Normal Modes of a Mass-Spring Chain

A fixed-end chain of $N$ masses on springs oscillating in a chosen normal mode, with the dispersion relation alongside. The monatomic chain has exactly $N$ modes at $\omega_n = 2\sqrt{K/m} \sin\left(\frac{n\pi}{2(N+1)}\right)$, each a standing wave with $n-1$ internal nodes. Switching to a diatomic chain with two alternating spring constants splits the spectrum into an acoustic and an optical branch separated by a zone-boundary band gap that closes exactly when the springs are equal. The scene animates the chain in the selected mode while the side panel draws $\omega(k)$ with the gap shaded; clicking the curve picks a mode. This is the one-dimensional origin of phonon bands and the acoustic/optical split in real crystals.