Why do different wave shapes appear and disappear?

Each pendulum completes a different integer number of oscillations in the cycle time T. At t=0 all are in phase (synchrony). As time passes, the faster pendulums race ahead; at t=T/2 they have accumulated phase differences that produce a standing-wave pattern. At t=T/4 and 3T/4 you see travelling waves. At t=T all pendulums have completed a whole number of cycles and return to synchrony — the pattern resets perfectly.

What determines the pendulum lengths?

Pendulum k is assigned nMin+k oscillations in time T, giving period T_k = T/(nMin+k). From the small-angle formula T_k = 2π√(L_k/g), we get L_k = g·T_k²/(4π²). Longer pendulums swing more slowly and hang lower in the display; shorter ones swing faster and hang higher.

Is the simulation physically accurate?

Yes, for small angles. The simulation uses the exact small-angle SHM solution x(t) = A·sin(ω·t) where ω = 2π·(nMin+k)/T. Real pendulum waves use this same design principle — the Harvard Natural Sciences Lecture Demonstrations set used 15 pendulums completing 51–65 cycles in 60 seconds and became a popular physics demonstration worldwide.

What is the connection to Fourier analysis?

The collective motion of all pendulums is a superposition of N sinusoids with consecutive integer frequencies. The visual patterns are related to the Fourier structure of the group — when consecutive phases align they reinforce into a wave; when they spread uniformly they cancel into apparent disorder, then reform. This is analogous to how a Fourier series can represent a pulse spreading and reconcentrating over time.

Home▸Physics & Mechanics▸Pendulum Waves | Emergent Wave Patterns from Simple Pendulums

🧪 Pendulum Waves | Emergent Wave Patterns from Simple Pendulums

Interactive pendulum wave simulation: watch 8–25 pendulums of carefully chosen lengths sweep through hypnotic wave patterns — travelling waves, standing waves, serpents, and back to synchrony. Adjust N, amplitude, and cycle time in real time.

Physics & Mechanics2DModerate60 FPS

⚙ Under the hood

This simulation explores wave phenomena generated by a simple pendulum. It demonstrates how emergent wave patterns arise due to the continuous oscillation and reflection of waves.

Pendulum WavesEmergent Waves

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install