A Question of Bores
I was at a lengthy meeting this morning so naturally there popped into my mind the subject of bores. The most prominent of these that will be familiar to British folk is the Severn Bore, but they happen in a variety of locations, including Morecambe Bay (which is in the Midlands):
As you can see, a bore consists of a steep wavefront that travels a long distance without disruption, and is one manifestation of a more general phenomenon called a hydraulic jump; in a coordinate frame that moves with the wavefront, a bore is basically identical to a stationary hydraulic jump.
Anyway, I while ago I decided to set an examination question about this, which I reproduce here in severely edited form for your amusement and edification; you can click on it to make it larger if you have difficulty reading the question. With the examination season over I’m sure there are many people out there missing the opportunity to grapple with physics problems! Or perhaps not…
If you need hints, I suggest first working out how the pressure P varies with depth and then using the result to work out to work out the balance of forces either side of the discontinuity. Then deploy Bernoulli’s theorem and Bob’s your uncle!
P.S. For another hint, try the yellow pages:
In the Dark 
June 7, 2013 at 2:03 pm
Boring, Oregon, is also a twin town with Dull in Perthshire. And now the two are set to link up with Bland in Australia:
http://www.scotsman.com/the-scotsman/scotland/dull-village-potential-link-with-boring-and-bland-1-2951534
June 7, 2013 at 6:20 pm
Solitons!
June 7, 2013 at 8:16 pm
I think the question is misleading as stated. Your equation for V_1 applies in the rest frame of the “shock”, so V_1 is the speed of propagation of a jump into a stationary body of water. If the river flows more slowly than this, the bore moves upstream. But the question implies that the upstream flow speed of the river determines the height ratio of the jump – which isn’t right if you make the reasonable assumption that an unqualified “flow speed” means relative to the bank.
June 7, 2013 at 9:43 pm
I take your point. That’s what happens when you edit things quickly! The actual exam question was much longer and had more intermediate steps..
June 8, 2013 at 12:16 pm
And actually, is Bernoulli applicable? It’s for streamline flow, whereas surely there is turbulent dissipation at the discontinuity. You normally solve a shock by conserving total mass flux and momentum flux across the boundary. The latter gives you a different equation from just applying Bernoulli to a streamline on the surface of the water (i.e. at constant P).
June 8, 2013 at 1:51 pm
It is applicable, but that depends where you apply it…
January 7, 2020 at 1:31 pm
This is such an interesting thing that I have never seen done before. A wave is essentially created, that never stops. I will have to do some research and see if I can find one near me, so I can see it with my own eyes.