• Black Nodal Lines physics hoodie, pixel-art Chladni figure design, worn by a model, front close-up
  • Black Nodal Lines physics hoodie, pixel-art Chladni figure design, worn by a model, back close-up
  • Black Nodal Lines physics hoodie, pixel-art Chladni figure design, front view
  • Black Nodal Lines physics hoodie, pixel-art Chladni figure design, back view
  • Black Nodal Lines physics hoodie, pixel-art Chladni figure design, turning on a model
  • Black Nodal Lines physics hoodie, ΨP Pixelated Physics neck label
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Nodal Lines HoodieSTILLNESS IS A PATTERN | Pixelated Physics

Nodal Lines Hoodie – STILLNESS IS A PATTERN | Pixelated Physics

$54.99 USD
$54.99 USD
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Shipping calculated at checkout. Free US shipping · $10 flat shipping outside the US.
Color
Black
Size
S–5XL
Made
To order · shipping times

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.

Color
Size

Size guideS–5XL · inches

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.

ShippingFree US

Free US shipping. $10 flat shipping outside the US.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

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)
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