Newton at Niagara Falls

Factual Questions
21

I’ll hold your ankles…but, seriously: I agree the formation can probably be ignored, but the brink of the American Falls is quite jagged with some places protruding out a little further than others…as opposed to the “T” profile. Of course, despite the jagged edge, the “curtain” of water appears rather uniform overall.

Also, the Horseshoe Falls carries the bulk of the load - so to speak. Due to erosion, the brink has a profile in keeping with the name. It could be said to somewhat resemble the “D” shape, or perhaps a backwards “C” shape.

22

Niagara river -212,000 cubic feet/second.

Check out the site… http://www.iaw.on.ca/~falls/origins.html

Check out the sections on Erosion and Power of Niagara for all the info on the underlying rock, rate of flow, and the rate of erosion etc.

23

Wow! What a great thread! And not a single flame! I’m going to nominate you guys as the, er, poster-boys for the SDMB! :wink:

Seriously, thanks for an interesting discussion. Just wanted to let you know there are lurkers out there (who wouldn’t DARE try to get involved discussing the math!) who appreciate you!

-Melin

24

Thanks for the link funnee. I was particularly interested by the first photo “The jagged edge of the American Falls,” which can be expanded by clicking on it.
In looking at the drops seen in the center of the photo it seems clear that the water is falling in some sort of arch pattern, but that the lower part of the arch quickly approaches the nearly vertical. This would be logical because the horizontal velocity of the water would eventually overwhelmed by the exponential effects of gravity on the vertical velocity. (How quickly, however, remains an important question at the heart of the OP.)

However, looking at the top portion of a drop pictured in the foreground, it appears that the underlying rock does have a curved aspect leading into the edge of the drop. This would be a logical pattern of erosion. (I was actually surprised at the relatively rapid rates of erosion and geological change described in the website.)

Thinking of the garden hose sitting on the top railing of the deck example in the OP, after you ran enough water out of it, the bottom of the metal or plastic hose fitting would become eroded away, and then the wood (or whatever) railing of the deck until rather quickly (in a geological sense) you would have a nice channel shaped like a parabolic arch at “deck edge falls,” and the water would run down the channel without any visible arching (i.e. there would be no air space between the arch of the water and the arch of the channel. In other words, given enough time, the underlying structure should erode to conform to the water’s “natural” parabolic arch (subject, of course to variations in materials, etc.).

Another potential impact is the reduced flow over the falls due to upstream dams and hydroelectric plants. This would reduce the horizontal velocity of the water in the falls. If the prior rock channels or arches had been cut (eroded) during a period of higher flow, they would be “wider” for any given vertical distance because of the greater horizontal distance. If the flow rate decreases, the horizontal velocity over the falls will consequently decrease (assuming a constant cross-sectional area). In the century or so that the river has been artificially impeded (an eyeblink or so in geological time), the water would most likely gently roll off the “too wide” channel, rather than go spurting off the edge of the “too narrow” drop in your garden hose example.

Another factor raised by the website is the seasonal variation in river flows. Perhaps you saw the falls at a time of relatively low flows, and there might be some noticable arching at a time of high water flow?

25

My natural inclination is to get radically offensive and denounce the lot of you as being imbeciles. My natural inclination is not FAR off that mark, but patience is a virtue, and mine with you must be a virtue worthy of the gods…

All that has been said about the wearing down or ‘rounding’ of the brink at Niagara may or may not be true. Likewise, speculation as to the depth of the water at the brink at any given point is just that – the true average depth is not accurately known. (I made the mistake of accepting the average of a poster’s estimates – 5’.)

But the flows I mentioned are verifiable, as are the lengths of the falls. Even if we were to assume a 1’ depth, the maximum x-axis velocity would be increased by a proportional value of 5, to 35 fps at most. (I strongly doubt that this sort of speed ever happens.) The water falling from the ‘brink’ would very quickly match that speed in the y-axis, and AGAIN the expected ‘arch’ would suffer.

Billdo and Jinx could very well be on to something, although I haven’t the slightest idea what that might be… :wink:

Yes, regardless of depth, regardless of flow, one could expect that the brink at Niagara would be somewhat more rounded than the right-angle precipice we all may have imagined. Theoretically, that rounding could/should/would make a difference in the configuration of the the arch. In fact, it most certainly does.

To the extent that the bedrock underlying the brink is rounded, that is to the extent that the riverbed drops in the last few feet, the apparent arch of the falls is reduced. With its own fall, the riverbed reduces what apparent arch there may have been, by accomodating a certain amount of vertical acceleration.

But in any case, I can’t abandon my position that the primary reason for the near-vertical appearance of a waterfall is the fact that the water is falling near-vertically!

I don’t know why fortune smiles on some and lets the rest go free…

T

26

Perhaps my “T” isn’t a good example. Maybe a “C” would be better, where the water is flowing across the top from left to right. This phenomenon is in fact very common for waterfalls across the globe. To explain this formation, you have to start from the beginning. I’m going to make an attempt with some crude ASCII drawing, so please don’t laugh. If you aren’t using fixed width font, this is going to look terrible. I suggest you copy and paste the whole thing to a small text editor like Notepad in that case.
Legend:
@ Water

  • Softer Rock

Harder Rock

Phase 1: River forms and begins to fall off an edge of rock formation.

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
------------------------------@@@
------------------------------@@@
##############################@@@
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------------------------------@@@
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##############################@@@
##############################@@@
Phase 2: The Soft rock is easily eroded…

@@@@@@@
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–@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
-------------@@@@@@@@@@@@@@@@@@@@
##############################@@@
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##############################@@@
----------------------------- @@@
---------------------------- @@@
----------------------------- @@@
##############################@@@
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Phase 3: Finally the riverbed is depleted of the soft rock and now consists mainly of hard rock, and the rock formation below the fall turns into a “C”.

@@@@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
##############################@@@
############################# @@@
############################ @@@
------------------------- @@@
------------------------ @@@
------------------------- @@@
############################ @@@
############################# @@@
So there… If there were more layers below, the softer rock will still be eroded faster, leaving the harder layers sticking out.

I hope that wasn’t too ugly…

27

Ewe… get your Notepad out I suppose…

28

TBone, I agree with you that the water falling off the waterfall quickly goes to a near-vertical drop for any reasonable value of horizontal velocity. However, my understanding of the part of the the question that I am answering (your conclusion about what I’m trying to say and whether it makes any sense may vary) is “why doesn’t the water cascade off of the falls with a large air pocket behind it?” I think that the roundedness of the cliff edge makes a big difference in the way the water behaves, changing it from a problem in pure ballistics into one involving some combination of ballistics, geology and hydrodynamics.

Which brings to Zor’s illustrations (which are much improved by putting them in a fixed width font). I hadn’t thought about a stratified geology of the river bank. My assumption was that the river was some theoretical homogeneous rock, though of course it isn’t. I’m not sure how this would affect the ballistics/hydrodynamics of of the waterfall. As I understand your illustration the rapid erosion of the soft rock and the adhesion of the water to the cliff face would actually draw the water back in for a bit to a more than vertical position, totally screwing up anything like an arch. Interesting.

I realize that I am hopelessly out of my depth in here in this combined discipline problem, and the only way to resolve the issue is to actually go over Niagra Falls in a barrel. You first.