IndexFale, chaos i materiaHoodie

Drum Mode Hoodie

SOME DRUMS SOUND IDENTICAL | Pixelated Physics

\[\nabla^2 u = \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}\]
$54.99 USD
$54.99 USD
W promocji Wyprzedane
Color
Size

Size guide S–5XL · inches

A drumhead fixed at its rim can only vibrate in certain shapes. Each one is a Bessel function in radius times a cosine in angle. Mode (3,2) of a circular drumhead, u = J₃(kr) cos 3θ: twelve lobes, three nodal diameters and one nodal ring, exactly. The slogan is about isospectral drums: Kac asked in 1966 whether you can hear a drum’s shape, and in 1992 Gordon, Webb and Wolpert found two different shapes with identical spectra. The art shows one vibrational mode of a circular drum, which is not one of them. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

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.

ShippingBy destination

Shipping calculated at checkout by destination.

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

A drumhead fixed at its rim can only vibrate in certain shapes, each a Bessel function in radius times a cosine in angle. The rim must sit at a zero of the Bessel function, which fixes the allowed frequencies and the pattern of still lines.

A drumhead fixed at its rim can only vibrate in certain shapes. Each one is a Bessel function in radius times a cosine in angle.

This is mode (3,2): cos 3θ gives three nodal diameters, and J₃(kr) with the rim at its second zero, kR = 9.7610, gives one nodal circle inside. The amber and cyan lobes move in opposite directions; the dark lines never move.

In 1966 Mark Kac asked whether you could hear the shape of a drum, that is, whether its set of frequencies fixes its shape. In 1992 Gordon, Webb and Wolpert showed you cannot: two differently shaped drums, now called isospectral, can have exactly the same frequencies.

A circle is not one of those shapes. Some drums sound identical; this one only sounds like itself.

Rings and spokes of silence.

Equations

\[\nabla^2 u = \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}\]
\[u(r,\theta,t) = J_3(kr)\cos 3\theta\cos\omega t\]
\[kR = j_{3,2} = 9.7610\]
SymbolMeaningUnit
\(u\) transverse displacement of the membrane \(\mathrm{m}\)
\(c\) wave speed on the membrane \(\mathrm{m/s}\)
\(J_3\) Bessel function of the first kind, order 3 \(\mathrm{1}\)
\(k\) wavenumber \(\mathrm{m^{-1}}\)
\(r, \theta\) polar coordinates on the drumhead \(\mathrm{m, rad}\)
\(R\) drum radius \(\mathrm{m}\)
\(\omega\) angular frequency, ck \(\mathrm{rad/s}\)
\(j_{3,2}\) second zero of J_3 \(\mathrm{1}\)

Sources

  1. Poisson (1829) Mémoire sur l’équilibre et le mouvement des corps élastiques, Mém. Acad. Sci. Paris 8, 357
  2. Rayleigh, The Theory of Sound, Vol. 1 (1877), Ch. IX: Vibrations of Membranes
  3. Kac (1966) Can One Hear the Shape of a Drum?, Am. Math. Monthly 73, 1
  4. Gordon, Webb & Wolpert (1992) One cannot hear the shape of a drum, Bull. Am. Math. Soc. 27, 134

Source Poisson, S. D. (1829). Mémoire sur l’équilibre et le mouvement des corps élastiques. Mémoires de l’Académie Royale des Sciences de l’Institut de France, 8, 357–570.

Field notes

Wyświetl wszystkie
1 3
Wyświetl wszystkie