Pendulum Waves - Mathematical Description cont. 2

If you missed the first two parts to this post, you can start here to get some context.



Well, perhaps the equation is not fantastically clear to you how the waves act at all (don't worry, it's not clear to me either). Let's actually graph this equation and figure out how to make sense of it.

This is a top view of the pendulum wave machine. The x-axis represents the pendulums location on the machine, the y-axis represents each pendulums displacement from equilibrium.

We can see that at the beginning, all pendula are exactly in phase at maximum amplitude, as expected. Afterwards, we see a series of patterns until we reach the middle of the "dance", during which all adjacent pendula are exactly out of phase. After this, the dance plays out in the reverse order until we get all the pendulums back in phase again.

If you want to explore this graph dynamically and play around with a simulation of the pendulum waves (and have Mathematica/Wolfram CDF Player), go to this interactive demonstration.

Whats wrong with the graphs we see above?

Well, whats what is that although the pendulum dance seems periodic, the equation we've figured out tells us that it's not. As you can see, the function $y(x,t)$ has increasingly smaller wavelengths as time progresses. It's only by looking at specific points on the function (i.e. by looking at where the pendula are located) that we get the illusion of a periodic function. 

The proper term for this phenomenon is aliasing (an example is below - from Wikipedia):

I think you get the idea of what type of confusion aliasing can cause.

With the function we've derived, we could check many things mathematically, such as the symmetry of the patterns before and after $\frac{\Gamma}{2}$ as well as whether we'll get the same patterns in the time from 0 to $\Gamma$ vs. $\Gamma$ to $2\Gamma$. If you want to see these properties shown from our function, the original paper that this derivation was distilled from can be found in the references below. If there are enough requests for me to show these properties on my blog, I'll be glad to write another post.

If you wish to apply the equation we've derive to make your own pendulum wave machine, see my next post.

***********

References:

[1] Harvard Demonstrations - Simple Harmonic Motion
[2] J. Flatten and K. Parendo, Pendulum Waves: A lesson in aliasing, American Journal of Physics 69, 778-782 (2001).
[3] R. Berg, Pendulum Waves: A demonstration of wave motion using pendula, American Journal of Physics 59, 186-187 (1991).