One step in an argument on uk.rec.waterways to show that when a boat moves along a canal, all other things being equal, the water is moved backwards a little. This fed into an argument about locks and the net water flow if boats are worked in one direction only.

It has only my contributions, not the rebuttals or skepticism.

|C ....| A |........|......................... ////|___________|////////|///////////////////////// /////////////////////////|///////////////////////// --------------------------------------------------- |C .........................|....| B |........ /////////////////////////|////|___________|//////// /////////////////////////|///////////////////////// ---------------------------------------------------

So I say that in the top picture there is more water to the right of the mark C than there is to the left. The boat then moves from A to B to make the second picture. There is now more water to the left of the mark C than there is to the right. So, some water must have flowed to the left past mark C.

OK, I ageee that it's not a continuous flow, like a river, and indeed the water only actually moves when the boat is passing. It's quite still the rest of the time. But all of the water has moved slightly backwards compared to where it was before the boat passed. Though that is a very small movement, it might be significant. Can we see now if we can agree how much the movement is?

Or a slightly different way of saying the same thing:

Let's imagine a canal with a simple rectangular section, 30 feet wide and 9 deep, a cross-section of 270 ft^{2}. Each 1-foot length contains 270 ft^{3} of water, so 1688 gallons, so 16880 pounds, so 7.5 tons of water.

If a 20-ton boat goes past, the water, on average, gets moved 20 / 7.5 = 2 ft 8 in.

So we know how far the water move when a boat goes past.

Now imagine a lockless length of canal with the generous section we used earlier, it's a nice day and we are having a picnic somewhere in the middle. We watch the boats going past, and we see that when each boat goes right to left, the water moves 2 ft 8 in to the right. And when each boat goes left to right, the water moves 2 ft 8 in to the left.

Now we doze off for a bit, then wake up and notice that while we were asleep the water has moved 2 ft 8 in to the left. Our first thought is that another boat must have passed to the right.

But it wasn't. For some odd reason what happened was that someone drained 20 tons of water off from the end of the canal off to our left and added another 20 tons to the end of the canal off to the right. This also moves the water 20 tons or 2 ft 8 in to the left.

The two things are of course quite different. In one the water all shuffles 2 ft 8 in to the left as the boat passes and is stationary the rest of the time, but if the boat goes from one end to the other the result is that all of the water along the length has moved to the left. In the other effectively 20 tons has moved down the canal sort of all together, but still the result is that all of the water has moved a little bit, the same little bit, to the right.

The question is, in our position in the middle of the section, seeing the water has moved that little 2 ft 8 in to the left but not seeing what caused it, can we tell which occurred from the effect on the water only?

Or to put it another way, is the movement of the water from a boat passing exactly the same as the movement of water from flowing the boat's weight of water along the section?

For those getting bored, the next step will be more interesting!

Now let's stay on the banks of our canal, but let there now be a lock going up to a higher level off to the right and a lock down to a lower level off to the left.

Numbers are easier than algebra, so let's choose a size for these locks. Say they are both 72 ft long 7 ft 6 in wide and have a fall of 7 ft. That's 72 * 7.5 * 7 = 370 ft^{3} = 23625 gallons = 236250 pounds = 105 tons roughly. Let's take 100 tons for our example. Each time this lock is cycled (filled and emptied again) 100 tons of water is put into it and then 100 tons are taken out of it on the other side.

So we sit by the canal watching the water. Forgetting boats for a moment, suppose the lock to the left is filled and the lock to the right is emptied. We do this carefully of course, setting the lock below first, so that we don't lose water over any overflow weirs. Now as one lock empties and the other lock fills, 100 tons of water leaves the one on the right and eventually 100 tons enters the lock to the left. The result is that all of the water between shuffles to the left. 100 tons of water at 7.5 tons per foot (at the cross-section we assumed) is 13 ft 4 in of the canal.

So when we work the locks at each end of this pound, all of the water in the pound shuffles left by 13 ft 4 in.

There are three things that we see, sitting by the middle of the pound

1) When a (20-ton) boat goes from left to right, going up the canal, the water moves to the left by 2 ft 8 in.

2) When a (20-ton) boat goes from right to left, going down the canal, the water moves to the right by 2 ft 8 in.

3) Each time the (100-ton) locks at each end are cycled, the water moves to the left by 13 ft 4 in.

On a normal day, boats go back and forth, pushing the water back and forth so that it averages out about the same place, but every time the locks are cycled it all moves to the left by that 13 ft 4 in, feeding 100 tons to the lock to the left.

But now imagine everyone is coming back down after a waterways festival, so all the boats are going right to left and no-one is going the other way. What happens?

Each time the locks are cycled, the water moves left by 13 ft 4 in. Each time a boat moves past going to the left, the water moves to the right by 2 ft 8 in. The net movement is 10 ft 8 in to the left.

Work the locks, 13-4 to the left. Boat passes 2-8 to the right, so now at 10-8 to the left. Work the locks again, another 13-4 to the left, so now at 24-0 to the left. Another boat passes 2-8 to the right, now at 21-4 to the left. And so on, average 10-8 per cycle.

When boats are passing back and forth, the water passes left on average 13-4 per cycle of the locks. But when the boats are all going the same way the average amount the water moves per cycle is different. Specifcally, if the boats are all going down, the average movement is just a little bit less, in this case 10-8.

What's that mean? Not much, but it is interesting (to me). I doesn't say anything about the detail of what's happening at the locks, but it's that movement of the water to the left that is feeding the lock to the left that is fascinating. That appears to be 100 tons per cycle on average, but with all the boats going down it appears to be 80 tons (10 ft 8 in at 7.5 tons per foot).

There's an anomaly. The calculated movement is not what seems obvious.

*nib*