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Resonance Rules

Supplement for chapter 9

Excellent introduction to the idea of resonance, with illustrations from the Smithsonian collection, by Curator Steven Turner.

The key concept behind resonance is adding amplitude constructively to undulatory amplitude which is already present. When you push a child on a swing such as to amplify their swing, you are pushing resonantly. You push as the swing begins to move away from you, adding more velocity, in the same direction, to what was already present without your push. You are constructively adding velocity amplitude to pre-existing velocity amplitude. Pushing even once is resonant addition but doing it repeatedly at the proper times is resonant driving of a system.

Nikola Tesla (1856 - 1943) - Master of Resonance

Tesla's famous lightning shows from his Tesla coils were spectacular and fearsome. The coils are based on resonance, and Tesla's inventions are the basis of high voltage AC power transmission. He understood resonance in electrical and also mechanical terms. Once he sent an entire Manhattan building into resonant vibration with a modest off-center spinning weight attached to a beam in the basement. The police arrived just as he reportedly was smashing the device with a sledgehammer to stop the vibration. Tesla dreamed of resonant power transmission over long distances (possible, but rife with practical problems), without wires. He was an extraordinary genius and also extraordinarily neurotic. If you want to know more about Tesla, watch the 43-minute video here:

The Russel Acoustics Manhattan skyscraper story

As mentioned in section 9.2, the problem of an older 40 story building in Manhattan newly swaying was solved by Norman Dotti; story from Physics Today here.

Music of the Spheres - Literally

Suppose we turn the motion of the planets into sound, i.e. try to create the music of the spheres or the music of at least some of them. Since the orbits are fairly circular and circular motion, as we know, generates sinusoids, (see chapter 3), we will assign a sinusoid a frequency according to a constant times the actual orbital frequency of the planet. To augment the realism, we take the amplitude of each frequency to be the proportional to the orbital radius of the planet from the sun.

In the below sound, only Mercury, Venus, Earth, Mars, Saturn, and Jupiter are used, the planets known to the ancients.

Music Of Spheres
00:00 / 00:07

The frequencies are, respectively, 33 (Jupiter), 84 (Saturn), 528 (Mars), 1004 (Earth), 1613 (Venus), and 4166 (Mercury) Hz. The corresponding relative amplitudes, based on 1 for Earth, are 9.5, 5.2, 1.52, 1, .72, .39. The resulting tone possesses a strong 33 Hz hum tone, with a much higher perceived pitch almost exactly that of Mars, 526 Hz.

This raises an interesting point. In chapter 23 we shall learn that pitch need not be one of the partials present, but that perceived pitches do correspond to certain prominent early peaks in the autocorrelation function. These peaks, we shall see, can arise as a combination effect of several prominent partials, but the resultant collective effect is to make a correlation peak at a time that agrees with none of the periods of the individual partials. But what if say 3 partials with periods t1, t2, t3, acting together with given amplitudes create an autocorrelation peak at a time t that coincides none of these three, quite likely. Then a fourth partial with period t4 is added that happens to coincide with t, i.e. t4 = t. Clearly the perceived pitch of period t can only be enhanced. This seems to have happened with Mars: there is a peak in the autocorrelation corresponding to 526 Hz even without Mars included, although not one that causes a pitch dominance. Nonetheless, this peak makes the mars pitch all the more prominent when Mars is added to the mix.

Project: Pitch and adding a partial.

According to this idea, if we had an effective pitch f as above (often called a residue or virtual pitch) , and we now at a new partial with pitch f', the impression of a strong, dominating pitch should loom up when f'=f. This is easy to set up in MAX Partials, including variation of the pitch.

Since this project comes up in our "Music of the Spheres" context, but really belongs in chapter 23, we shall repeat it again there.


Resonance and the glass harmonica.

The glass harmonica was invented by Ben Franklin in 1761. Rather than trying to describe it, here is one being played by a skilled musician:

*** looks like an image is missing here on the live site ***


The glass bowls are rotating on a spindle, and set into motion by the fingers in a manner we shall describe shortly. They are not touching each other. The bowls flex and produce sound in a way that is described in chapter 20. The smaller bowls on the spindle you see in the video produce the higher frequency tones, that you can hear are quite pure, with little overtone intensity. The pitch is not dependent on the rate of the spindle rotation, except that there is an optimum speed to get the tone to form quickly.

