Making your own pendulum wave machine

This post is a continuation of a three part post started here.



To make your own pendulum wave machine, you can see from the video that you just need to place 15 pendula or so beside each other at equal separations. What is hard to see however, is how long the length of string should be. Below we'll derive a function for you to calculate the lengths of each pendulum on your machine (or just scroll to the bottom to see some precalculated lengths).

In the small angle approximation, the period of a pendulum is \[T=2\pi\sqrt{\frac{l}{g}}\implies \omega=\frac{2\pi}{T}=\sqrt{\frac{g}{l}}\] where $g$ is 9.8 m/s and $l$ is the length of the pendulum.

Since length of the pendulum depends on its location on the apparatus, we have $l(n)$ instead of a constant $l$ (where $n$ is the pendulum number). Putting it all together, \[\omega(x)=\sqrt{\frac{g}{l(x)}}.\]

But we also know from the previous post that \[\omega(x)=\frac{2\pi}{T}=\frac{2\pi \left(N+\frac{x}{d}\right)}{\Gamma}=\frac{2\pi \left(N+n\right)}{\Gamma}.\]

Isolating for $l(n)$, we have \[l(n)=g\left(\frac{\Gamma}{2\pi\left(N+n\right)}\right)^2\] where $\Gamma$ is the duration for the entire dance (60s in the video), $N$ is the number of oscillations the longest pendulum performs in the cycle (51 in the video - I always mistake this for the number of pendula for some reason), $n$ is the pendulum index (starting from 0).

As an example, we can calculate the lengths of the pendula in the video of the previous post.

Pendulum Index	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Length (m)	0.343	0.330	0.318	0.306	0.295	0.285	0.275	0.265	0.256	0.248	0.240	0.232	0.225	0.218	0.211

You can calculate your own pendulum lengths on the online spreadsheet file I've posted here.

UPDATE: I've newer much better simulations and calculators. You can find them at https://cs.stanford.edu/people/paulliu/webapps/pendulumcalc.html,
https://cs.stanford.edu/people/paulliu/webapps/pendulum.html

All that left to do is decide on the dimensions of your pendulum wave machine. For example, if I wanted to make the machine above, I'd place each pendulum above 5 cm apart, making the entire machine at least 70 cm long and 34.3 cm tall. However, one thing to note is that these pendulum wave demonstrations are made perfect by careful timing of the periods on each pendulum, not by careful measurement of the pendulum lengths. As such, make the individual pendulums on your machine adjustable! The very first design of this demonstration used 15 adjustable screws to which each pendulum tied on to. By carefully adjusting each screw and timing each pendulum, the correct number of oscillations can be tuned for each individual pendulum.

That's all there is to it, I'll leave the nitty-gritty details of the design to all you talented folks out there. They make for great gifts! I made my girlfriend one for her birthday and she loved it! OK, that was a lie. But it did get her parents to like me.