The Fourier View

Is there a basis in which the coupled chain becomes N independent oscillators — and what does the energy in each mode tell us?

Interactive · live simulation

The top panel is the chain; the bottom is its live mode-energy spectrum. Energy stays locked in each mode (decoupled oscillators) — a triangle pluck lights up only the odd modes, falling off like 1/j².

The Fourier View — simulation render — fig 1. A triangular pluck (a plucked guitar string): only odd modes survive by symmetry, and their energies fall off as roughly 1/j² — the physics of musical timbre as a bar chart.

Diagonalizing the chain

Project the chain onto orthonormal sine mode shapes \varphi_j(n) = \sqrt{2/(N-1)}\,\sin\!\big(\pi j n/(N-1)\big). Each modal amplitude obeys a_j'' = -\omega_j^2 a_j — a plain harmonic oscillator, completely decoupled from every other mode. The coupling in the position basis was an illusion of coordinates.

fig 2. Seed a single mode and the spectrum is one spike that stays put forever — decoupled oscillators never leak energy between each other.

Parseval, exactly

Because the mode shapes are orthonormal and the Hamiltonian is diagonal in this basis, total energy splits exactly into a sum of per-mode energies. That identity is the lab's correctness check: if the mode energies don't sum to the chain's total, the projection is wrong.

fig 3. A localized Gaussian pulse: narrow in space means broad in mode space — the discrete uncertainty principle.

Why a string holds its tone

In an undamped run each mode's energy is constant in time — energy does not leak between modes. That is precisely what "decoupled oscillators" means, and it is why a struck string keeps its timbre instead of decaying into noise. The triangle pluck's 1/j^2 odd-mode falloff is the same spectrum that makes a guitar sound musical.

fig 4. A deliberate three-mode mix holding its 1 : 0.36 : 0.09 energy ratio constant in time.

This chapter is drawn from the physics-lab study notes and renders. The longer write-ups and project essays live on the blog.