Question 1

What are Chladni figures?

Accepted Answer

Chladni figures are the geometric nodal patterns that form on a vibrating plate. When a plate is bowed or driven at a resonant frequency, sand or salt sprinkled on the surface migrates away from areas of maximum vibration and collects along the nodal lines  the places where the plate has zero displacement. The patterns were first studied systematically by Ernst Chladni in 1787 and are now understood as the zero-sets of the plate's eigenfunction solutions.

Question 2

What determines the shape of the pattern?

Accepted Answer

For a simply-supported (hinged) square plate of side L, the eigenfunctions are u(x,y)=sin(mpx/L)sin(npy/L). The integers m and n count the number of nodal lines parallel to each edge (not counting the fixed edges). The nodal lines are a grid of m-1 lines parallel to the y-axis and n-1 lines parallel to the x-axis, forming rectangular cells. When m?n the two modes (m,n) and (n,m) have the same eigenfrequency (degenerate), and any linear combination u=Asin(mpx)sin(npy)+Bsin(npx)sin(mpy) is also an eigenfunction; the mixing ratio =B/A controls the orientation and shape of the resulting nodal pattern.

Question 3

Why do degenerate modes produce more complex patterns?

Accepted Answer

When m=n only one eigenfunction exists (up to scaling) and the nodal lines are a simple grid. When m?n, modes (m,n) and (n,m) share the same eigenfrequency because the frequency is proportional to v(m+n), which is symmetric in m and n. Any superposition of the two with mixing parameter ?[-1,+1] is a valid resonant motion, and the nodal pattern  the zero-set of the superposition  changes continuously from a grid through diagonal lines, chevrons, and intricate curves as  varies. This degeneracy is why Chladni often observed multiple distinct patterns at the same driving frequency depending on where he bowed the plate.

Question 4

How does this simulation model the plate?

Accepted Answer

The simulation uses the analytical eigenfunctions for a simply-supported square plate: u(x,y)=sin(mpx/W)sin(npy/H)+߷sin(npx/W)sin(mpy/H). Nodal lines are rendered where |u|<threshold (rendered bright). The animated wave mode multiplies u by cos(?t) to show the full oscillation cycle. To keep frame rates smooth, sine tables are precomputed whenever m, n, or  change, and the pixel loop uses only table lookups and arithmetic.

🧪 Chladni Figures Simulator

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