For the Love of Physics (20 page)

When singers start demonstrating the physics of resonance by sending air through their vocal cords (“vocal folds” would be a more descriptive term), membranes vibrate and create sound waves. I ask a student to sing too, and the oscilloscope tells the same story, as similarly complicated curves pile up on the screen.

When you play the piano, the key that you press makes a hammer hit a string—a wire—whose length, weight, and tension have been set to oscillate at a given first harmonic frequency. But somehow, just like violin strings and vocal cords, the piano strings also vibrate simultaneously at higher harmonics.

Now take a tremendous thought-leap into the subatomic world and imagine super-tiny violinlike strings, much, much smaller than an atomic nucleus, that oscillate at different frequencies and different harmonics. In other words, consider the possibility that the fundamental building blocks of matter are these tiny vibrating strings, which produce all the so-called elementary particles—such as quarks, gluons, neutrinos, electrons—by vibrating at different harmonic frequencies, and in many dimensions. If you’ve managed to take this step, you’ve just grasped the fundamental proposition of string theory, the catchall term to describe the efforts of theoretical physicists over the past forty years to come up with a single theory accounting for all elementary particles and all the forces in the universe. In a way, it’s a theory of “everything.”

No one has the slightest idea whether string theory will succeed, and the Nobel laureate Sheldon Glashow has wondered whether it’s “a theory of physics or a philosophy.” But if it’s true that the most basic units of the universe are the different resonance levels of unimaginably tiny strings, then the universe, and its forces and elementary particles, may resemble
a cosmic version of Mozart’s wonderful, increasingly complex variations on “Twinkle, Twinkle Little Star.”

All objects have resonant frequencies, from the bottle of ketchup in your refrigerator to the tallest building in the world; many are mysterious and very hard to predict. If you have a car, you’ve heard resonances, and they didn’t make you happy. Surely you’ve had the experience of hearing a noise while driving, and hearing it disappear when you go faster.

On my last car the dashboard seemed to hit its fundamental frequency whenever I idled at a traffic light. If I hit the gas, speeding up the engine, even if I wasn’t moving, I changed the frequency of the car’s vibrations, and the noise disappeared. Sometimes I would hear a new noise for a while, which usually went away when I drove faster or slower. At different speeds, which is to say different vibrating frequencies, the car—and its thousands of parts, some of which were loose, alas—hit a resonant frequency of, say, its loose muffler or deteriorating motor mounts, and they talked to me. They all said the same thing—“Take me to the mechanic; take me to the mechanic”—which I too often ignored, only to discover later the damage that these resonances had done. When I finally took the car in, I could not reproduce the awful sounds and I felt kind of stupid.

I remember when I was a student, when we had an after-dinner speaker in my fraternity we didn’t like, we would take our wineglasses and run our wet fingers around the rim, something you can do at home easily, and generate a sound. This was the fundamental frequency of our wineglasses. When we got a hundred students doing it at once, it was very annoying, to be sure (this was a fraternity, after all)—but it was also very effective, and the speakers got the message.

Everyone has heard that an opera singer singing the right note loud enough can break a wineglass. Now that you know about resonance, how could that happen? It’s simple, at least in theory, right? If you took a wineglass, measured the frequency of its fundamental, and then generated a sound at that frequency, what would happen? Well, most of the time, in my experience, nothing at all. I’ve never seen an opera singer do
it. I therefore don’t use an opera singer in my class. I select a wineglass, tap on it, and measure its fundamental frequency with an oscilloscope—of course it varies from glass to glass, but for the glasses I use it’s always somewhere in the range of 440 to 480 hertz. I then generate electronically a sound with the
exact
same frequency of the fundamental of the wineglass (well
exact
, of course, is never possible, but I try to get very close). I connect it to an amplifier, and slowly crank up the volume. Why increase the volume? Because the louder the sound, the more energy in the sound wave will be beating against the glass. And the greater the amplitude of the vibrations in the wineglass, the more and more the glass will bend in and out, until it breaks (we hope).