With clean fingers and water on the surface of the glass, the "squeaky clean" effect is put to use in setting up a cooperative resonance. Cooperative resonance is considered in several later chapters. However this is a particularly easy case to understand, so we will attempt it here. The stick-slip motion of a clean finger on clean glass with water is caused by the natural adhesion of skin to glass, overcome by accumulating force if the finger is moved along by muscle power. That is, the finger jumps forward, re-adheres, only to succumb a moment later to growing force trying to move the finger forward. This jumpy motion flexes the glass, finger, and water and gives rise to an emitted squeaky tone.

The glass deformation created by the finger stopping or starting its motion will propagate as a wave away from the point where finger touches glass. If that wave encounters barrier or other impedance changes, it may return to the vicinity of the finger: an echo. A finger transiently stuck to the glass might be coaxed to release at a certain moment as the returning vibration echo passes by and the surface . Soon, the finger and the glass agree on a periodic motion at a frequency determined almost entirely in this case by the glass - how fast does the echo return? That periodic motion becomes a tone that is heard - the deformation of the glass bowl resonantly built up, with the softer "squeaking" noise taking a back seat. This is a perfect example of cooperative resonance: it is resonance, because the finger ends up driving the bowl at the frequency of one of its natural modes of vibration.

This is exactly the mechanism at work in Benjamin Franklin's glass harmonica - a whole set of glass bowls of different pitch all skewered on the same rotating rod.

Eigler's quantum corrals

In Why You Hear What You Hear, we show a picture of quantum electron wave interference. Electrons jump off a tiny metal needle and crash into Iron atoms put into a closed pattern on the smooth surface of a copper crystal. The electrons partly escape, and partly reflect off the "walls" made of Iron atoms. The electrons are really wave like, making the situation exactly analogous to sound wave from a loudspeaker moved around a room. The sound is partly absorbed and partly reflected by the wallls of the room. What is plotted as height (here shown in a computer drawn perspective plot) perfectly corresponds to the power output of a loudspeaker as a function of its position in the room, only here what is plotted is electron current as a function of the position tip of the metal needle as it moves just above the surface.

The bumps are iron atoms, and the ripples show the rate of electrons jumping off the tip. The ripples are there due ot wave interference.

These "quantum corral" experiments by Don Eigler's group at IBM Almaden, Ca., are some of the most astonishing and beautiful experiments of the 20th century: They developed the ability to move atoms at will, one by one, and discovered the electron wave interference.

In the image above from the Eigler group the elliptical corrals are shown under construction at various stages, atom by atom. The needle moves atoms one at a time, but first they must be found - they start out at random positions.

Intriguingly, sound arises literally in this searching process, since the experimentalists turned the electron current into sound (using an electronic audio system) in order to conveniently monitor how the experiment was going. If anything went wrong, this process of real time "sonification" of the incoming data would quickly reveal that they had a problem. What they heard usually was the sound of the needle scanning back and forth looking for iron atoms (they had been randomly sputtered onto the surface). They heard the needle creeping up to an atom if it found one (slowly rising and falling pitch - pitch is keyed to the ripples that form near an atom due to constructive and destructive electron interference, i.e. the electron wave just coming off the needle is interfering with the wave that has bounced off the nearby atom). If you study quantum mechanics, it is shown that this is a single electron interfering with itself! Then they heard a pulse as the atom is picked up by the needle with a voltage jolt they supply, followed by a scratchy sound as the single atom is dragged along the bumpy surface to its new position in the corral wall. Another pulse succeeds in dropping the atom in place. Then, more raster scanning for new atoms is heard. It is wonderful of the Eigler IBM group to let us present this image, and the sound file just described.

Wonderderful demonstration of collective resonance

Several metronomes, not perfectly synchronous (not keeping the same time) are able to do so and fall in lock step when linked together by a base that moves back and forth, and thus assists their motion, encouraging lock step motion. The base (when it is placed on the rolling drink cans) is not driven externally but only by the metronomes, which makes this example of collective resonance remarkable. You might want to watch the related and masterful lecture and demonstration by Steven Strogatz of MIT doing a TED talk on synchrony, which appears as a selection at the end.

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