Chladni Plates

by Deborah Stout

I recently watched The Rings of Power, the prequel series to The Lord of the Rings. As the first episode fired up, the opening sequence rolled and I saw something that was only recently familiar to me.

For the last several years, I have helped create and run a summer camp for the mathematics department at THE Ohio State University. Aimed at middle-school and high-school students, we attempt to investigate math from a non-academic, recreational viewpoint to pique their interest in a topic that sometimes has an unfun stereotype.

This summer’s topic was Math & Music. One aim each year is to discover interactive items to illustrate topics, as they help engage our campers. In looking for compelling, physical ways to play with sound, we discovered something called Chladni Plates.

What are Chladni plates? I’m glad you asked!

They are metal plates raised by a thin, fixed rod in the center, which is connected to a flat surface. In their original usage, someone would rub a violin bow along the edge of the plate, causing it to vibrate. With improved technology, however, we can actually use the rod as the source of vibration.

Why vibrate a metal plate? Sound is essentially composed of vibrations traveling through the air (or other media), so, by shaking the plate at certain frequencies. we can “see” the sound waves on the plate (with a little help)!

The plate vibrates too fast for our eyes to see distinctly. If we sprinkle the plate with sand, it will “pool” in certain areas of the plate which do not vibrate, allowing us to “see” the oscillations.

Different frequencies produce different patters in the sand; some of them are quite fascinating:

You may be more familiar with one-dimensional representations of waves, created by shaking a string. Though this movement is not sound, it can illustrate the concept of a sound wave moving through a string. When employed at a specific frequency, we can create a standing wave, where the string oscillates back and forth. Certain points on the string, called nodes, remained fixed during this process.

Chladni plates are two-dimensional representations of what we see in a string! The one-dimensional object becomes a two-dimensional plate. Similarly, the zero-dimension nodes (points lack dimension) in a standing wave on a string become one-dimensional “node lines” on the plate.

Isn’t math great?

Check out a Chladni plate in action:

You may recall we began the article with a reference to the opening sequence of The Rings of Power. What does it have to do with Chladni plates?

Check out the intro:

Immediately, I recognized the employment of Chladni plates. Some of these designs are rather intricate, so the montage could be high-level usage of the plates, clever computer graphics inspired by the plates, or a mixture of both. My guess is the latter option, as some of the patterns do not seem likely to be produced by mere vibration.

My skepticism is not unique, as I discovered one of my favorite science explainers, Steve Mould, shared my suspicions. His video on the introduction sequence is long, but probably does a better job of explaining Chladni plates than my article, so it’s worth a watch.

For any readers who are fellow math nerds, I took the equations Steve used in his video to graphically show some possible Chladni designs and put them into an online program called Desmos. You can play with them yourself.

Here’s the demo where the center of the plate is fixed (and the edge is vibrated with a bow or some other instrument):

Here’s the demo where the center of the plate is creating the vibrations:

You can click on the links in the paragraphs above or the images above to mess around with the equations that produced these designs.

Turns out Chladni plates are more than just fun. As is usually the case in math and physics, when we discover something cool or fun, someone else finds a way to make it useful.

One such application of Chladni plates helped create the best violins. Violins (and pretty much all instruments) work based on resonance, which is a reinforcement and amplification of sound vibrations. Violins sound best if the front and back pieces of the body resonate. By treating the instrument’s wooden pieces as Chladni plates, you can shape them so the designs produced by metal filings (in place of sand) produce as much symmetry as possible. The more symmetrical, the better the violin sounds.