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[GIF] The Band Plays On (Chladni figures for a square drum)


Nodal lines for a square drum

The Band Plays On

Following up on Drumbeat, which shows one of the vibration modes of a circular drum, here are the nodal lines of a family of vibration nodes of a square drum. 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 the animation, I'm taking the combination $u = u_{7,9} + c u_{9,7}$ 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 $u=0$.

Anyway, here's the code:

With[{p = 7,
  q = 9,
  A = 1,
  B = 0,
  cols = RGBColor /@ {"#F66095", "#2BCDC1", "#393E46"}},
   ψ[x, y, 0., p, q, A, B] + 
     Tan[(π/2 - .0001) (Haversine[Mod[2 θ, π]] + 
          Floor[2 θ/π] - 1)] ψ[x, y, 0., q, p, A, B] == 0,
   {x, .01, .99}, {y, .01, .99},
   Axes -> False, Frame -> False, 
   ContourStyle -> 
    Directive[CapForm["Round"], Thickness[.01], 
     Blend[cols[[;; -2]], Haversine[2 θ]]], 
   PlotRangePadding -> -0.01, ImageSize -> 540, 
   Background -> cols[[-1]]
   ], {θ, 0., π}]
POSTED BY: Clayton Shonkwiler
1 year ago

I'm glad you caught the Chladni bug. They really are pretty pictures... :)

1 year ago

enter image description here - another post of yours has been selected for the Staff Picks group, congratulations !

We are happy to see you at the top of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!

POSTED BY: Moderation Team
1 year ago

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