Chladni Figures: Visual Music in Vibration

When a metal or glass plate is set into vibration and lightly dusted with fine sand, the grains migrate along specific paths and settle into striking, symmetric patterns. These shapes — known as Chladni figures, after the German physicist Ernst Florens Friedrich Chladni (1756–1827) — are a direct visual map of the plate’s vibrational mode shapes, revealing the nodal lines where the plate remains stationary while the rest oscillates.

Historical Background

Chladni systematically documented these patterns in 1787 by drawing a violin bow across the edge of clamped metal plates, recording dozens of distinct figures in his treatise Entdeckungen über die Theorie des Klanges. When he demonstrated the experiment in Paris in 1809, Napoleon Bonaparte was so impressed that he offered a prize for a rigorous mathematical explanation. Sophie Germain eventually derived the governing plate equation after several attempts. Jules Lissajous later extended the study to coupled oscillations, and Michael Faraday’s closely related “Faraday crispations” on fluid surfaces deepened understanding of surface wave patterns driven by vertical vibration.

Physics and Equations

A thin elastic plate obeys the biharmonic (fourth-order) wave equation:

$$nabla^4 w + frac{rho h}{D_{text{flex}}} frac{partial^2 w}{partial t^2} = 0$$

where $w(x,y,t)$ is the transverse displacement, $rho$ the plate density, $h$ the thickness, and $D_{text{flex}}$ the flexural rigidity:

$$D_{text{flex}} = frac{E h^3}{12(1 – nu^2)}$$

Here $E$ is Young’s modulus and $nu$ is Poisson’s ratio of the plate material. Note that $nabla^4 = nabla^2(nabla^2)$ is the biharmonic operator, not the ordinary Laplacian of the wave equation for membranes — this reflects the restoring force being bending stiffness rather than tension.

Eigenfrequencies — Rectangular Plate

For a rectangular plate of dimensions $a times b$ with free edges (the standard Chladni experiment boundary condition), the resonance frequencies take the approximate form:

$$f_{mn} = frac{pi}{2} sqrt{frac{D_{text{flex}}}{rho h}} left[ left(frac{m}{a}right)^2 + left(frac{n}{b}right)^2 right]$$

The integers $m$ and $n$ are mode numbers counting the number of half-waves along the $x$- and $y$-directions respectively. The frequency ratio $f_{mn}/f_{11}$ depends only on plate geometry and boundary conditions, not on material — changing the material shifts all frequencies by the same factor $sqrt{E/rho}$.

Eigenfrequencies — Circular Plate

For a circular plate of radius $R$, the mode shapes involve Bessel functions:

$$w_{mn}(r,theta) = J_n!left(frac{k_{mn}, r}{R}right) cos(ntheta)$$

where $k_{mn}$ are the characteristic roots determined by the boundary condition (clamped or free edge). The corresponding resonance frequency is:

$$f_{mn} = frac{k_{mn}^2}{2pi R^2} sqrt{frac{D_{text{flex}}}{rho h}}$$

Here $n$ counts the number of nodal diameters and $m$ counts the nodal circles (excluding the boundary). The simplest mode ($n=0$, $m=1$) produces a single nodal circle; higher modes yield the characteristic “sunburst” and “ring” patterns.

Why Sand Collects at Nodal Lines

At antinodal regions, the plate vibrates with maximum amplitude, accelerating grains upward and downward many times per second. Grains in these regions are repeatedly launched into the air and land displaced slightly from their original position — a biased random walk that on average pushes them away from regions of high acceleration. At nodal lines, the plate displacement is zero (or nearly so), providing a stable resting place. Grains driven off antinodes gradually drift toward the nearest nodal line, where they accumulate and remain.

The optimum grain size is approximately 100–300 µm. Grains finer than ~50 µm are too light and become suspended in the air currents generated by the vibrating plate; grains coarser than ~500 µm carry enough inertia that typical plate velocities cannot displace them efficiently.

Key Parameters

Experiment Setup

To observe Chladni figures in the lab you need: a square or circular metal or glass plate (1–3 mm thick), a support stand gripping the plate at its center or one corner, a violin bow or a speaker driven by a function generator coupled to the plate, and fine dry sand or table salt. Slowly sweep the driving frequency upward; at each resonance, a stable nodal pattern snaps into place within a few seconds. Changing the support location or slightly altering the plate dimensions shifts the resonance spectrum and produces different families of patterns. Comparing measured $f_{mn}/f_{11}$ ratios with the theoretical predictions above provides a clean quantitative test of thin-plate theory.

References

1. Chladni, E. F. F. (1787). Entdeckungen über die Theorie des Klanges. Leipzig: Weidmanns Erben und Reich.
2. Rayleigh, J. W. S. (1894). The Theory of Sound, Vol. 1, 2nd ed. London: Macmillan. §306–§312.
3. Germain, S. (1821). Recherches sur la théorie des surfaces élastiques. Paris: Mme Ve Courcier.
4. Leissa, A. W. (1969). Vibration of Plates. NASA SP-160. Washington D.C.: NASA.
5. Faraday, M. (1831). “On a Peculiar Class of Acoustical Figures, and on Certain Forms Assumed by Groups of Particles upon Vibrating Elastic Surfaces.” Philosophical Transactions of the Royal Society, 121, 299–340.