In order to show the glass vibrating, I zoom in on it with a camera and illuminate it with a strobe light, set to a slightly different frequency than the sound. It’s fantastic! You see the bowl of the wineglass beginning to vibrate; the two opposite sides first contract, then push apart, and the distance they move grows and grows as I increase the volume of the speaker, and sometimes I have to tweak the frequency slightly and then—
poof!
—the glass shatters. That’s always the best part for the students; they can’t wait for the glass to break. (You can see this online about six minutes into lecture 27 of my Electricity and Magnetism course, 8.02, at:
http://ocw.mit.edu/courses/physics/8-02-electricity-and-magnetism-spring-2002/video-lectures/lecture-27-resonance-and-destructive-resonance/
.)

I also love to show students something called Chladni plates, which demonstrate, in the oddest and most beautiful ways, the effects of resonance. These metal plates are about a foot across, and they can be square, rectangular, or even circular, but the best are square. They are fastened to a rod or a base at their centers. We sprinkle some fine powder on the plate and then rub a violin bow along one of the sides, the whole length of the bow. The plate will start to oscillate in one or more of its resonance frequencies. At the peaks and valleys of the vibrating waves on the plate, the powder will shake off and leave bare metal; it will accumulate at the nodes, where the plate does not vibrate at all. (Strings have nodal
points, but two-dimensional objects, like the Chladini plate, have nodal lines.)

Depending on how and where you “play” the plate by rubbing it with the bow, you will excite different resonance frequencies and make amazing, completely unpredictable patterns on its surface. In class I use a more efficient—but far less romantic—technique and hook the plate up to a vibrator. By changing the frequency of the vibrator, we see the most remarkable patterns come and go. You can see what I mean here, on YouTube:
www.youtube.com/watch?v=6wmFAwqQB0g
. Just try to imagine the math behind these patterns!

In the public lectures I do for kids and families, I invite the little ones to rub the plate edges with the bow—they love making such beautiful and mysterious patterns.
That’s
what I’m trying to get across about physics.

The Music of the Winds

But we’ve left out half the orchestra! How about a flute or oboe or trombone? After all, they don’t have a string to vibrate, or a soundboard to project their sound. Even though they are ancient—I saw a photograph of a 35,000-year-old flute carved out of vulture bone in the newspaper a little while ago—wind instruments are a little more mysterious than strings, partly because their mechanism is invisible.

There are different kinds of winds, of course. Some, like flutes and recorders, are open at both ends, while clarinets and oboes and trombones are closed at one end (even though they have openings for someone to blow in). But all of them make music when an infusion of air, usually from your mouth, causes a vibration of the air column
inside
the instrument.

When you blow or force air inside a wind instrument it’s like plucking a guitar string or exciting a violin string with a bow—by imparting energy to the air column, you are dumping a whole spectrum of frequencies
into that air cavity, and the air column itself chooses the frequency at which it wants to resonate, depending mostly on its length. In a way that is hard to imagine, but with a result that’s relatively easy to calculate, the air column inside the instrument will pick out its fundamental frequency, and some of the higher harmonics as well, and start vibrating at those frequencies. Once the air column starts vibrating, it pushes and pulls on the air, just like vibrating tuning fork prongs, sending sound waves toward the ears of the listeners.

With oboes, clarinets, and saxophones, you blow on a reed, which transfers energy to the air column and makes it resonate. For flutes and piccolos and recorders, it’s the way the player blows across a hole or into a mouthpiece that creates the resonance. And for brass instruments, you have to put your lips together tightly and blow a kind of buzz into the instrument—if you haven’t been trained to do it, it’s all but impossible. I end up just spitting into the damn thing!

If the instrument is open at both ends, like a flute or piccolo, the air column can vibrate at its harmonics, each of which is a multiple of the fundamental frequency, as was the case with the strings.  For woodwind instruments that are closed at one end and open at the other, the shape of the tube matters. If the bore is conical, such as the oboe or saxophone, the instruments will produce all harmonics like the flute. However, if the bore is cylindrical, such as the clarinet, the air column will only resonate at the odd-number multiples of the fundamental: three times, five times, seven times, and so on. For complicated reasons, all brass instruments resonate at all harmonics, like the flute.

