Chladni figures

Example

Chladni figures are the patterns that sand forms on a thin plate made to vibrate in one of its modes. The sand is thrown off the moving parts and collects along the nodal lines, where the mode shape $W$ is zero. So the figure is a picture of the nodal set of a mode of the vibration of plates.

A computable case is a simply supported square plate $0 < x, y < a$, whose modes include $W_{mn} = \sin(m\pi x/a) \sin(n\pi y/a)$. The frequencies of $W_{12}$ and $W_{21}$ are equal, so every combination $W_{12} + c\,W_{21}$ is also a mode. Using $\sin 2\theta = 2 \sin\theta \cos\theta$, $$W_{12} - W_{21} = 2 \sin\frac{\pi x}{a} \sin\frac{\pi y}{a} \left( \cos\frac{\pi y}{a} - \cos\frac{\pi x}{a} \right),$$ which vanishes inside the square exactly on the diagonal $y = x$. In the same way $W_{12} + W_{21}$ vanishes exactly on the other diagonal $x + y = a$. So the sand gathers on a diagonal, although neither $W_{12}$ nor $W_{21}$ alone has a diagonal nodal line.