Making a stretched string
Supplement for chapter 8
There is no better way to gain a grasp of the many principles and concepts already encountered than with Paul Falstad's Loaded Strings Java applet,
This applet is the perfect companion for our investigations. You can change the number of masses (called "loads'' in the applet), change tension, set the initial displacements of the masses arbitrarily (i.e. shaping them or plucking them) and view the subsequent motion at any speed. The motion can be analyzed in several different ways, and what is more it can be heard by checking the sound button. The frictional damping can be adjusted, stimulating the frictional effects of air and internal friction in the string. Not only can you pluck and shape the string, but you can also drive the system sinusoidally, either resonantly or nonresonantly, from the center. We'll have much more to say about this in the chapter on resonance.
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Overall, this applet is far more useful than having a real stretched string at hand, because of its capacity to decompose the motion into its individual component parts or modes, its ability to build up the string from an ever-increasing number of small masses, to shape the string arbitrarily, to specify the amplitudes in individual modes, and to watch the motion of the string in slow motion. Still, having a real stretched string at hand, in the form of a violin, guitar, etc., in almost any condition, is invaluable.
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We will follow the logical development with a parallel investigation in Loaded String, beginning with only one bead, setting the Damping and the Driving Force options to 0. There is not much that can be done with such a simple system, but it is still worth watching the motion, changing the tension, and adjusting other controls. It is somewhat premature, but you can turn the sound on and hear a sine tone, i.e. a simple tone, that consists only of sinusoidal oscillation at one frequency. This tone corresponds to the sound that the vibrating bead would make, if you assume that somehow it was connected to walls that could move a little as the bead vibrated or connected to a resonator. These issues are a little ahead of ourselves, but it is worth pointing out again that small beads on filaments make almost no sound unless the vibrations are somehow connected with things that push air around better than a bead and string does.
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The pure modes involve one stalk raised only. Any shape can be reproduced as a combination of all the pure modes, if the correct amplitudes are used. Compound oscillation, wherein two or more pure modes are excited at once, can easily be produced with the "Mouse = pluck string'' option. Or, you can pull up two or more amplitude stalks at the left bottom corner, creating a combination of pure modes. Finally, you can select the "Mouse = shape string'' option and make any initial shape for the string. As you pluck the beads, thus shaping the "string'', the applet automatically finds the right combination of the pure modes, and reveals that combination as a set of amplitudes in the lower left of the panel. If the mouse is held over one of the stalks, it turns yellow, and the appropriate pure mode panel above also turns yellow, revealing the pattern of vibration associated to that stalk, i.e., normal mode. If the mouse is held over the stalk while the application is animating the vibration, the phase and amplitude of the individual normal mode taking part in the combination is shown. It is difficult to imagine a more instructive applet to illustrate these points.
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2 Beads
Once you have grasped 2 beads, it is almost true that you understand any number of beads on a stretched filament. Each new bead means a new mode is created; the modes that cause adjacent beads to go in opposite directions at the same time cause a stronger restoring force and a resultant higher frequency.
This 2 bead video isn't perfect; the higher frequency mode has a little bit of the lower frequency mode in it - that was a matter of an imperfect "pluck". But you get the idea below. This video mirrors the 2 bead case discussed in Why You Hear What You Hear.
Project: 30 Bead strand - experiments in recording and analysis
Caption: Experiments with a 30 bead string. Top left, A: Waveform of the sound recording for the case of a sharp pluck of the last bead on the string. It is highly non repetitive. Lower left: 30 bead system showing the initial pluck and the resulting amplitudes, all of which are quite low but non zero. Upper right, A: spectrum of the sound recorded for the sharp pluck. Note the presence of many peaks at high frequency, and also that the peaks are roughly equally spaced at lower frequency, but pile up at high frequency. This plot is linear on the frequency scale, but logarithmic in the sound amplitudes, so that very small amplitudes are magnified compared to larger ones.
At low frequency, a 60 Hz peak due to ambient noise i n the room, a quite common source of noise coming from alternating current power, is seen. We shall discuss logarithmic decibel scales in succeeding chapters. Top left, B: Waveform of the sound recording for the case of only five of the lowest modes excited. Notice that the waveform is quite nearly periodic. Lower right: screen shot of the 30 bead system with this initial amplitudes and displacement. Upper right, B: spectrum showing four strong peaks corresponding to the four amplitudes initially selected. noticed that these peaks are nearly equally spaced. The peak near 60 Hz, closest to the origin, should be discounted as electrical hum. Listen to the sound below:
It is instructive to actually measure the spectrum of frequencies, and see the connection between which modes are excited and the type of pluck causing the excitation'. This is good practice in sound recording and analysis. Loaded String applet can be used as a sound source; most computers can record themselves on their own built in microphone at he same time.
