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[GIF] Dance (Chladni figures for a square drum)

Posted 1 month ago
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Chladni figures for a square drum


Speaking of vibration modes of a square drum, here are the Chladni figures for linear combinations of the $(20,21)$ and $(21,20)$ vibration modes of the square.

This is entirely in the spirit of The Band Plays On; since that was a while ago, let me quote from that description:

Cribbing from the MathWorld article, the vertical displacement of the $(p,q)$ vibration mode of a $1 \times 1$ square drum is

$u_{pq}(x,y) = (A \cos \omega_{pq} t + B \sin \omega_{pq} t) \sin(p \pi x) \sin (q \pi y)$,

where $\omega_{pq} = \pi \sqrt{p^2 + q^2}$. This is easy enough to turn into a function:

 ω[p_, q_] := π Sqrt[p^2 + q^2];
 ψ[x_, y_, t_, p_, q_, A_, B_] := (A Cos[ω[p, q] t] + B Sin[ω[p, q] t]) Sin[p π x] Sin[q π y];

In fact, the same holds for arbitrary rectangles, so long as the product of sines becomes $\sin(p\pi x/L_x)\sin(p \pi y/L_y)$ where $L_x$ and $L_y$ are the lengths of the sides of the rectangle. The nice thing about squares is that you get an extra symmetry: the $(p,q)$ mode and the $(q,p)$ mode have the same frequency, so any linear combination will also form a standing wave.

In this GIF, I'm taking the combination $u = u_{20,21} + c u_{21,20}$ and (by letting $c=\tan \theta$) varying $c$ from $-\infty$ to $\infty$. The curves in the animation are the so-called Chladni figures, or nodal lines of the vibration, meaning the solutions of $u=0$.

Here's the code which does not produce a Manipulate[] that updates nicely in real time, unfortunately):

With[{p = 20, q = 21, A = 1, B = 0, 
  cols = RGBColor /@ {"#a2ef44", "#08085e"}},
   ψ[x, y, 0., p, q, A, B] + Tan[(π/2 - .0001) 2 (θ - π/2)/π] ψ[x, y, 0., q, p, A, B] == 0,
   {x, .01, .99}, {y, .01, .99},
   Axes -> False, Frame -> False, PlotPoints -> 50, 
   ContourStyle -> Directive[CapForm["Round"], Thickness[.01], cols[[1]]], 
   PlotRangePadding -> -0.01, ImageSize -> 540, 
   Background -> cols[[-1]]],
  {θ, 0, π}]

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