Cymatic/Chlandi simulation

http://local.wasp.uwa.edu.au/~pbourke/surfaces_curves/chladni/

Paul Bourke is the man.

I'm not sure my math is up to it. But here's a first thought:

If you represent the surface of a (square) plate as a matrix of
dimension LxL, the value of the cell [x,y] is:

cos(n pi x / L) cos(m pi y / L) - cos(m pi x / L) cos(n pi y / L)

Might this be in the scope of jit.expr?

The sand should move towards areas where the value is nearest zero
(you could do some processing on the matrix to get a topology or
field out of it, and then put your sand particle system through it).
Differentiate the matrix (jit.op @op -, with feedback) and you will
get the slope at x,y; then move the sand particle by a fraction of
that xy (jit.repos?)

n and m I believe are two frequency components of the input signal,
not entirely sure. Putting arbitrary signals through might be hard;
maybe do an FFT and pick the peak bins? Or use fiddle~/analyze~?

I'm actually tempted to try it.

For the circular plate, I think I would use a matrix of r & theta,
then do a cartesian to polar conversion to represent it visually. No
idea how to implement a Bessel function.

Chladni Plate Mathematics, 2D

Written by Paul Bourke
March 2003

The basic experiment that is given the name "Chladni" consists of a
plate or drum of some shape, possibly constrained at the edges or at
a point in the center, and forced to vibrate historically with a
violin bow or more recently with a speaker. A fine sand or powder is
sprinkled on the surface and it is allowed to settle. It will do so
at those parts of the surface that are not vibrating, namely at the
nodes of vibration.

The 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. The Chladni patterns for n,m between 1
and 5 are shown below, click on the image for a larger version or
click on the "continuous" link for the standing wave amplitude maps.
Note that the solution is uninteresting for n = m and the lower half
of the table is the same as the upper half, namely (n1,m2) = (n2,m1).

Circular plate

For a circular plate with radius R the solution is given in terms of
polar coordinates (r,theta) by

Jn(K r) (C1 cos(n theta) + C2 sin(n theta))

Where Jn is the n'th order Bessel function. If the plate is fixed
around the rim (eg: a drum) then K = Znm / R, Znm is the m'th zero of
the n'th order Bessel function. The term "Znm r / R" means the Bessel
function term goes to zero at the rim as required by the constraint
of the rim being fixed.

On Mar 27, 2007, at 6:39 AM, Carl Knott wrote:

>
> I'd like to write a simulation of a Cymatic Tonoscope.
>
> I plan to create a physical model of a circular membrane, sand will
> be sprinkled on top of it. When a person speaks into a microphone
> the membrane will vibrate causing the sand to move into the nodal
> lines forming standing wave patterns.
>
> Obviously this will be done in real time.
>
> I have looked at PMPD in PD and I dont think it's possible.
>
> Any ideas guys? I really want to write this :)

grrr waaa
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