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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 StringModesLaunch.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 is the way that the bowed and plucked vibrating string motions shown in the Vibrating Strings applet can be built up out of normal mode motions, or standing waves. The window contains three graphs, and in every case the top graph is the mathematical sum of the bottom two graphs. Following is a general interpretation of the graphs.
Every mechanical system with linear restoring forces has a number of special patterns of vibration known as normal modes. In a normal mode vibration, all parts of the system vibrate periodically with simple harmonic motion at a common frequency. All parts of the system pass through their equilibrium positions simultaneously. A specific normal mode is characterized by its shape, frequency, and damping time (the time for the amplitude of vibration to die down to a specific fraction of its initial amplitude).
In general, when the system is given some initial disturbance and released, it will commence vibrating in some way that is definitely NOT a normal mode and usually not even periodic. However, the importance of normal modes is that any arbitrary vibration of the system can be expressed as a mathematical sum of normal mode motions, if each mode is given an appropriate amplitude and phase.
In the case of an ideal string tied down at both ends, the normal mode shapes are sinusoidal curves with wavelengths such that zeros of the sine functions occur at the ends of the string. The frequencies of the normal modes are integer multiples of the lowest-mode frequency. The actual motions of a bowed or plucked string are definitely not pure normal mode motions, as it is clear that individual points are not moving with simple harmonic motion. However, as you will see the actual motions of the strings can be obtained by adding normal mode motions.
1. At the bottom of the window is a button labeled “Go/Stop.” Clicking this button at any time stops or restarts the motion.
For comparison, the other button labeled “Ideal” toggles a display of the actual vibration of the string under either bowing or plucking, as shown in the Vibrating Strings applet. It is presented as a fine line added to the first two graphs. In reality, here as in the Vibrating Strings applet the “actual” vibration is approximated by a sum of the first 40 modes.
2. At the top of the window is a menu bar that controls the display. Available menu choices and their effects are the following:
| Menu Selection | Action |
|---|---|
| String Modes > About String Modes | Reveals the name of the author and the year the applet was written. |
| Excitation > Bowed | Shows the modes that may be summed to yield bowed-string motion. |
| Excitation > Plucked | Shows the modes that may be summed to yield plucked-string motion. |
| Modes in Sum > 1+2 | The top graph shows the sum of the first two modes; the middle graph shows mode 1; the bottom graph shows mode 2. |
| Modes in Sum > 1+2+3 | The top graph shows the sum of the first three modes; the middle graph shows the sum of the first two modes; the bottom graph shows mode 3. |
| Modes in Sum > 1+2+3+4 | The top graph shows the sum of the first four modes; the middle graph shows the sum of the first three modes; the bottom graph shows mode 4. |
1. Examine individual modes. When “Modes in Sum > 1+2” is selected you can view modes 1 and 2, and modes 3 and 4 are shown when the next two menu options are selected. Note the sinusoidal shape of each mode, with each higher mode having one more antinode than the previous one. The modes are depicted with the proper shapes, and relative frequencies, amplitudes, and phases to yield the desired overall motion for bowed or plucked strings. The relative amplitudes and frequencies of the first two modes are virtually the same for the two excitation cases. But, compare the relative phases of the first two modes for the bowed and plucked strings. Do this when “Modes in Sum > 1+2” is selected by stopping the motion at a time when the string in mode 1 is passing through equilibrium. At that instant the bottom graph showing the mode 2 contribution to the sum will be at a different point in its cycle for a bowed string than it is for a plucked string.
2. General inspection without the “Ideal” graph. Generally notice that by the time the first four modes are combined in each case, their sum yields a motion that is getting close to what is expected for a bowed or plucked string.
3. Careful inspection. Select “Modes in Sum > 1+2+3,” and stop the motion when the bottom mode 3 graph is clearly non-zero. Then click the “Ideal” button to turn on the “actual” shape of the string. Notice places in the middle graph of modes 1+2 where the sum of the first two modes deviates significantly from the actual shape. See that the contribution from adding mode 3 at those places will bring the sum closer to the actual shape, as shown in the top graph. What is most amazing is that the improvement from adding mode 3 holds at all times, not just the instant when you stopped the action.
4. Overall satisfaction. It is remarkable how close one can get to the actual string motions with just four modes. The places where the four-mode sum is most inadequate are places where there is some sort of sharp bend in the actual string. It takes higher modes, with sharper curvatures, to develop those features.