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:
?[p_, q_] := ? Sqrt[p^2 + q^2];
?[x_, y_, t_, p_, q_, A_, B_] := (A Cos[?[p, q] t] + B Sin[?[p, q] t]) Sin[p ? x] Sin[q ? 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"}},
Manipulate[
ContourPlot[
?[x, y, 0., p, q, A, B] +
Tan[(?/2 - .0001) (Haversine[Mod[2 ?, ?]] +
Floor[2 ?/?] - 1)] ?[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 ?]]],
PlotRangePadding -> -0.01, ImageSize -> 540,
Background -> cols[[-1]]
], {?, 0., ?}]
]
