This worksheet shows Chladni patterns using the trace-method with list and point coloring. Here is used wave equation of 3D standing waves.
The well known equation for the zeros of the standing wave on a square Chladni plate (side length L) constrained at the center 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 ( [url]http://paulbourke.net/geometry/chladni/[/url]).
[i]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]
