The Mass-Spring Chain

How does a chain of point masses turn into a medium that carries waves, and what controls their speed, modes, and reflections?

Interactive · live simulation

Send a Gaussian pulse down the chain, or excite a pure normal mode. Toggle the ends between fixed (pulses reflect inverted) and free (no inversion), and add damping to watch the wave decay.

The Mass-Spring Chain — simulation render — fig 1. A Gaussian pulse on a fixed-fixed chain, drawn as a space-time waterfall: it splits into left- and right-going halves that each reflect inverted off the walls.

A discrete wave equation

Each interior mass feels m\,\ddot u_n = k(u_{n+1} - 2u_n + u_{n-1}) - \gamma\,\dot u_n. That second difference is a discrete second spatial derivative, so in the long-wavelength limit the chain is exactly the 1D wave equation u_{tt} = c^2 u_{xx} with speed c = a\sqrt{k/m}: stiffer springs and lighter masses make faster waves.

fig 2. The same pulse with free ends — reflection without inversion. Side by side, this is the cleanest way to see what a boundary condition does.

Boundaries are physics

Fixed ends reflect pulses inverted; free ends reflect them upright; periodic ends wrap the chain into a ring where waves travel forever; a driven end is an oscillating source. Running the fixed and free pulses side by side makes the inversion unmistakable.

fig 3. Pure mode 1 oscillating in place, inverting at half a period — a standing wave seeded directly from the dispersion relation.

Lattice dispersion is real

A fixed-fixed chain of N nodes has N-2 standing modes with \omega_j = 2\sqrt{k/m}\,\sin\!\big(\pi j/2(N-1)\big). For long wavelengths this is the familiar evenly-spaced harmonic series; for short wavelengths the sine saturates and modes travel slower than the wave equation predicts. That gap is numerical dispersion — and it is a genuine, physical feature of any discrete medium, not a bug.

fig 4. Driving the left end at the fundamental: with damping the energy climbs to a steady state set by drive-versus-dissipation balance.

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