Simulations & Interactive Learning
Two live widgets, aimed at the two places where PHY621's abstraction most needs something to look at: a basis expansion converging, and an eigenvalue problem being solved.
Live
- Fourier series builder — watch harmonics assemble a square wave, and meet the Gibbs phenomenon
- Normal modes lab — a real generalised-eigenvalue solver (Jacobi rotations) on coupled oscillators: the CO₂ triatomic, the uniform chain, and a heavy defect that localises a mode
What the normal-modes lab is checked against
| Case | Exact result | Agreement |
|---|---|---|
| Triatomic, translation | \(\omega_1 = 0\) | \(10^{-9}\) |
| Triatomic, antisymmetric | \(\omega_2 = \sqrt{k/m}\) | machine precision |
| Triatomic, symmetric | \(\omega_3 = \sqrt{(k/m)(1+2/\mu)}\) | machine precision |
| Centre-mass amplitude in mode 2 | exactly zero | \(10^{-16}\) |
| Uniform chain | \(\omega_n = 2\sqrt{k/m}\sin\frac{n\pi}{2(N+1)}\) | \(10^{-16}\) |
| Mass-weighted orthogonality | \(\mathbf{a}_i^{\mathsf T}A\mathbf{a}_j = 0\) | \(10^{-16}\) |
The two ideas these illustrate
Both widgets are the same mathematical operation wearing different clothes: expand in the eigenbasis of an operator.
- The Fourier builder expands a function in the eigenfunctions of \(d^2/dx^2\) on an interval.
- The normal-modes lab expands a displacement in the eigenvectors of the stiffness matrix.
Take \(N\to\infty\) in the second and you get the first: the discrete chain's eigenvectors are sampled sines, and its frequencies \(2\sin(n\pi/2(N+1))\) tend to \(n\pi/(N+1)\) — the linear dispersion of a continuous string. The normal modes lab lets you watch that limit happen by raising \(N\).
Related
Eigenvalues & eigenvectors · Fourier series · Normal modes · Inner product spaces