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[Note: The Java Applet for this animation can no longer be run in a web browser. Clicking this link, however, will download to your computer a Java Web Start file named StringHarmonicsLaunch.jnlp. This file can launch the animation in a separate window on your computer. Downloading and executing the file may be blocked by local Java permissions, but persevere! You may finally have to launch the file by right-clicking (control-clicking) it and choosing Open, instead of simply double-clicking it.]
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Illustrated here in slow motion are three versions of the transverse motion of an ideal bowed string, either an open string, or a string touched lightly in the middle or one-third of the way from one end as a string player would do to play harmonics of the string. It is seen that the string is vibrating on both sides of the point where the string is touched, but with a period that is one-half or one-third that of the open string.
This animation is closely related to the bowed string case of the Vibrating Strings applet, the description of which gives useful information for this applet as well. As in that demonstration, in addition to showing the Helmholtz motion of the bowed string this applet may be used to plot time-dependent graphs of the displacement and/or velocity of any point on the string, and the transverse component of the force exerted on the left-hand support by the string.
Aside from one detail, the motions pictured here are computed from a sum of normal mode vibrations, using exactly the same modes, relative amplitudes, and phases that were used for the open bowed string in the Vibrating Strings applet and shown in the String Modes applet. A touch to the center of a bowed string will damp out any modes (the odd-numbered ones) possessing an antinode in the center of the string. But that touch will not affect modes (the even-numbered ones) with a node at that point, because the latter modes do not involve any motion of the center of the string; touching the string does not adversely affect those modes. Thus the illustration of a string touched in the midle is computed by taking the sum of modes for an open bowed string and simply omitting all the odd-numbered modes from the sum. Similarly, to obtain the illustration for a bowed string touched one-third of the way from one end, one retains from the modal sum for an open bowed string only the modes numbered 3, 6, 9, ..., since these are the only ones having a node one-third of the way from one end.
1. At the bottom of the window are several controls for the display. On the left is a button labeled “Go/Stop.” Clicking the button at any time stops or restarts the motion. This control must be used to start some of the motions of the string.
The remaining controls at the bottom of the window are only effective when a graph has been chosen from the menu bar. The button labeled “Clear” restarts the graphing if desired, when system changes in mid-graph have modified the character of the graph. The right-hand slider may be used to move the point whose motion is being graphed to any location on the right half of the string.
2. At the top of the window is a menu bar that affects the display. Available menu choices and their effects are the following:
| Menu Choice | Action |
|---|---|
| Harmonics > About Harmonics | Reveals the name of the author and the year the applet was written. |
| Touch > None | Selecting this shows the motion of a bowed open string. |
| Touch > MIdpoint | Selecting this shows the motion of a bowed string when a finger lightly touches the center of the string, thus stopping motion there. |
| Touch > 1/3 or 2/3 | Selecting this shows the motion of a bowed string when a finger lightly touches the string one-third of the way from either end, thus stopping motion there. |
| Graph > Displacement | Turns on or off a graph of the transverse position of the observation point as a function of time. The observation point is shown by a red dot on the string and may be moved using the slider at the bottom. This graph and the other graphs may be displayed simultaneously. |
| Graph > Force | Shows or blanks a graph of the transverse force of the string on the left support as a function of time. This and the other graphs may be displayed simultaneously. |
| Graph > Velocity | Toggles a graph of the transverse velocity of the observation point on the string as a function of time. This and the other graphs may be displayed simultaneously. |
Looking at one or more of the graphs, compare the periods of vibration of a point on the string for open or touched strings, and verify the statement in the first paragraph, above. The kink travels along the length of the string at the same speed in all cases, but for the harmonics effectively makes a round trip along only half or one-third the length of the string.