Spring-Mass Lattice Wave Propagation

A 1D chain of masses connected by springs. Drive one end and watch transverse waves propagate — reflecting, interfering, and forming standing waves.

Drive freq: 1.0 Hz
Drive amp: 30
Spring k: 800
Damping: 0.02
Boundary: Fixed

Phonons & Wave Mechanics

Each mass obeys Newton's second law: mẍᵢ = k(xᵢ₊₁ − 2xᵢ + xᵢ₋₁) − γẋᵢ. The dispersion relation for this lattice is ω² = (4k/m)sin²(qa/2), where q is the wavenumber and a the lattice spacing. This departs from linear (sound-like) behavior near the Brillouin zone edge q=π/a — a hallmark of discrete lattices. Standing waves form when the wavelength is commensurate with the chain length. Toggle the right boundary between fixed and free to see different reflection conditions.