distance.

Correct, because that is the direction of gravity, so there is no force coming from any other direction.

 

[OTF]

It helps to think of the weight of the water above as "squeezing" all of the water below it rather than "pushing down".

 

[lowwall]

What I'm struggling to understand is that the actual amount of water above doesn't affect the force of pressure below - only the vertical distance.

I don't think this is correct.

For example, let's assume a 10m tall cylindrical tank the top of which is buried 30m below ground level. The tank is completely filled with water and is connected to the surface only by a very narrow pipe. If the pipe is full of air, then I think we can agree the absolute pressure at the bottom of the tank is very close to 2 atmosphere (surface pressure + 1 atmosphere of water).

Now if we fill the pipe with water, what happens to the absolute pressure at the bottom of the tank? Does it suddenly become 5 atmospheres? It seems more likely to me that it becomes 2 atmospheres + a little bit where the little bit is the weight of the water in the pipe divided by the surface area of the tank.

If I'm wrong about this, I can't wait to find out how.

 

[tursiops]

I don't think this is correct.

For example, let's assume a 10m tall cylindrical tank the top of which is buried 30m below ground level. The tank is completely filled with water and is connected to the surface only by a very narrow pipe. If the pipe is full of air, then I think we can agree the absolute pressure at the bottom of the tank is very close to 2 atmosphere (surface pressure + 1 atmosphere of water).

Now if we fill the pipe with water, what happens to the absolute pressure at the bottom of the tank? Does it suddenly become 5 atmospheres? It seems more likely to me that it becomes 2 atmospheres + a little bit where the little bit is the weight of the water in the pipe divided by the surface area of the tank.

If I'm wrong about this, I can't wait to find out how.

What if your 10m tall cylindrical tank were just the diameter of the little pipe connecting it to the surface. Would your conclusion about the pressure at the bottom of the tank once the pipe is full of water change?

 

[gmerick]

How close to the water tower do you live? Water 130 feet up in the tower exerts a pressure of 4 bar or 60psi at the hydrant at the base of the tower. That same 60psi is at your kitchen sink 2 miles away. The only pump is the one that fills up the water tower.