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Part 1
Ron Larham
Abstract: These papers describe a method of measuring the speed of the domino effect using the apparatus that was available at zero marginal cost in the Larham household. The basic technique is to record the sound of the domino wave using a microphone attached to an old laptop running the windows sound recorder under Windows 98. The frequency of domino collapse was then determined using Fourier analysis of the resulting wav file. Part 1 describes some background and the experimental set-up. Part 2 will present and discuss the result.
Introduction
While browsing through the arXive.com pre-print service, I came across a pre-print of a paper by Efthimiou and Johnson [1] on the speed of the domino effect published in SIAM Review.
This paper seemed strange in a number of ways, one of which was that it proposed a model of the sequential collapse of a linear array of dominoes, and used this to calculate the speed of the wave of collapse of the array. But the paper did not give or cite the results of any experiments to measure the speed with real dominoes. Now I could not believe that no one had ever measured the speed of the domino effect. So I started a search for other papers where speeds were reported and started planning to make my own measurements.
It was no surprise that there was indeed an extensive literature on the speed of domino waves. Some of this literature is: [2], [3], [4], [5] and [6], the references in these sources seem to comprise a near complete bibliography of domino wave research. A number of these do quote experimentally determined speeds. However, suspicions were raised about the number of mistakes and omissions in some of these reports. (This does not include the authors of [2], [3] and [4], which are rather good).
In the mean time I had procured some dominoes from our local hypermarket and was planning to record the domino wave as a video using a digital camera, then to determine the wave speed by analyzing the frames of the video. I have used this technique before for a number of phenomena so knew this would work (probably), but this is a relatively cumbersome process. While thinking about how to proceed, I realized that my computers have built-in sound recorders, and we also have an old PC mike laying around. This suggested that maybe the frequency of the domino impacts could be extracted from a sound recording of the domino wave, then, with knowledge of the domino spacing, I could derive the wave speed. The signal processing involved in the analysis of this data was to be done using the Euler [7] numerical analysis package, which is open source software and so also incurs no additional cost.
This approach had the added attraction of being very similar to what I do for my day job (sonar signal processing algorithm design). Another satisfying aspect of making these measurements was that I already was in possession of all the necessary equipment. This conforms to one of the main principles that I apply when conducting kitchen table experiments; the acquisition of apparatus should have negligible impact on our household finances. This is an important consideration as the raising of smallish children represents a black hole for money, and I have minimal amounts left for expenditure on my interests.
Experimental Set Up
The measurements were conducted on our kitchen table, which provided for a domino array of length up to about 1.2 meters and also space for the laptop and microphone. Two sets of double six dominos were available, so the maximum number of dominoes that could be used was 56. However, the practical limit was determined by the number that could be fitted in the available space at the inter-domino interval required for each run, so the actual number ranged from 56 down to about 27. The general arrangement can be seen in Fig. 1.1, which is a photo of the set up for an early proving run with three different domino spacings in the same array. The table was covered with a vinyl table cover, which provides a sufficiently high friction surface to stop the dominoes slipping to any significant extent during collapse.
Figure 1.1. The experimental set up.
The microphone was placed about 0.3 meter from the axis of the domino array, close to the mid-point. The computer placed conveniently so the experimenter could reach both the keyboard and one end of the array. Sound levels were checked by clicking fingers at an appropriate point while monitoring the recorder display (and also checking the levels subsequently in the data analysis stage). A run consisted of setting up the array using a meter rule to position the dominoes at the desired spacing, starting the recorder, then toppling the first domino. When all the dominoes had fallen, the recorder was stopped.
Data Analysis
Once a run was done the data was saved to a "wav" file for subsequent analysis. Figure 1.2 is a plot of an unprocessed recording, which shows a number of recorder introduced artifacts. The most obvious of these are the zero offset and the AGC (Automatic Gain Control) action. It would be nice to be able to switch the AGC off, but, unfortunately, at present I am unable to do so. For the form of analysis that I perform on this data, these artifacts are not show stoppers, but may limit use of this recorder in future experiments where they may be a problem. Note that due to a different interpretation of the wav file format specification between the windows sound recorder and data analysis language, I had to modify the provided wav file utilities so that the wav files produced by the recorder could be read. The modified code is available at the Euler help group [8] on Google Groups.
