Wednesday, May 8, 2019

Desmos Chladni

Like Lissajous figures, Chladni figures provide a surprising and aesthetically engaging example of wave interaction.

Named for Ernst Chladni, these figures represent nodal patterns formed by vibrating surfaces. Traditionally, these are formed placing fine particles on a surface, like a sheet of metal that is set vibrating (a violin bow against an edge of the metal plate is one popular method). The particles settle in the areas of the surface that have the least motion - the nodes. When you achieve a resonant frequency, a characteristic pattern emerges.

In past posts I've pointed to code that draws Chladni figures using R (here and here), and using JavaScript. Maybe not surprisingly, you can also play around with Chladni-like figures using Desmos, and this may be the most accessible way to explore them them and appreciate how they are generated from the sinusoidal functions.

Chladni-like figure generated in R

Chladni-like figure generated using JavaScript

In Desmos, you can create images similar to these using inequalities. The equations are reasonably straight forward - the graph here will draw the figure across the whole plane - best results are seen when zooming in on a small region.

Chladni-like figure generated in Desmos,
graph here

More Chladni-like figures in Desmos

Try playing around with the desmos graph here, R scripts for generating figures are found here, the JavaScript Chladni generating page is here.

After posting on Twitter, the desmos sketches were improved by   and @PaulaKrieg. Here are some other graphs inspired by their changes:

A Chladni-like pattern with
two distinct inequalities

A Chladni-like pattern with three distinct

With the added layers, the Chladni patterns are approaching an abstract William Morris appearance.

Another Update
This other graph allows you to experiment more directly with the Chladni figures, similar to the web page mentioned above.

graph for building 
Chladni figures