Materials:

• Pendulum Wave Apparatus
• Initiator Stick

Demo:

The apparatus shown above is known as a wave pendulum, where nine steel balls of equal mass are suspended by strings of varying length to form nine separate pendulums. To begin the demonstration, a flat “initiator stick” is pushed against each of the nine weights to where they are level, and equally distanced from their initial positions by an angle θ. By pulling the stick away so that each pendulum starts instantaneously, the nine pendulums begin to oscillate at unique periods. Together, they form a wave-like pattern that undergoes multiple phase relationships. This demo is currently assembled, but in the case of needing assembly instructions, please reference this manual: Wave Pendulum Assembly

Explanation:

Each of the nine pendulums, although suspended by two strings rather than one, can be treated as a simple pendulum system interacting with gravity. This is because the forces of tension (T1 and T2)  resulting from the two strings can be combined and utilized as one force of tension (T) that is perpendicular to the pendulum’s motion. Additionally, the distance from the steel ball to the acrylic stand can replace what is normally the length (L) of a single-string system.

Displacing a simple pendulum by an angle θ results in a vertical displacement  h = L(1-cos(θ)). The force of gravity acts as a restoring force in this position, compelling the ball to return to its equilibrium point at θ = 0. As a result, the pendulum swings back towards its original position, decreasing height h as θ decreases, and gaining a velocity. The velocity is maximized as it passes the equilibrium point as its potential energy is converted into kinetic energy. Consequently, the pendulum enters simple harmonic motion.

The restoring force due to gravity can be described as f = -mgsin(θ), which can be applied to Newton’s second law in order to find the period as T = 2π√(L/g). We can see that a pendulum’s period is only dependent on its length and gravity.

In a wave pendulum, the lengths of each pendulum are chosen to influence their periods. From the shortest string to the longest string, the periods become increasingly longer. Since the difference in lengths increases only slightly between each pendulum, they start out relatively in sync, but quickly start to differ to where a sinusoidal wave seems to appear. This harmonic wave seems to “travel” perpendicular to the direction of the pendulums’ oscillations.

As time goes on, the wave created undergoes multiple phase relationships. When two steel balls are oscillating along the wave with a distance equal to a wavelength, they are considered in-phase. Conversely, two steel balls are out of phase if they are distanced by more or less than a wavelength.

When all of the steel balls are in phase, this indicates the beginning of the cycle, such as the position before they are released. A single wave-like pattern is created when adjacent weights are all slightly out of phase, while a few located at the peaks of the wave are in phase with one another. As the distance between adjacent steel balls becomes closer to half a wavelength, two separate waves begin to form, interfering in a way that appears destructive, until the distance reaches a half wavelength and two completely straight lines are formed. At this point, the cycle is reversed, and the weights work their way back into being completely in phase with one another. In transitional stages, where the phase differences have no significance, the motion could be described as complex, or chaotic.