In this problem, you'll use a geometrical method to construct standing waves for a plucked string. For simplicity, we'll assume equal and constant tension throughout the string. We'll also assume that the string is plucked at its center. This situation can be modeled as that of triangular component waves moving in opposite directions on the string to produce standing waves. At the instant that the string is released, these component waves begin moving in opposite directions with equal speeds, frequencies, wavelengths, and amplitudes. They reflect from the fixed ends and continue moving back-and-forth in that manner. At any instant of time, the shape of the string is determined by the superposition of the component triangular waves.
A photograph of the component waveform is shown below. The string is shown being held in the release position. After it's released, it takes on a trapezoidal shape. (A flash unit discharged twice in succession in order to capture the two images overlaid in one photo. The tape recorded at the bottom was used to detect the twang of the release and discharge the flash unit automatically.)
The photograph below shows 4 successive positions of the string after release.
Your assignment is to verify by wave superposition that the waveform is trapezoidal. Open the graph on this page. It shows triangular waves moving in opposite directions at equal speeds. In the first frame, the two waves overlap perfectly. Thus, the superposition at any point has double the displacement of the component waves. This is like the situation when the string is first released. In the second frame, the waves are shown one-eighth of a period later. Each wave has moved a distance of λ/8, one to the right and the other to the left. Do the following. Use colored pencils if you have them in order to distinguish different waveforms.
If you play a stringed instrument, you may be interested to know that these waveforms are characteristic of those on your instrument, whether the string is plucked or bowed. For an instrument, however, the string is set into oscillation near the bridge, which is far from the midpoint of the string. This results in much richer tones than one would get by plucking or bowing at the center. The waveforms for an instrument would look more nearly like those in the figure to the right. The shape of the waves is angular as before, but the plateau region isn't horizontal. Here's a short video clip of an actual vibrating string plucked off-center.