One of my favorite physics demonstrations to perform at local schools, conventions, and expos is the production of Chladni patterns, such as the one shown below.

I’ve blogged about these patterns before. They are formed by vibrating a metal plate at one of its special resonance frequencies, which causes the plate to form standing waves. These waves have some locations — antinodes — where they vibrate a lot, and other locations — nodes — where they don’t vibrate at all. By sprinkling sand over the plate, the sand will be pushed to the nodes allowing the otherwise invisible vibrations to be visualized.

This technique is remarkably old, first published by German physicist Ernst Chladni in 1787; the patterns created are therefore known as *Chladni figures*. Chladni used a violin bow to excite his plate, but today we can use a speaker and frequency generator to produce the effect more readily.

I’ve been doing Chladni pattern demos for nearly five years, and when I recently did them again at a local school, I decided to spice things up with colored sand, to produce multicolored patterns. This results in lovely things such as the pattern below!

This became a bit of an art project for me, and I spent a couple of hours over the past few days making pretty colored Chladni patterns! I thought I would share the results here. In addition, I learned a little bit more about the physics of these patterns that I will share along the way.

I thought of five different ways to use colors in interesting ways in forming the Chladni patterns. As I went up in frequency, generating more elaborate patterns, I experimented with these different techniques.

The first technique is to generate a pattern with one color of sand, and then place different colored sand on either side of the nodes, allowing colored bands.

The effects weren’t spectacular, but I got some satisfaction from them!

The second strategy was combined with the first. By only firing up the vibrations for a second or two, I can scatter the second or third deposit of sand to “fill” in a region with color.

The third technique is the one I used the most, and that is simply to place the sand in strategic locations in one go in order to produce colors in appealing places. For example:

Different size grains of sand can add some variation to the patterns, as technique four.

At this stage, I started naming some of my patterns. The one above I call “The Eye of Horus.” The one below I call “Fireworks.”

The “Fireworks” pattern is typically very difficult to generate — it requires a much higher amplitude vibration in order to get it to appear at all. While I was playing around this past week, I suddenly realized why!

Can you see a key difference between this pattern and all the ones that came before it? I’ll start a new paragraph before answering, in case you want to look.

The “Fireworks” pattern has nodes that pass directly through the center post, a feature none of the other patterns have. The problem is that that center post is also the point at which the vibrations are being generated. So we are trying to generate a pattern with no vibrations in the middle using vibrations that start in the middle! In principle, we should not be able to generate this pattern at all by vibrating the center post — in physics terms, we are trying to generate an asymmetric mode using a symmetric excitation. However, the center post is not perfectly vibrating up and down, and probably has a little side-to-side motion, so a little bit of energy couples into the “Fireworks” pattern, though it is very inefficient. That is why it is so difficult to generate.

The reason that I figured this out is that I realized that I had never seen the two lowest-frequency Chladni patterns, which would look like a simple “+” and a simple “X” on the plate. My guess is that I was unable to find them because they are also patterns which have a node right in the middle.

Anyway… here’s a “before” and “after” picture of the next pattern, to show how I laid the sand down.

Keep in mind: when you hit a resonant frequency, the plate is basically moving up and down at all locations, so the sand will tend to stay close to where it was originally placed, settling into nearby nodes. If you know what the pattern looks like, it is straightforward to place the sand to make cool patterns.

Here are a couple of additional patterns.

The last thing I did on that first day of experimenting was to hunt down two higher-frequency patterns that I had never seen before; their images are below.

I’ve been wondering for a while why I haven’t seen many discussions of the mathematical physics of Chladni patterns — it seems like they should be a relatively simple wave phenomenon, right? I was wrong.

In an ordinary wave equation, the acceleration of the wave at any point is dictated by the *curvature* of the wave at that point — in calculus terms, it is proportional to the second spatial derivative of the wave. For waves on a rigid plate, however, the acceleration of the wave is dictated by the *fourth* spatial derivative of the wave — the *curvature of the curvature*. This makes the solutions extremely complicated, and very difficult to find.

It is an interesting bit of trivia I came across that the derivation of the Chladni patterns is the only historical physics problem I know of that had major contributions provided by three women! When Chladni’s technique became famous in the early 1800s, the Paris Academy of Sciences sponsored a competition “to give the mathematical theory of the vibration of an elastic surface and to compare the theory to experimental evidence.” The only person to even submit an attempt at a solution was the brilliant Sophie Germain.

Germain’s first attempt at solution was rejected when she submitted it in 1811, but she revised and improved it and resubmitted in 1813. She still did not win, but received an honorable mention for her attempts. In 1816, she made a third attempt, and became the first woman to win a prize from the Paris Academy!

Her solution was not complete, however; she had derived the proper mathematical equation that describes the waves, but had not figured out the correct *boundary conditions*, which describe what happens to the wave on the edges of the plate. Approximate boundary conditions for a square plate with free edges (edges unsupported in any way) were only given in 1909 by the Swiss theoretical physicist Walther Ritz.

The two other women who made a contribution to the problem were Alice Lemke, who in 1928 repeated Ritz’s approximate calculation with much greater accuracy, and Mary Waller, who in 1939 studied how different combinations of modes may produce a large variety of patterns.

Speaking of patterns, let’s get back to some! A few days after my first experiments, I found a few more colors of sand, allowing me to make more colorful patterns and try out technique five: using black sand to produce the illusion of space between different colors!

With five (non-black) colors, I was able to make truly lovely patterns.

But the best thing? I also took videos of the patterns forming, and the transitions between different patterns. The video is included below.

I’m sure I’ll have more to say about Chladni patterns in the future — I’m kind of obsessed with understanding the physics of them! But that’s going to take some time, so I leave you with a few gifs to share!

I don’t understand any of that… but the artwork is AMAZING!!