Functional Analysis: Eigenfunction Spectrum

Initial shape f(x)

Eigenmodes N 10

Time speed 1.0×

Energy captured

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Dominant mode

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Sim time t

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Modes shown

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How it works

The functions eₙ(x)=sin(nx) are eigenfunctions of the self-adjoint operator −d²/dx² on (0,π) with Dirichlet boundary conditions: −eₙ'' = n²·eₙ. Because the operator is self-adjoint, its eigenfunctions form an orthogonal basis of L²(0,π) and each one evolves independently in time — this is the spectral theorem in action. A plucked string with initial shape f(x) is first decomposed into that eigenbasis, then each mode is played forward at its own eigenfrequency and the modes are summed back together.

−eₙ''(x) = n²·eₙ(x),  eₙ(x) = sin(nx)     (eigenfunction, eigenvalue n²)
f(x) = Σ bₙ·eₙ(x)                         (spectral decomposition)
u(x,t) = Σ bₙ·eₙ(x)·cos(n·t)              (each mode oscillates at √(eigenvalue))

Initial shape — the plucked string profile f(x) being decomposed into eigenmodes.
Eigenmodes N — how many terms of the spectral sum are kept; each is drawn as its own ribbon receding into depth by frequency index.
Time speed — scales how fast the modes oscillate, so slow speeds make the phase relationships between eigenfrequencies easy to see.
Auto-orbit — rotates the camera around the mode stack; drag to orbit manually at any time.

Real-world relevance: this eigenfunction expansion is exactly how a vibrating guitar string, a quantum particle in a box, or a heat-conducting rod evolve — decompose the initial state into the operator's eigenbasis, evolve each eigencomponent independently, then recombine.

Individual eigenmodes (by depth) Resultant u(x,t)

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Functional Analysis: 3D Eigenfunction Spectrum