Cymatics

Scatter sand on a metal plate and bow it at the right pitch. The grains flee the shaking parts and pile up along the still ones, drawing a sharp geometric figure out of pure noise. Change the frequency and the figure reorganises into another. You are seeing sound stand still.

The concept

A driven plate does not vibrate everywhere equally. At certain frequencies it rings in a standing-wave mode, with regions that heave up and down and curves between them, the nodal lines, that never move at all. Sand bounces off the moving regions and settles on the motionless ones, so the nodal set becomes visible. Each resonant pitch has its own signature figure.

The math

A thin square plate obeys the biharmonic plate equation; its standing modes are shapes $W(x,y)$ satisfying

$\nabla^4 W = \frac{\rho h}{D}\, \omega^2\, W$

where $D$ is the bending stiffness and $\omega$ a resonant frequency. The sand collects on the nodal set, the curve where $W(x,y) = 0$. For an idealised square plate these modes are well approximated by combinations of $\cos\!\frac{m\pi x}{L}\cos\!\frac{n\pi y}{L}$, and a Chladni figure is essentially the zero set

$\cos\frac{m\pi x}{L}\cos\frac{n\pi y}{L} \pm \cos\frac{n\pi x}{L}\cos\frac{m\pi y}{L} = 0$

whose two integers $(m,n)$ pick the pattern.

Why it stays strange

The pattern is not painted onto the plate; it is the plate's own resonance made of the few places that hold still while everything else shakes. The same standing-wave logic sizes an organ pipe, colors a soap film, and quantises an electron in a box. Stillness, here, is the structure.

Further reading

• Chladni (1787), Entdeckungen über die Theorie des Klanges.
• Rossing, The Science of Sound.

@book{rossing2002sound,
  author    = {Rossing, Thomas D. and Fletcher, Neville H.},
  title     = {Principles of Vibration and Sound},
  edition   = {2nd},
  publisher = {Springer},
  year      = {2004}
}