# Mathematics / Signal theory behind billiard ball 'wave pendulum' effect

This YouTube video shows a very interesting effect.

What is the underlying science?

I have recently started studying Fourier theory and DSP, and and trying to understand what is going on in terms of the material I've been learning. I would like to understand this from as many angles as possible.

For example, at 0:39 the signal forms two lines moving against one another, and shortly before that I can clearly see two sine waves in the signal.

This reminds me of the illustrations at http://www.dspguide.com/ch3/2.htm

PS not quite sure how to tag this

• see: en.wikipedia.org/wiki/Pendulum_(mathematics). note that each pendulum have different "rod" size and thus different oscillation – Dov Mar 8 '12 at 10:48
• Can someone edit the link in the above comment? The closing ) should be part of the link – P i Mar 8 '12 at 14:30

As was pointed out in a comment above, this is a simple consequence of the dynamics of a pendulum. There's nothing particularly signal-processing-related about this problem, just some simple physics and trigonometry.

For a pendulum that is displaced from its angular equilibrium point by an angle of $\theta_0$ with an angular velocity of $\omega_0 = 0$ at $t = 0$, for small initial angles, the motion of the pendulum can be approximated by simple harmonic motion:

$$\theta(t) = \theta_0 \cos\left(\sqrt{\frac{g}{l}}t\right)$$

where $g$ is the magnitude of the force of gravity and $l$ is the length of the pendulum. In the video, a number of pendulums of varying lengths are given an approximately equal initial displacement $\theta_0$ and released simultaneously. Once they are released, each pendulum traces out a path whose displacement angle is a sinusoid with frequency related to the pendulum's length. The lengths were carefully chosen to give the interesting visual effect that you see in the video; the frequencies of the sinusoids are multiples of some fundamental frequency, so for particular values of $t$, they line up in interesting ways.

You'll notice toward the end, all of the balls "line up" again; this is because each of the sinusoids that describe their motion have moved through an integer number of periods, so the balls are back at a common angular displacement. If there were no effects of friction, air resistance, etc., the oscillation would continue, with the same patterns repeating indefinitely.

• Thanks for the answer -- I kind of realised that each ball is like a sinusoidal oscillator of k*f_0, where k might run from 50 to 75 or something, which is sounding a lot like Fourier decomposition... – P i Mar 8 '12 at 14:33
• Of course it's signal processing related. It's a discrete-time sine wave frequency sweep which bounces off the Nyquist frequency and aliases back to DC. :) – endolith Mar 8 '12 at 14:38
• @endolith: I agree. It's just the physics that aren't really DSP-related. It's a really neat demo that can be used to teach a lot of signal processing concepts, though. Makes me want to try to build one with my kids. – Jason R Mar 8 '12 at 15:09
• @endolith, RIGHT! Now we are getting somewhere! Would you be up for elaborating on that in a separate answer? Forget the physics, I'm interested in how this can teach signal processing concepts -- how much DSP can be learned through this demonstration? – P i Mar 8 '12 at 21:19