For the Love of Physics

by Walter Lewin

  But here’s the magic. When the violinist rubs the string with a bow, she is imparting energy to the string, which somehow picks out its own resonance frequencies (from all the vibrations possible), and—here’s the even more mind-blowing part—even though we cannot see it, it vibrates simultaneously in several different resonance frequencies (several harmonics). It’s not like a tuning fork, which can only vibrate at a single frequency.

  These additional harmonics (with frequencies higher than the fundamental) are often called overtones. The interplay of the varied resonant frequencies, some stronger, some weaker—the cocktail of harmonics—is what gives a violin or cello note what is known technically as its color or timbre, but what we recognize as its distinctive sound. That’s the difference between the sound made by the single frequency of the tuning fork or audiometer or emergency broadcast message on the radio and the far more complex sound of musical instruments, which vibrate at several harmonic frequencies simultaneously. The characteristic sounds of a trumpet, oboe, banjo, piano, or violin are due to the distinct cocktail of harmonic frequencies that each instrument produces. I love the image of an invisible cosmic bartender, expert in creating hundreds of different harmonic cocktails, who can serve up a banjo to this customer, a kettledrum to the next, and an erhu or a trombone to the one after that.

  Those who developed the first musical instruments were ingenious in crafting another vital feature of instruments that allows us to enjoy their sound. In order to hear music, the sound waves not only have to be within the frequency range you can hear, but they also must be loud enough for you to hear them. Simply plucking a string, for instance, doesn’t produce enough sound to be heard at a distance. You can impart more energy to a string (and hence to the sound waves it produces) by plucking it harder, but you still may not produce a very robust sound. Fortunately, a great many years ago, millennia at least, human beings figured out how to make string instruments loud enough to be heard across a clearing or room.

  You can reproduce the precise problem our ancestors faced—and then solve it. Take a foot-long piece of string, tie one end to a doorknob or drawer handle, pull on the other end until it’s tight, and then pluck it with your other hand. Not much happens, right? You can hear it, and depending on the length of the string, how thick it is, and how tight you hold it, you might be able to make a recognizable note. But the sound isn’t very strong, right? No one would hear it in the next room. Now, if you take a plastic cup and run the string through the cup, hold the string up at an angle away from the knob or handle (so it doesn’t slide toward your hand), and pluck the string, you’ll hear more sound. Why? Because the string transmits some of its energy to the cup, which now vibrates at the same frequency, only it’s got a much larger surface area through which to impart its vibrations to the air. As a result, you hear louder sound.

  With your cup you have demonstrated the principle of a sounding board—which is absolutely essential to all stringed instruments, from guitars and bass fiddles to violins and the piano. They’re usually made of wood, and they pick up the vibrations of the strings and transmit these frequencies to the air, greatly amplifying the sound of the strings.

  The sounding boards are easy to see in guitars and violins. On a grand piano, the sounding board is flat, horizontal, and located underneath the strings, which are mounted on the sounding board; it stands vertically behind the strings on an upright. On a harp, the sounding board is usually the base where the strings are attached.

  In class I demonstrate the workings of a sounding board in different ways. In one demonstration I use a musical instrument my daughter Emma made in kindergarten. It’s one ordinary string attached to a Kentucky Fried Chicken cardboard box. You can change the tension in the string using a piece of wood. It’s really great fun; as I increase the tension the pitch goes up. The KFC box is a perfect sounding board, and my students can hear the plucking of the string from quite far away. Another one of my favorite demos is with a music box that I bought many years ago in Austria; it’s no bigger than a matchbox and it has no sounding board attached to it. When you turn the crank, it makes music produced by vibrating prongs. I turn the crank in class and no one can hear it, not even I! Then I place it on my lab table and turn the crank again. All the students can now hear it, even those way in the back of my large lecture hall. It always amazes me how very effective even a very simple sounding board can be.

  That doesn’t mean that they’re not sometimes works of real art. There is a lot of secrecy about building high-quality musical instruments, and Steinway & Sons are not likely to tell you how they build the sounding boards of their world-famous pianos! You may have heard of the famous Stradivarius family in the seventeenth and eighteenth centuries who built the most fantastic and most desirable violins. Only about 540 Stradivarius violins are known to exist; one was sold in 2006 for $3.5 million. Several physicists have done extensive research on these violins in an effort to uncover the “Stradivarius secrets” in the hope that they would be able to build cheap violins with the same magic voice. You can read about some of this research at www.sciencedaily.com/releases/2009/01/090122141228.htm.

  A good deal of what makes certain combinations of notes sound more or less pleasing to us has to do with frequencies and harmonics. The best-known type of note pairing, at least in Western music, is of notes where one is exactly twice the frequency of the other. We say that these notes are separated by an octave. But there are many other pleasing combinations as well: chords, thirds, fifths, and so on.

  Mathematicians and “natural philosophers” have been fascinated by the beautiful numerical relationships between different frequencies since the time of Pythagoras in ancient Greece. Historians disagree over just how much Pythagoras figured out, how much he borrowed from the Babylonians, and how much his followers discovered, but he seems to get the credit for figuring out that strings of different lengths and tensions produce different pitches in predictable and pleasing ratios. Many physicists delight in calling him the very first string theorist.

  Instrument makers have made brilliant use of this knowledge. The different strings on a violin, for example, all have different weights and tensions, which enable them to produce higher and lower frequencies and harmonics even though they all have about the same length. Violinists change the length of their strings by moving their fingers up and down the violin neck. When their fingers walk toward their chins, they shorten the length of any given string, increasing the frequency (thus the pitch) of the first harmonic as well as all the higher harmonics. This can get quite complex. Some stringed instruments, like the Indian sitar, have what are called sympathetic strings, extra strings alongside or underneath the playing strings that vibrate at their own resonance frequencies when the instrument is being played.

  It’s difficult if not impossible to see the multiple harmonic frequencies on the strings of an instrument, but I can show them dramatically by connecting a microphone to an oscilloscope, which you have probably seen on TV, if not in person. An oscilloscope shows a vibration—or oscillation—over time, on a screen, in the form of a line going up and down, above and below a central horizontal line. Intensive care units and emergency rooms are filled with them for measuring patients’ heartbeats.

  I always invite my students to bring their musical instruments to class so that we can see the various cocktails of harmonics that each produces.

  When I hold a tuning fork for concert A up to the microphone, the screen shows a simple sine curve of 440 hertz. The line is clean and extremely regular because, as we’ve seen, the tuning fork produces just one frequency. But when I invite a student who brought her violin to play the same A, the screen gets a whole lot more interesting. The fundamental is still there—you can see it on the screen as the dominant sine curve—but the curve is now much more complex due to the higher harmonics, and it’s different again when a student plays his cello. Imagine what happens when a violinist plays two notes at once!

  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 p
ulls 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.