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Pendulum wave 🌊

\[\begin{equation} \theta(t)=\theta_0 \cos \bigg( \sqrt{ \frac {g} {L}} t \bigg)\text{, where }L = \frac{g T^2}{4\pi^2} \end{equation}\]

Click to start the animation!

⭐ Based on the VPython PendulumWave and inspired by www.dynamicmath.xyz
👉 Related to Newton's cradle simulation

🧠 Pendulum wave synchronization

At first glance the pendulum wave looks chaotic, but it is actually a perfectly designed periodic motion!

The key idea:

👉 Each pendulum has a slightly different period, chosen very carefully.

Each of the pendulum’s period is carefully chosen

\[\begin{equation} T_i = \frac{T_{pw}}{N + i} \end{equation}\]

where $T_{pw}$ is the period of the pendulum wave. As a consequence, after a time $T_{pw}$:

\[\begin{equation} \text{number of oscillations} = \frac{T_{pw}}{T_i} = N + i \end{equation}\]

👉 That is an integer.

So every pendulum has completed 15 swings, 16 swings, 17 swings, etc.

Because each pendulum completes an integer number of cycles, they all return to the same position with the same phase, i.e. they re-synchronize perfectly.

What happens in between?

Between $t = 0$ and $t = T_{pw}$:

Phases drift apart
The pattern looks like a traveling wave
Then becomes messy / “random”
Then reorganizes again

This is due to phase differences evolving linearly:

\[\begin{equation} \theta_i(t) \sim \cos(\omega_i t)\text{, with } \omega_i = \frac{2\pi}{T_i} \end{equation}\]

🎼 A many-body “beat” phenomenon

Although here you’re seeing many frequencies interfering in time, it’s similar to:

musical harmonics
interference patterns
Fourier superposition

🧠 Why the wave pattern appears

At intermediate times:

Neighboring pendulums are slightly out of phase
This creates spatial phase gradients

Our brain interprets this as a wave traveling through the system. However, nothing is actually traveling — it’s a mere illusion.

🧩 The deeper math

This is a system of pendulums, where for each individual pendulum we have:

\[\begin{equation} \theta_i(t) = A \cos\left(\frac{2\pi}{T_i} t\right) \end{equation}\]

Because all $T_i$ are chosen as:

\[\begin{equation} T_i = \frac{T_{pw}}{N+i} \end{equation}\]

all frequencies are commensurate (rationally related). This guarantees a common period and an exact recurrence, since after a time $T_{pw}$:

\[\begin{equation} \theta_i(t + T_{pw}) = \theta_i(t) \end{equation}\]

for all pendulums.

Summarizing:

👉 The wave comes back because every pendulum “keeps perfect count” and they all land back in sync after completing whole-number cycles.

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