Chladni figures

Parametric equations

The nodal lines are where the amplitude on the square is zero, for the mode the controls select:

A (x, y) = sin (\frac{mπ x}{L}) sin (\frac{nπ y}{L}) - sin (\frac{nπ x}{L}) sin (\frac{mπ y}{L})

Mode m: choose that integer. The nodal lines change, and the points scatter, then gather on the new zeros.
Mode n: choose the other integer. Together, m and n set the nodal pattern in both directions; neither belongs to one side alone.
Vibration speed ω: speed up or slow down the oscillation that pushes the points. It does not move the places where the mode’s amplitude is zero.

The points rest where the selected mode’s amplitude is zero and are pushed away from the antinodes.
Swapping $(m, n)$ with $(n, m)$ flips the sign of the amplitude and leaves the nodal lines in place.
The picture is the nodal set of this idealized square mode with fixed edges (a real free-edge plate has more complicated modes), and if $m = n$ the amplitude is zero everywhere, so there is no nodal curve.