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Archive for February 2009

Mathematical Modelling Validation and Dominoes

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Some time ago now I was browsing on ArXiv.org and came across a paper on the speed of propagation of the collapse front of a domino wave (ref 1). Now what initially attracted my attention to this paper, other than the intrinsically interesting phenomenon of the domino effect, was that the paper presented a plot of the non-dimensional wave speed against a non-dimensional spacing parameter but no mention was made of any practical measurements of the speed of domino waves, and in fact in the text of the paper there was a challenge to the reader to make such measurements. In addition to this challenge the plot showing the results of the model showed what I though implausibly high wave speed at small but non-negligible domino spacings.

Not being one to pass up a challenge I thought that I might make my own measurements so I started collecting materials. Initially I thought I would video the domino wave on a digital camera, extract the frames and analyse them by hand (or rather in a graphics package) to extract the data that I needed. This seemed a natural approach as I had used it previously to settle a couple of friendly disagreements with topsquark (another moderator on the Math Help Forum). Anyway before I actually made any measurements I realised that it would be easier to record the sound of the wave and extract the frequency of domino impacts, and so deduce the wave speed. These experiments and their results ware published in two part bythe Citizen Scientist (refs 2 and 3), though there are some problems with the rendering of the equations in the second part for some reason.

However interesting the experiments and analysis technique is that is not what I want to discuss here. But this: while doing the experiments I also spent some time chasing what references I could find on the speed of domino waves. Doing this I discovered that while there was not an overabundance of published measurements of domino wave speeds there were some (ref 5, 6, 7 and probably more by now) and moreover that some of these were references in the original paper that lead to these investigations. So we are left with a paper that proposes a model of domino wave speeds that looks implausible and presents no experimental data in validation of the model, despite the authors knowing the location of such data. Furthermore when the available data is compared with the model it shows poor agreement with the model (see fig 1).

Not only does this look like at best sloppy work by the authors, it also represents sloppy work by the referee/s at SIAM Review who should have asked for some attempt at validation.

(of the models presented in the references that in ref 4 is of comparable complexity (simplicity?) as that in the paper being discussed, while those in refs  6, 7 and  8 model the physics in significantly greater detail)

Figure 1: Plot showing ref 1's model predictions (solid line) and measurements from various sources
Figure 1: Plot showing ref 1’s model predictions (solid line) and measurements from various sources including my experiments (broken line is from the model in ref 4)


1.  C.J. Efthimiou C.J. , M.D. Johnson M.D. ,   Domino Waves,  SIAM Review 49 (2007) 111-120, also  arXiv:0707.2618  (July 2007)

2.  Larham R., Measuring the Speed of the Domino Effect Using the Windows Sound Recorder, Part 1, Citizen Scientist, Nov 2007, http://www.sas.org/tcs/weeklyIssues_2007/2007-11-02/project2/index.html

3.  Larham R., Measuring the Speed of the Domino Effect Using the Windows Sound Recorder, Part 2, Citizen Scientist, Dec 2007, http://www.sas.org/tcs/weeklyIssues_2007/2007-12-07/project2/index.html

4. Banks R.B.,  Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics, Princeton University Press 1998.

5. McLachlan B.G., Beaupre G., Cox A.B., Gore L., SIAM Review 25 (1983) 403-404

6. Stronge W.J., The domino effect: a wave of destabilizing collisions in a periodic array, Proc R. Soc Lond. A 409 (1987), 199-208

7. Stronge W.J, Shu D.}, The Domino Effect: Successive Destabilisation by Cooperative Neighbours, Proc R. Soc Lond. A 418 (1988), 155-163

8. van Leeuwen J.M.J, The Domino Effect,  arXiv:physics/0401018v1 (2004)


Written by CaptainBlack

February 25, 2009 at 15:13