Chladni patterned surfaces
- Author:
- Roman Chijner
Here is the next step from painting of implicit functions
https://www.geogebra.org/material/show/id/pRCY9r5T to Physics.
I offer You the worksheet shows "Chladni" patterned surfaces https://www.geogebra.org/material/show/id/1267579
For me, standing waves are the most amazing topics in Physics. Who has not admired Chladni's sound figures- Amazing Resonance Experiments?
https://www.geogebra.org/m/c4NBuJnb
It is a well known equation for the zeros of the standing wave on a square Chladni plate (side length L) is given by the following:
cos(n pi x/L) cos(m pi y/L) - cos(m pi x/L)cos(n pi y/L) = 0, where n and m are integers ( http://paulbourke.net/geometry/chladni/).
I generalised this equation for the three-dimensional case:
cos(k*x π/L)[cos(l*y π/L) cos(m*z π/L)+s*cos(m*y π/L) cos(l*z π/L)]+
cos(l*x π/L)[cos(k*y π/L) cos(m*z π/L)+s*cos(m*y π/L) cos(k*z π/L)]+
cos(m*x π/L)[cos(k*y π/L) cos(l*z π/L)+s*cos(l*y π/L )cos(k*z π/L)]=0, where k, l and m are integers, s=∓ 1.
I managed to get not only already known 2D Chladni patterns: https://www.geogebra.org/material/show/id/kxXpKDaw,
https://www.geogebra.org/m/RD6tuxru,
but also 3D Spacial Chladni patterned surfaces. By taking advantage of all the Geogebra possibilities of today, we can build not only trace of surfaces-
https://www.geogebra.org/material/show/id/tfsu4uuW,
but also Network of rotatable implicit curves:
https://www.geogebra.org/material/show/id/PzBug5SMhttps://www.geogebra.org/material/show/id/JXtDvVjc
Video:https://www.geogebra.org/m/Ayu65AEj
Pictures of spacial Chladni patterns:
https://www.geogebra.org/m/kxXpKDaw
Thanks to the Geogebra developers!
Regards,
Roman Chijner
