# Pendulum Waves

This example shows interesting wave patterns that emerge among an array of simple pendulums with carefully chosen lengths. It is based on the physical system that can be viewed at www.youtube.com/watch?v=yVkdfJ9PkRQ

Under a common approximation of pendulum motion, the period of a pendulum depends only upon the length of the pendulum and the acceleration of gravity. The pendulum system in this example comprises 24 pendulums with lengths chosen so that each pendulum completes a different integral number of oscillations during a 60 second "system period". The number of oscillations of the pendulums during the system period lie among the 24 integers in the range [68, 91]: the longest pendulum completes 68 oscillations while the shortest one completes 91. When the pendulums are released from rest at a common initial angle, wave patterns emerge in their motion. These patterns are most evident when the system is viewed from the side or top.

The entire simulation runs for two minutes, or two system periods; at the one minute mark, the pendulums return to their initial state, and the pattern repeats. The formula for computing pendulum periods assumes a pendulum executes simple harmonic motion. In fact, this is an approximation that becomes less accurate as the angular amplitude of the swing increases. As a result, slight discrepancies emerge in the motion patterns, and it is evident that after two system periods, the system has not quite returned to its initial state. (The true period of a simple pendulum is longer than the one computed under the harmonic motion assumption.) Reducing the initial angle improves the repeatability of the wave patterns. 