Standing waves on a plucked string

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.

1. Construct the superpositions for Frames 1 and 2.
2. In Frame 3, draw the component waves, assuming that each has moved another λ/8. Then construct the superposition.
3. Repeat step 2 for Frame 4.
4. Now examine the 4 superpositions that you've constructed. Did you obtain a standing wave? How can you tell?
5. On a second sheet of graph paper (you can download a template here), draw the 4 superpositions on the same set of displacement axes so that they overlap. Draw them exactly the same size as in the original graphs. The result shows you the standing wave as a function of time. As a check on your work, you can view this animation.
6. Let's use the following terminology to describe the waveforms. The slanted sides will simply be called the sides. The flattened top (or bottom) will be called the plateau. How does the slope of either side change with time? (Trace this through a half-cycle of the motion.)
7. How does the slope of the plateau change with time?
8. Download and view this animation of the waveforms through one-half cycle. The animation was created using actual data from photos. The string is released from the bottom. Play the animation several times to see how the waveform changes. How does it differ from your graph?
9. The differences are due to the fact that the assumption of constant tension isn't realistic. The amount that the string is stretched changes as the string oscillates. The linear density of the string changes, too. When it's stretched more it has smaller linear density.  As a result of these factors, the wave speed isn't constant. When is the wave speed greatest? Explain using the equation v = (FT/μ)½.

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.