Chladni Figures
Sprinkle sand on a metal plate, bow a violin across its edge, and the grains migrate to the nodal lines — the loci where the plate does not move. Ernst Chladni demonstrated this to Napoleon in 1809; it was described as the most beautiful experiment in physics. Below, 2 000 simulated grains are pushed by the gradient of the time-averaged displacement field toward the nodes of mode (m, n). Change the harmonic indices, watch the pattern reconstruct. Mix in a second mode for richer superposition geometry.
The displacement field
A simply-supported rectangular plate vibrating in mode (m, n) has the instantaneous displacement
where $\omega_{mn} = c \pi \sqrt{m^2/L^2 + n^2/L^2}$ is the resonant angular frequency, proportional to $\sqrt{m^2 + n^2}$ for a square plate. The nodal lines are the curves where $Z = 0$ for all $t$: straight lines at $x = kL/m$ ($k = 1 \dots m-1$) and $y = kL/n$ ($k = 1 \dots n-1$). Mode $(m, n)$ has $(m+n-2)$ nodal lines in total.
Why sand moves to the nodes
Sand (or salt) grains sitting on the plate are accelerated by its surface. At an antinode the plate kicks them repeatedly, imparting net energy; they hop outward. At a node the plate is stationary, so grains accumulate there. The net driving force is proportional to the gradient of the time-averaged displacement energy $\langle Z^2 \rangle$. Because the time average of $\cos^2(\omega_{mn} t)$ is $\frac{1}{2}$, we have:
Using $2\sin(\theta)\cos(\theta) = \sin(2\theta)$ and scaling $L=1$, the gradients are:
Near a nodal line this is a restoring spring force; the simulation adds velocity damping (friction) and a tiny thermal noise to prevent particles locking at unstable saddle points.
Mode superposition
When two modes with incommensurable frequencies are mixed, their time-averaged energy densities add independently (cross-terms vanish). The combined force landscape is a sum of the individual gradients, weighted by the squared amplitudes $A^2$ and $B^2 = \alpha^2 A^2$. The resulting Chladni pattern is no longer composed of straight nodal lines — it becomes a curve defined by the level set $A^2 \sin^2(m_1\pi x)\sin^2(n_1\pi y) + B^2 \sin^2(m_2\pi x)\sin^2(n_2\pi y) = 0$, which has no closed-form solution and must be found numerically (or, experimentally, by the sand itself).
Historical note
Ernst Chladni (1756–1827) published his figures in Entdeckungen über die Theorie des Klanges (1787). He toured Europe demonstrating them and presented to Napoleon in 1808. Napoleon was sufficiently impressed that he offered a prize of 3000 francs for a satisfactory mathematical theory. Sophie Germain eventually won it (after three attempts, 1816) with the biharmonic plate equation $\nabla^4 w + \rho h \ddot{w} = 0$ — one of the first major contributions to mathematical physics by a woman.