## Wave Pendulum Analysis

Thanks to both Frank Noschese and John Burk, I’ve been reminded recently of the very cool wave pendulum demonstration apparatus:

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 $\omega=\sqrt{\frac{g}{l}}$ 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 $\text{Period}=\frac{2\pi}{\omega}$ and that $\text{Period}=\frac{T}{n}$ we find that

$\omega_n=\frac{2\pi n}{T}$

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:

Low angle wave pendulum

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.

Nonlinearity

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 $d\theta$. Ok, so we know the energy of the pendulum is the gravitational potential energy it has right before you let go:

$E_\text{total}=m g h_\text{max}=m g l(1-\cos\theta_0)=\text{KE}_\text{anywhere}+\text{PE}_\text{anywhere}=\frac{1}{2}m v^2+m g l(1-\cos\theta)$

Great, with that we should be able to get the speed as a function of angle:

$v(\theta)=\sqrt{2 g l\left(\cos\theta-\cos\theta_0\right)}$

Awesome! Now we can figure out the time to traverse a small $d\theta$ and then integrate for the quarter period:

$\text{Period}=4 \int_0^{\theta_0} dt=4 \int_0^{\theta_0} \frac{l\,d\theta}{v(\theta)}=4 l\int_0^{\theta_0}\frac{1}{v(\theta)}\,d\theta=\frac{4 l}{\sqrt{2 g l}}\int_0^{\theta_0 }\frac{1}{\sqrt{\left(\cos\theta-\cos\theta_0\right)}}\,d\theta$

or, more simply:

$\text{Period}=\sqrt{\frac{l}{g}}\frac{4}{\sqrt{2}}\int_0^{\theta_0 }\frac{1}{\sqrt{\left(\cos\theta-\cos\theta_0\right)}}\,d\theta$

The cool thing about that equation is that the familiar $\sqrt{\frac{l}{g}}$ 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 $2\pi \sqrt{\frac{l}{g}}$. Well, take a look at this plot of $\frac{4}{\sqrt{2}}$ times the crazy integral:

Plot of ugly integral as a function of initial angle

The pink line is at $2\pi$ 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:

all pulled back to 90 degrees

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:

wave pendulum model with high initial amplitude

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)

## About Andy "SuperFly" Rundquist

Professor of physics at Hamline University in St. Paul, MN
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### 19 Responses to Wave Pendulum Analysis

1. Andy "SuperFly" Rundquist says:

Here’s a cool Mathematica Demonstrations Project about the wave pendulum:
http://demonstrations.wolfram.com/PendulumWaves/

2. matthew mcshane says:

Hello, I plan to build this to make the coolest executive ball clacker and would love to get 5 minutes of your time to talk about what the best lengths and materials you think should be used.

• Andy "SuperFly" Rundquist says:

sure, no problem. We can either discuss it here in these comments or you can shoot me an email (andy.rundquist@that-google-email-that’s-so-cool)

-Andy

3. hello~
i want to build a wave pendulum for my physics summative and i was wondering if you would take some time to answer some questions as to how to build the apparatus and what materials would be most appropiate. thank you!

• Andy "SuperFly" Rundquist says:

you bet, just put your thoughts/questions/ideas here and I’d be happy to help.

• Andy "SuperFly" Rundquist says:

I would probably use simple materials like wood and string, but being careful to mount them so that the string is free to swing a full <90 degrees each way. The trick it to get all the lengths right so making them adjustable would be smart.

• Estella says:

thanks! if you were to build one, what kind of materials would you use to build the apparatus and how would you go about it?

4. William says:

I’m building a wave pendulum as well. I was wondering what kind of set up you guys are using to make the individual pendulum adjustable in length.

Here’s a link for the length’s of the strings: http://hippomath.blogspot.com/2011/06/making-your-own-pendulum-wave-machine.html

• Andy "SuperFly" Rundquist says:

One idea I have for making the lengths adjustable would be to drill holes through the top piece and have something like a clamp above each hole that can adjust the string length (the string then feeds down through the hole).

5. William says:

oops, typo. *lengths.

6. Fran says:

I would like to build one, too. What should the distance be between the 2 strings of the bob at the top and also the distance between the next bob’s string? Is that important?

• Andy "SuperFly" Rundquist says:

Hi Fran. I’m not sure what you mean by the first questions. As far as the distance between bobs, I don’t think that really matters. -Andy

7. ericboy says:

The non-linearity you recognized can be taken corrected with a power function that takes into account the release angle. Here is a link: http://www.instructables.com/id/Unique-Pendulum-Wave-and-Release-Mechanism/step3/The-Pendulum-Period-Equation/

• ericboy says:

Oops, I posted this too soon (i.e., before reading anything). I see that you already derived the correction function for the release angle. Here is a much better link that reaffirms you observation. See equation 8.

http://fy.chalmers.se/~f7xiz/TIF080/pendulum.pdf

I created a simulation for a pendulum wave meant to have a constant amplitude as seen from above. Consequently, each pendulum has a different release angle. Using the corrective function (from #8 of the link above) in my calculations resulted in a near-perfect simulation.

• Andy "SuperFly" Rundquist says:

Cool, thanks Eric.

8. Ng Chee Lun says:

what wave did a simple pendulum use?longitudinal or transverse wave?
pls explain to me:) thx:)

9. Nita says:

Hi sir,
I’m building a Pendulum for my school’s science fair, would you please help me chose the right materials for it.

10. ike manlick says:

what is the angle that the pendulums balls are at.