As I touched on already, a wave pendulum is a series of pendulums with incremented frequencies. One way to think of the pendulum wave is as a series of points used to sample a wave of increasing frequency. This effect is shown in the video below, which I created using Matlab:
The Nyquist Sampling Theory (NST)
states that to sample a wave of a given frequency, one needs to measure points at one-half cycle of that frequency. Or, for a given frequency f, one needs points spaced 1/(2*f) units of time apart (since T = 1/f). When the number of sampling points becomes less than the number required by the NST, aliasing occurs
. This aliasing is the reason for the alternating waveforms present in a pendulum wave.