Once a sound is recorded, it can be analyzed for its amplitude and frequency content. This is done as follows: Using Audacity, or Amadeus, etc. record a pluck of the string in Loaded String applet for a few seconds. If you use Audacity, set the sample rate for 8000 or 16,000 in the preferences. This has the effect of emphasizing the lower frequency regions in the spectrum to be calculated later. Create a system of beads, and then pluck the system after pressing the Clear button, and setting Mouse = shape string. Note the difference in tone when different beads are plucked, and note too that the amplitude stalks are different in each case. Starting with your recording in Audacity, select Analyze... Plot Spectrum from the top menu bar. You should get a series of peaks, as in figure~{fig:30beads}, each one corresponding to the frequency of one of the normal modes of the bead system that have all been simultaneously excited with your pluck. You can check the veracity of the whole arrangement by going back to the applet and using the amplitude stalks to create the initial conditions for the bead system. The corresponding spectrum in Audacity should reflect the stalks you picked. This will build confidence in the capabilities of the sound analysis program.
A complete spectrum for 1 to 30 beads (no. of beads along the horizontal axis, mode frequencies along the vertical). Notice the lower frequencies becomes nearly equally spaced as the number of beads gets large. The entire spectrum becomes exactly equally spaced in the limit of an infinite number of beads.
PHET linear chain
PHET linear chain interactive Java applet simulates a string of 1 to 10 masses connected by springs. This is not quite the same as a stretched string of beads, but it is analogous, and for "vertical polarization" (the masses move up and down vertically, perpendicularly to the line of masses) in the applet, it is treated the same way mathematically.
PHET string
In a useful, hands-on demonstration of a stretched string, which includes adjusting boundary conditions, damping, and tension, plus the ability to shake the string from one end. This helps to get a tactile feeling for resonance.
Normal modes of a beaded string CDF applet
Here you can explore the way modes proliferate and adopt various positions along a sinusoid as more beads are added.
Normal Modes of a Beaded String from the Wolfram Demonstrations Project by Enrique Zeleny
Project: The metal chair and other miscellaneous objects
The chair has many modes, like the string (actually many more), but they are rather disorganized in frequency - no systematic spacing of the frequencies of vibration. Listen to the sound recording of a hit with a knuckle on the seat of the open chair below. This is the file that is displayed in Chapter 8, along with its Fourier power spectrum, obtained just as with the string, but with results that confirm the statement about "rather disorganized" above.
Find your own object and do the same sort of analysis; try analyzing different places to hit, and/or with hit with different "hammers".
While we are on the subject of the vibrations of weird and arbitrary objects, the analysis of the modes of a beer bottle by Prof. Daniel Russell is of interest.
Exciting modes by the way you strike
Below is an elementary, two bead example "hammered" so as to excite only the second mode (top), or hammered on a single bead only, this excites both modes at once (since they both shake that bead).
Project: The sound of a string
It is rather difficult to show that a vibrating string make very little sound in and of itself, since whatever the string is attached to is likely to deform as the string oscillates, owing to the oscillating tension that the string exerts on the object. Oscillations of microns or even, at higher frequencies, nanometers can be heard. Even thick steel walls would deform and vibrate near the points of attachment. One partial solution is to use people holding the string in their hands. A fleshy hand transduces sound poorly, especially at moderate to low frequencies, owing to its small size and strong damping qualities. Try holding the string tightly using two people with a third person plucking the string. It will be heard faintly.
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Idea: use a mic and try to find out where most of the sound is coming from. HOWEVER, watch out for evanescent wave effects, see especially Chapter 7. Evanescent waves can be loud near the source, but die off as you get away from it due to short circuiting of the sound. Essentially, the wavelengths in close to the source are mostly shorter than the true wavelength at the frequency of vibration; these cannot propagate away from the source. If you hold the microphone close to the string or hands (within about one wavelength or even further away) the sound may be artificially loud compared to what is heard further away! Note this effect when you are listening too! Maybe you need a fourth person to judge loudness.
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Project: String with a "defect"
Consider a stretched string made of zillions of beads, so that we may consider it to be continuous. We break the perfection with an extra bead in the center of the string. It adds mass there, but every other mode (the second, fourth, etc.) has a node in the middle, and doesn't "notice" the bead sitting there, since it does not move. The first, third, etc. modes however are saddled with an extra mass to move up and down, but the with same restoring force, so their frequency decreases. The result, when such a weighted string is plucked, is sour sounding, even for a very modest extra mass (try this with putty, gum, etc. on a violin, ukulele, ... etc.). The Helmholtz kink oscillation requires periodic motion, that in turn requires the mode frequencies to be multiples of a fundamental frequency. With half the modes lowered in frequency, this is no longer the case, and indeed a violin string becomes impossible to bow successfully when it is weighted in this way.
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The experiment with a little putty is entirely nondestructive and it is recommended that you make you own experiments, including measuring the frequencies with the weight on and off (pluck the string in various places, record it, and find the peaks in the power spectrum).
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The affected modes are lowered in frequency and shown in red; before the load was placed in the middle, they would have been 220, 660, 1100, etc. Hz. The even numbered modes are unaffected since the weight is at a node and the mode doesn't "notice" it is there.
