As the dowel turns on the motor shaft, the end of the string that is taped to the dowel moves in a circle. If you think of circular motion as a combination of vertical and horizontal motion, you can visualize the string as being shaken up and down at the same time as it is being shaken right and left. The shaking of the string causes wave pulses to travel along the string.
You and a partner can use a jump rope to produce wave pulses very similar to those in the string. If you hold one end tightly in place while your partner shakes the rope, the pulses will bounce off your end and travel back. These returning pulses will travel through the ones your partner continues to make. Since any particular piece of the rope can only be in one place at one time, the two waves traveling in opposite directions combine with each other, adding together to produce a single overall shape for the rope. This is still true even if you both shake the rope to produce wave pulses. If you both shake the right way, you can produce an overall shape that is stable and exhibits nodes and antinodes. This stable overall shape is called a standing wave. The various stable patterns you produced with your String Machine are also standing waves.
The simplest standing wave that you produced was the fundamental, or first harmonic. The fundamental is one-half of a whole wave, or one-half a wavelength.
The second standing wave you made, the one with three nodes, is a second harmonic wave, and it is a whole wavelength. The third wave you made, with four nodes, is a third harmonic wave, which is one-and-a-half wavelengths.
For waves in a string, standing wave formation normally depends on a number of factors, including the frequency with which the string is shaken and the tension in the string. If any of these factors were changed, then the standing wave pattern would change. In your String Machine, however, it's possible to change the frequency (motor speed) and tension (spread between ends of the motor arms) without changing the pattern because of the circular motion of the string. This motion introduces forces on the string not present in waves generated in a single plane, as in the simple case of the jump rope being shaken up and down.
When you “strobed” the string by waving your hand back and forth in front of it, you were able to get successive views of the string over short time intervals. If the time it took for adjacent spaces between your fingers to change places in front of your eyes were exactly the time it took for the string to go through one whole cycle (or any whole number of its cycles) then you would see the wave in the same position each time, and it would appear to be standing still.
On occasion—possibly due to slightly different motor speeds—the moving string may seem to develop sub-patterns within the main pattern, which will vary slowly in a regular way. This behavior is the result of waves with slightly different frequencies interacting in a complex way to produce a regular alternating or oscillating pattern called a beat. Perhaps the simplest and most well-known example of a beat is the loud-soft-loud-soft tone produced when musical instruments (or two different strings on the same instrument) are being tuned. The two instruments are in tune—each producing the same frequency note—when the variable tone is no longer noticeable.