Software Musings

White Water Relaxation

Posted on: 26/06/2006

Just north of where I live is a small village called Wakefield. Although Wakefield is largely occupied by artists, it also has a system which solves very large sets of highly complex and linked partial differential equations in real time.The problems it solves are ones of fluid mechanics, including highly turbulent flow. Whenever I’m visiting Wakefield I always try to find time to go to stand and watch it at work, marvelling at its sophistication, simplicity, accuracy and method of displaying its results.

Equations associated with turbulent flow are known to be computationally intensive and readers who have grown up since digital computers started to dominate the desktop might be puzzled at having so powerful a machine in so remote a community. The answer is, of course, that it’s not a digital computer. It’s an analogue computer. And it’s not the type of analogue computer we all used to use (here for example, is a desktop—I don’t remember many laptops).

It’s a river. I stand on the bridge and watch the water flowing down over the rocks marvelling at how the water knows ahead of time that a rock is coming and moves to one side to avoid it. The world may be all that is the case but all that is the case is all that can be expressed as differential equations. If the world is the totality of the equations then my river is solving those equations in real-time and displaying the answer to me in a graphical form.

So two of our most difficult mathematical problems are being solved by an inanimate river with no program other than the initialisation of the equations. I was reminded of the graphical display of complex functions, something with which the river seems to have no problem, the other day when reading Don and Walker’s paper Time-Frequency Analysis of Music. This paper compares the spectrograms of the last 60 seconds of the Firebird Suite with the song Buenos Aires, a warbler’s “song” and a piece of fractal music. For each of these, it is necessary to display the essentially three-dimensional spectrogram (time, frequency, intensity) on a two-dimensional page—with limited success.

So, my river acting as an analogue computer, of which more in a later blog, correctly solves the complex set of differential equations which represent its part of the universe and elegantly and accurately displays the results.

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7 Responses to "White Water Relaxation"

This is a response to this and the later blog (of July 3rd) which you mention here.

My first response to this idea of reality as an instance of a class was that of the old dilemma of a fist trying to grasp itself. If (the class is defined by the equations and is inaccessible to us) and if (these equations are that class [i.e., the real world]) then these equations are also inaccessible to us and the equations we have must be only an instance of the underlying class of equations. This is not to belittle the equations we have. It’s just that the fist which entirely consists of itself cannot actually grasp itself and in the same way these equations which we have, and even if they are all we have, cannot quite encompass the underlying class of equations which are the real world.

My next response was about the river as an analogue computer solving the equations. As a piece of the real world the river has no choice in what happens to it. By taking many measurements, we can describe that certainty with equations. (Some of the equations – possibly all of them – will have been derived from sources other than the measurements taken from this particular river.) We can then take these equations and write a piece of software which will input data about the water and the obstacles encountered by it, process the data via the equations and output a graphical display which closely imitates what the river does.

We can go a stage further and imagine the software output in the form of another river just like the original (3D, wet display – there’s a thought!) However close we get to the original it will never quite be identical. More to the point, we won’t be able to tell if it is or not. However much we measure, we are up against the infinite and the infinitesimal. The real world does them effortlessly; our equations won’t produce the real world, just an approximation. How do we know how close that approximation is?

Another stray thought on this one – the water doesn’t know ahead of time that a rock is coming. It doesn’t move to one side to avoid it, it crashes into it and never learns not to. But the river wreaks its revenge given long enough, by wearing the rock away and having the last laugh. You can probably hear it laughing from where you stand on the bridge.

I think, David, that you need to spend more time leaning over the bridge. If you watch debris floating down the river you will see it (and the water carrying it) move aside to avoid the stone well before it reaches it. Now, scientists have several ways of describing this (note that scientists have no tools to *explain* anything: only to *describe* it) including a concept called pressure which acts through the water in a way they describe but do not explain. This pressure can apparently be detected by the oncoming water which can then take the path to avoid the rock.

While this description is adequate for people designing bridge supports or ships I don’t see how it can be used to *explain* my analogue computer in Wakefield.

On the more general comment of my analogue computer in Wakefield, I think that you’ve missed a point. The differential equations are not a description of the world: the differential equations *are* the world and what we perceive is an expression (“instance” if you like) of those equations.

I built a computer at university that could do two dimensional Fourier transforms at the speed of light. It consisted of an aperture shaped like the thing you want to transform, two lenses and a screen.

We’ve just spent a fair bit of time in North Wales watching water crashing over and into rocks. So on comment 3, which molecules are in the know, which ones have to find out the hard way (ouch!) and which ones have to be told?

And on comment 4, comment 1 para 2 postulates exactly what you are saying but suggests that we do not actually have those equations, just instances (approximations?) of them.

[…] water seemed to me to have no choice in the matter of where it was going, despite what Chris put in one of his blogs last June. The water knows ahead of time that a rock is coming and moves to one side to avoid […]

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Disclaimer

The author of this blog used to be an employee of Nortel. Even when he worked for Nortel the views expressed in the blog did not represent the views of Nortel. Now that he has left, the chances are even smaller that his views match those of Nortel.
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