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Nodal Lines Hoodie

STILLNESS IS A PATTERN | Pixelated Physics

\[D\,\nabla^4 w = \rho h\,\omega^2 w,\qquad D = \frac{E h^3}{12(1-\nu^2)}\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Bow the edge of a metal plate dusted with sand and the sand moves. It runs off the parts that shake and settles on the lines that stay still. A Chladni figure: sand on a vibrating square plate gathers on the lines that do not move. Mode (7,3), drawn from the plate equation. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Returns & replacements21 days

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The physics

A vibrating plate obeys an equation in which its bending stiffness balances its inertia. At a resonance, some lines on the plate stay perfectly still; sand sprinkled on the plate gathers there, drawing Chladni's nodal patterns.

Bow the edge of a metal plate dusted with sand and the sand moves. It runs off the parts that shake and settles on the lines that stay still.

Ernst Chladni published these figures in 1787. The plate obeys the equation on the back: stiffness times the fourth derivative of the displacement balances inertia, D∇⁴w = ρhω²w. At each resonant frequency the plate takes one shape, and the sand traces its nodal lines, where w = 0.

This is mode (7,3) of a square plate with free edges. The pattern is the classic approximation, cos 7πx cos 3πy − cos 3πx cos 7πy, the form Rayleigh used and Walther Ritz refined in 1909. It is close to the true free-plate mode, not exact; the exact one has no simple closed form.

Explaining the figures was hard enough that the French Academy offered a prize for it. Sophie Germain won it in 1816.

The sand goes where nothing moves.

Equations

\[D\,\nabla^4 w = \rho h\,\omega^2 w,\qquad D = \frac{E h^3}{12(1-\nu^2)}\]
\[w(x,y) \approx \cos 7\pi x\,\cos 3\pi y - \cos 3\pi x\,\cos 7\pi y\]
\[\text{sand stays where } w = 0\]
SymbolMeaningUnit
\(w\) transverse displacement of the plate at a point \(\mathrm{m}\)
\(D\) flexural rigidity of the plate \(\mathrm{N·m}\)
\(\nabla^4\) biharmonic operator, the Laplacian applied twice \(\mathrm{m^{-4}}\)
\(\rho\) density of the plate material \(\mathrm{kg/m^3}\)
\(h\) plate thickness \(\mathrm{m}\)
\(\omega\) angular frequency of the vibration \(\mathrm{rad/s}\)
\(E\) Young’s modulus \(\mathrm{Pa}\)
\(\nu\) Poisson’s ratio \(\mathrm{1}\)
\(x, y\) position on the plate, in units of the side length \(\mathrm{1}\)

Sources

  1. Chladni (1787) Entdeckungen über die Theorie des Klanges, Leipzig
  2. Rayleigh, The Theory of Sound, Vol. 1 (1877), Ch. X: Vibrations of Plates
  3. Ritz (1909) Theorie der Transversalschwingungen einer quadratischen Platte mit freien Rändern, Ann. Phys. 28, 737
  4. Timoshenko & Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (1959)

Source Kirchhoff, G. (1850). Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für die reine und angewandte Mathematik 40, 51–88.doi.org/10.1515/crll.1850.40.51

Field notes

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