Sketch · 2026-08-26
Chladni patterns, on a square plate.
Sixteen thousand grains scattered on a driven plate. Two integers pick which standing wave shakes it, and the grains, jittering harder where the plate moves more, drift off the anti-nodes and collect on the still lines between them. Ernst Chladni was showing this trick with a violin bow in 1787.
The rule that draws it
The plate is a unit square with its edges free. A single driven
mode gives every point (x, y) a signed displacement
f(x, y) = cos(m·π·x)·cos(n·π·y) − cos(n·π·x)·cos(m·π·y),
zero along a set of nodal lines. Each grain takes a small
random step whose length is proportional to
|f(x, y)|. Grains sitting
on quiet cells barely move; grains on loud cells bounce until they
fall onto a quieter one, and the pattern is the equilibrium of that
walk.
Where the formula comes from
A square plate with free edges has standing-wave solutions of the
form cos(m·π·x)·cos(n·π·y).
Whenever m ≠ n the two
modes (m, n) and
(n, m) are degenerate:
they share a frequency and any weighted sum of them is another
valid vibration. The anti-symmetric combination used here is the
one that gives the clean grid Chladni sketched by hand. Different
plate shapes, different boundary conditions, different formulas,
same trick.
Sources
- Chladni, E. F. F. (1787). Entdeckungen über die Theorie des Klanges. Leipzig. The bow, the plate, and the first published catalogue of the patterns.
- Kirchhoff, G. (1850). Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math., 40, 51–88. The plate equation whose modes are being drawn here.
- van Gils, W. (2013). Chladni figures and the Tacoma bridge: motivating PDE eigenvalue problems via vibrations. SIAM Review, 55(3), 601–612. A modern read on the boundary conditions that pick which patterns appear.