Laplacian Eigenfunctions

You can get very interesting result when plotting different linear combination of eigenfunctions of the Laplacian on a given (convex) shape. Here I use a square plate [0,1] by [0,1] and show the zero iso-value of the some of such linear combination. As time evolves, the linear combination changes.

Since the Eigenfunctions are symmetric, I only plot them on the [0, 0.5] interval. In that case, the eigenfunction are of the form

With (n1,n2) such that

The associated eigenvalue l= 340*pi^2 is of multiplicity 4 (12,14) (18,4) (14,12) (4,18). Below, same thing with l = 5525 which is of multiplicity 12.

Those isocontours are called nodal lines. By the unique continuation property, those nodal lives consist of curves in the interior of the domain (her a square, but the result is valid for any convex shape) which are infinetly differentiable. You may also note, but that is a well known theorem, that when nodal lines interesect a boundary (except may be at corner points), then they form egal angle. A single nodal line intersects the boundary with right angles, two intersect it at 60° angles, and so forth. This also explain why traction/compression cracks always cut boundaries of their domain at right angles, but that's another story.

Well, my point here was not simply to show funny stuff, but also to point out that using gnuplot and image magick is really simple to create animations.

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