Mathematics / Signal theory behind billiard ball 'wave pendulum' effect
As was pointed out in a comment above, this is a simple consequence of the dynamics of a pendulum. There's nothing particularly signal-processing-related about this problem, just some simple physics and trigonometry.

For a pendulum that is displaced from its angular equilibrium point by an angle of $\theta_0$ with an angular velocity of $\omega_0 = 0$ at $t = 0$, for small initial angles, the motion of the pendulum can be approximated by simple harmonic motion:

$$
\theta(t) = \theta_0 \cos\left(\sqrt{\frac{g}{l}}t\right)
$$

where $g$ is the magnitude of the force of gravity and $l$ is the length of the pendulum. In the video, a number of pendulums of varying lengths are given an approximately equal initial displacement $\theta_0$ and released simultaneously. Once they are released, each pendulum traces out a path whose displacement angle is a sinusoid with frequency related to the pendulum's length. The lengths were carefully chosen to give the interesting visual effect that you see in the video; the frequencies of the sinusoids are multiples of some fundamental frequency, so for particular values of $t$, they line up in interesting ways. 

You'll notice toward the end, all of the balls "line up" again; this is because each of the sinusoids that describe their motion have moved through an integer number of periods, so the balls are back at a common angular displacement. If there were no effects of friction, air resistance, etc., the oscillation would continue, with the same patterns repeating indefinitely.