Figure 1.2. Plot of raw data for run with domino pitch of 0.045 m.
The AGC problem could also be overcome by deducing the form of the AGC, and applying an inverse AGC to the signal to recover an estimate of the signal before AGCing (this will be corrupted by the effects of applying the inverse AGC to the digitization noise). How well this will work I don't know, but it may be interesting to try. On the other hand, I may upgrade my laptop before I have time.
The first thing done with the data was to clip out the part containing the domino signature for further analysis. Figure 1.3 shows a plot of the result.
Figure 1.3. Data of interest clipped from that shown in Fig.1.2.
From the plots we can see that the signature of the dominoes falling comprises a sequence of near periodic clicks as one domino hits the next. The clicks are embedded in the additional noise generated by the collapse of the array. This is clear near the beginning of the plot in Fig. 1.3 but becomes confused later. The signature is still present in the confused part of the spectrum, as is demonstrated by deleting the earlier part of the recording and repeating the analysis. The clicks, on detailed examination, appear to be at the domino resonant frequency near 8 kHz, or this may be an aliased image of the resonant frequency above the Nyquist rate of ~11kHz. In addition, there is a significant amount of 50 Hz hum, as can be seen in Fig. 1.4, which is the low frequency part of the spectrum of the signal shown in Fig. 1.3. This is probably due to the low energy fluorescent lighting in the room.
Figure 1.4. Low frequency part of the spectral power density plot of the data in Fig. 1.3.
We wish to measure the frequency of occurrence of this signature, and one traditional way of doing this is DEMON analysis (Detection of Envelope Modulation On Noise). It is difficult to find an easily available reference on DEMON. They are either non-existent in the open literature or in journals that are going to charge you an arm and a leg for a copy. The best freely and openly available reference I can find that covers DEMON is reference [9].
The frequency is measured by rectifying the signal and then performing a Fourier analysis of the rectified signal. This works as the information encoding the near periodic pulses or clicks is transferred from being a modulation of the frequencies emitted by the dominoes to base band. The analysis can be done exactly as described above, and the measurements can be extracted from this relatively unprocessed data. However, the wanted signal is clearer after some additional filtering. The clicks are close to a resonant frequency of the dominoes at about 8 kHz, so we filter out all frequencies other than a band around this frequency before rectification. Figures 1.5 and 1.6 show the DEMON spectra for the unfiltered and filtered data of Fig. 1.3, respectively.
While the amplitudes of the peaks corresponding to the frequency of dominoes falling are the same order in the plots, they are much clearer in the second (note in both plots the harmonics of the frequency spike near 20Hz are clearly visible).
Figure 1.5. Demon spectra of data in Fig.1.3.
Figure 1.6. DEMON spectrum of data in Fig.1.3 after filtering to a band centered on resonant frequency
I conclude this section by observing that we have a method of measuring the speed of a domino wave that we can use to generate validation data for mathematical models of domino array collapse speed. In Part 2 I will report my experimental results and compare them with similar results from the literature and with the model predictions from references [1] and [6].
References
[1] C. J. Efthimiou, M. D. Johnson, Domino Waves, SIAM Review 49 (2007) 111-120.
[2] W. J. Stronge, The domino effect: a wave of destabilizing collisions in a periodic array, Proc R. Soc Lond. A 409 (1987), 199-208.
[3] W. J. Stronge, D. Shu, The Domino Effect: Successive Destabilisation by Cooperative Neighbours, Proc R. Soc Lond. A 418 (1988), 155-163.
[4] J. M. J van Leeuwen, The Domino Effect, arXiv:physics/0401018v1 (2004).
[5] B. G. McLachlan, G. Beaupre, A. B. Cox, L. Gore, SIAM Review 25 (1983) 403-404.
[6] R. B. Banks, Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics, Princeton University Press 1998.
[7] EULER home page: http://mathsrv.ku-eichstaett.de/MGF/homes/grothmann/euler/.
[8] Google Groups: Euler group, http://groups.google.co.uk/group/EulerHelp?hl=en.
[9] A. Kummert, Fuzzy technology implemented in sonar systems, IEEE Journal of Oceanic Engineering 18 (1993), 483-490. (Also available at: http://www.fuzzytech.com/e/e_a_kumm.html) 
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