Recently I've seen various videos showing the pendulum wave effect. All of the videos which I have found have a pattern which repeats every 60s.
I am trying to work out the relationship between the period of the overall pattern and the difference in length between each of the pendulums.
From the small-angle approximation for the period of a simple pendulum, for each pendulum with period T we have:
T≈2π√Lg,
Where L is the length of the pendulum and g is the acceleration due to gravity.
If we take (say) n pendulums, with lengths L,L+d,…,L+(n−1)d, then the pattern repeats at a period t when all of tL∈Z∗, tL+d∈Z∗, …, tL+(n−1)d∈Z∗. where Z∗=N∪{0}.
However, I'm not sure how to develop a direct relationship between t, d and L.
Answer
Taking equally spaced lengths will not reproduce the pendulum wave effect, simply because it's the periods that you want to fulfil commesurability conditions, and lengths and periods are not linearly related.
To get a pendulum wave, say you have N pendula with lengths ln. Then, as you know, the nth pendulum will have a period Tn=2π√lng.
Given the above relationship between Tn and ln, the commesurability condition reads ln=g(T2πn)2
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