Geometry of Waterfalls

By Tom Brown

Fri, 07 Nov 2025
  • 145 reads
12 comments
 

A challenge, for our budding young engineers. Given are two problems on free falling liquid, the first being straightforward. The second I  found not so easy. Let us for terminology take water as our fluid.

We consider a waterfall, a free flowing falling stream as a (balanced) steady-state system i.e. unchanging with time. Reasonable assumptions may be made, such as uniform density and that cross-sections are circular. Our model is simplified and we will need only elementary (high-school level) mathematics and physics.
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The first question is, is there a temperature difference between the top and the bottom of a waterfall? Why? Can you calculate it?
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For the second, would the diameter d increase, or decrease with increasing h (distance as measured from above) like a “funnel”? Why?

Our problem is to find the diameter of the stream at a given h. It is given by

      d = 2 SqRt ( pi.A ) where A = V / [ SqRt (2gh) ].

A is the area of the cross section at h, V the volume of water passing through this section in one second. SqRt here stands for the square root.

As a hint the following: Use conservation of energy, conservation of mass, and an equation that relates to the geometry.
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In fact if you open the tap just a little you can already see the effect. It would actually take very simple experiments to test our findings.

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Comments

Submitted by GlosKat on

Don't Go Chasing Waterfalls ..

 

Submitted by Tom Brown on

... why not?

 

Submitted by GlosKat on

It's a song !  American R&B band TLC.

 

Submitted by Tom Brown on

Ok I thought it was Adelle! Beautiful powerful voice, asset to England. What she has against Waterfalls and Rainbows don't ask me, hope is always a good thing!

See you! Tom

 

Submitted by GlosKat on

Very close, I think perhaps you're thinking of her song 'Water Under the Bridge'.  And yes, she has an amazing voice.  Co-wrote the song too.  A very talented lady.

Can I take a stab at answering your first question ?  I think that there is a temperature difference.  I think the waterfall acts as a heat exchanger, so the falling water will be pulling the heat out of the air.  This will make the AIR temperature at the bottom lower than at the top. (Not sure whether you meant the temperature of the water or the air, or both).

 

Submitted by Tom Brown on

I was speaking of the water temperature I don't think the air temperature would differ, since the air would be "circulating" freely and remain at the constant surrounding (ambient) temperature.

My idea was more in terms of conservation of energy.

I discussed this problem on the forums recently I found the calculations difficult. All remarks are welcome as contributions, as well as those that confirm or disprove my own solutions.

If you want we could always have a discussion on the blogs there one can make sketches too and it is hard to try and do mathematics without proper typesetting.

Thanks very much for your contribution! Excellent! Tom

 

Submitted by Jane Hyphen on

Interesting, Tom. One of my daughters is a maths graduate, I'll ask her about all the equations. I'd say it's all the swishy-wishy, rushy-gushy energy on the way down, heating things up but that's as accurate as I can be. Unless of course it gets colder on the way down, in which case I'm way off..

 

Submitted by Tom Brown on

Really closer to engineering, but I think do ask your daughter Jane!

Thank-you for the comment and for taking interest! Tom

 

Submitted by Rhiannonw on

The tap stream seems to spread out. Rhiannon

 

Submitted by Tom Brown on

happy duckling!

You are referring to my blog! This here now is an idealised situation, but yes the diameter in fact decreases. The stream goes faster, and thinner.

All the best! Tom

 

Submitted by Tom Brown on

The temperature measured would increase with greater height, my reasoning is as follows:

I was thinking more in terms of the potential energy of the water at the top changing into kinetic energy as the water falls, and then flowing all faster, the potential energy lost thus changed to heat energy, (agitating particles) at the bottom, the "Pool" as such. Can't go anywhere else! Resulting in a small increase in water temperature.

As a system in practice you have constant zero netto flow of fluid mass and energy resulting in an unchanging state. One needs the assumption of steady flow of mass and energy to do the calculations, but the solution is easily adapted.

This reasoning, the height of the fall together with the specific heat capacity c and density of the fluid (water) would enable you to calculate the actual rise in temperature, as due to conservation of mechanical energy.

I have written out my explanations in a lot of detail. Oh yes, I was speaking of the water temperature at the beginning and the end, the air temperature would not differ.

The second problem, calculating the diameters is quite a bit harder.

Thanks very much, really appreciate all your contributions!

Keep well! Tom

 

Submitted by Tom Brown on

I was just wondering, as I said I only used first-year work at most, there might be much better ways to solve the second problem although I am sure my answers are correct. The calculations were very hard, but needing only the hints I gave and very elementary calculus and physics.

However the “Calculus of Variations” which is very advanced mathematics, I think could simplify finding the actual shape of the stream and with considerably less effort.

Note that the stream goes faster as it falls due to gravity, which means that since the rate of water volume passing stays the same the diameter of the stream must decrease resulting in the funnel shape. You can see this for yourself by opening the tap.

Tom Brown