What’s more intuitive is that the longer the air column is, the lower the frequency and the lower the pitch of the sound produced. If the length of a tube is halved, the frequency of the first harmonic will double. That’s why the piccolo plays such high notes, a bassoon plays such low ones. This general principle also explains why a pipe organ has such a range of pipe lengths—some organs can produce sounds across nine octaves. It takes an enormous tube—64 feet long (19.5 meters long, open on both sides) to produce a fundamental of
about 8.7 hertz, literally below what the human ear can hear, though you can feel the vibrations. There are just two of these enormous pipes in the world, since they aren’t very practical at all. A tube ten times shorter will produce a fundamental ten times higher, thus 87 hertz. A tube a hundred times shorter will produce a fundamental of about 870 hertz.

Wind instrumentalists don’t just blow into their instruments. They also close or open holes in their instruments that serve to effectively shorten or lengthen the air column, thereby raising or lowering the frequency it produces. That’s why, when you play around with a child’s whistle, the lower tones come when you put your fingers over all the holes, lengthening the air column. The same principle holds for brass instruments. The longer the air column, even if it has to go around in circles, the lower the pitch, which is to say, the lower the frequencies of all the harmonies. The lowest-pitched tuba, known as the B-flat or BB-flat tuba, has an 18-foot-long tube with a fundamental of about 30 hertz; additional, so-called rotary valves can lower the tone to 20 hertz; the tube of a B-flat trumpet is just 4.5 feet long. The buttons on a trumpet or tuba open or close additional tubes, changing the pitch of the resonant frequencies. The trombone is the simplest to grasp visually. Pulling the slide out increases the length of the air column, lowering its resonant frequencies.

I play “Jingle Bells” on a wooden slide trombone in my class, and the students love it—I never tell them it’s the only tune I can play. In fact, I’m so challenged as a musician that no matter how many times I’ve given the lecture, I still have to practice beforehand. I’ve even made marks on the slide—notes, really—numbered 1, 2, 3, and so forth; I can’t even read musical notes. But as I said before, my complete lack of musical talent hasn’t stopped me from appreciating music’s beauty, or from having lots of fun experimenting with it.

While I’m writing this, I’m having some fun experimenting with the air column inside a one-liter plastic seltzer bottle. It’s not at all a perfect column, since the bottleneck gradually widens to the full diameter of
the bottle. The physics of a bottleneck can get really complicated, as you might imagine. But the basic principle of wind instrument music—the longer the air column, the lower the resonant frequencies—still holds. You can try this easily.

Fill up an empty soda or wine bottle nearly to the top (with water!) and try blowing across the top. It takes some practice, but pretty soon you will get the air column to vibrate at its resonance frequencies. The sound will be high pitched at first, but the more you drink (you see why I suggested water), the longer the column of air becomes, and the pitch of the fundamental goes down. I also find that the longer I make the air column, the more pleasing the sound is. The lower the frequency of the first harmonic, the more likely it is that I will generate additional harmonics at higher frequencies, and the sound will have a more complex timbre.

You might be thinking that it’s the bottle vibrating, just as the string did, that makes the sound, and you do in fact feel the bottle vibrating, just the way you might feel a saxophone vibrate. But again, it’s the air column inside that resonates. To drive home this point, consider this puzzle. If you take two identical wineglasses, one empty and one half full, and excite the first harmonic of each by tapping each glass lightly with a spoon or by rubbing its rim with a wet finger, which frequency will be higher, and why? It’s not fair of me to ask this question as I have been setting you up to give the wrong answer—sorry! But perhaps you’ll work it out.

The same principle is at play with those 30-inch flexible corrugated colored plastic tubes, called whirling tubes or something similar, which you’ve probably seen or played with. Do you remember how they work? When you start by whirling one around your head, you first hear a low-frequency tone. Of course, you expect this to be the first harmonic, just like I did when I first played with this toy. However, somehow I have never succeeded in exciting the first harmonic. It’s always the second that I hear first. As you go faster, you can excite higher and higher harmonics. Advertisements online claim you can get four tones from these tubes, but you may only get three—the fourth tone, which is the fifth harmonic,
takes some really, really fast whirling. I calculated the frequencies of the first five harmonics for a tube length of 30 inches and find 223 hertz (I’ve never gotten this one), 446 hertz, 669 hertz, 892 hertz, and 1,115 hertz. The pitch gets pretty high pretty quickly.