What I wanted to figure out was how the inherent nonlinearities in pendula would affect the analysis of the motion.
First the basics
First I’ll assume that the pendula are true Simple Harmonic Oscillators with angular frequencies given by where l is the length of the simple pendulum. To get the cool wave effect, especially a recurrence of all the pendulum in their initial states, you need to have all the periods be related in a special way. The easiest way to think about this is to consider the time between recurrences, T, to be an integer number of the period of every pendulum, though the specific integer can (and will) be different for each one. That way, after time T, each one will have undergone an integer number of oscillations bringing them back to the same initial state.
Now, given that and that we find that
Ok, now we’re in business. All we have to do is pick some reasonable n’s and we’ll have a set of pendula that will exhibit the recurrences we want. Typically the n’s are chosen to be successive integers, and where to start and end has to do with the range of lengths you want. I’ve found some cool simulations with n between 40 and 50 seem to bare a resemblance to the real life versions:
You can see the recurrence works well and then the gif simply loops (there’s 200 frames in the animated gif). By the way, I cheated a little and actually ran a simulation of true pendula instead of SHO ones but kept the initial angles quite low.
Ok, so now for the nonlinearity. What do I mean by that? It’s really just that the potential energy well for the pendulum bob is not a parabola but rather the bottom of a sine function that mostly looks like a parabola but isn’t. If you move up far enough, you get far enough away from the true parabola that the isochronism of the pendulum no longer holds.
So how can we show that? The most straightfoward way (not counting actually doing it with a real pendulum, of course) is to calculate the time it takes for the bob to move from the bottom to the edge of its motion and then multiply by four to get the full period length. If that calculation gives you a different answer for different initial angles, then you don’t have an isochronic system.
The next bit is a little nasty, so if you’re convinced that the nonlinearity exists and just want to see the simulation results, skip down to “I can’t be bothered by the math”.
How do you do the calculation mentioned above? One way is to figure out the speed the bob is moving on its circular orbit as a function of angle and then add up the times it takes to move every little . Ok, so we know the energy of the pendulum is the gravitational potential energy it has right before you let go:
Great, with that we should be able to get the speed as a function of angle:
Awesome! Now we can figure out the time to traverse a small and then integrate for the quarter period:
or, more simply:
The cool thing about that equation is that the familiar is present but the rest is ugly, especially the integral. Note that the integral doesn’t have an analytical form, but it does have a name. Mathematica calls it the EllipticF function with some crazy constants thrown in.
For those of you following along, you probably were hoping for something like . Well, take a look at this plot of times the crazy integral:
The pink line is at and the blue curve is the actual result. It’s clear (I hope) that at low angles you get the result we’re used to. But note how much it differs as you get up to 90 degrees for the initial angle!
I can’t be bothered by the math
Ok, so where does that get us? Well, at first I figured that the nasty nonlinearity would make the wave pendulum not really work for large initial angles. However, I realized that it depends on how you pull the various pendula back. If you pull them all the same angle, you still get recurrence! Why? Because they all get an adjustment to their periods, but, since they all will start at the same angle, the ugly integral contribution will be the same multiplicative contribution for everyone! It just means that the recurrence will happen later but it’ll still happen.
Here’s an example where I pull them all out to 90 degrees. It goes by fast but you can see a hint, at least, of the recurrence:
But what if you pull it back like it looks like they do in the video above. It looks to me like they pull them all back the same horizontal distance. Since they all have different lengths, they all achieve different angles. Well, then you’re in trouble:
There is never a recurrence because there’s no longer an integer relationship among the periods because the ugly integral contribution is different for every pendula.
Where to go from here
I’d love it if someone who has taken the time to build one of these awesome demonstrations to try to pull their pendula back the two ways I’ve described here but with large angles to see if these calculated predictions can be seen in the real world (where it’s much more interesting)