by Walter Lewin
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.
Dangerous Resonance
The physics of resonance reaches far beyond classroom demonstrations. Think of the different moods that music can produce with these different instruments. Musical resonance speaks to our emotions, bringing us gaiety, anxiety, calm, awe, fear, joy, sorrow, and more. No wonder we talk of experiencing emotional resonance, which can create a relationship filled with richness and depth, and overtones of understanding and tenderness and desire. It’s hardly accidental that we want to be “in tune” with someone else. And how painful when we lose that resonance, either temporarily or forever, and what had felt like harmony turns into discordant interference and emotional noise. Think of the characters George and Martha in Edward Albee’s Who’s Afraid of Virginia Woolf? They fight atrociously. When the fight is one on one, they create heat, and they remain just a show for their guests. They’re much more dangerous when they join forces to play get the guest.
Resonance can become powerfully destructive in physics too. The most spectacular example of destructive resonance in recent history occurred in November 1940, when a crosswind hit the main span of the Tacoma Narrows Bridge just right. This engineering marvel (which had become known as Galloping Gertie for its oscillations up and down) started to resonate powerfully. As the crosswind increased the amplitude of the bridge oscillations, the structure began to vibrate and twist, and as the twisting grew more and more extreme, the span tore apart, crashing into the water. You can watch this spectacular collapse at www.youtube.com/watch?v=j-zczJXSxnw.
Ninety years earlier, in Angers, France, a suspension bridge over the Maine River collapsed when 478 soldiers crossed it in military formation, marching in step. Their marching excited a resonance in the bridge, which snapped some corroded cables; more than 200 soldiers died when they fell into the river below. The disaster stopped suspension bridge building in France for twenty years. In 1831, British troops marching in step across the Broughton Suspension Bridge caused the bridge deck to resonate, pull out a bolt at one end of the bridge, and collapse. No one was killed, but the British army instructed all troops crossing bridges from then on to do so by breaking their marching step.
The Millennium Bridge in London opened in 2000, and many thousands of pedestrians discovered that it wobbled a good bit (it had what engineers call lateral resonance); after just a few days authorities closed the bridge for two embarrassing years while they installed dampers to control the movement generated by pedestrian footsteps. Even the great Brooklyn Bridge in New York City frightened pedestrians who packed the bridge during a 2003 electrical blackout and felt a lateral swaying in the deck that made some of them sick.
In such situations pedestrians put more weight on a bridge than the cars that are usually crossing them, and the combined motion of their feet, even if they are not in step, can start to excite a resonance vibration—a wobble—on the bridge deck. When the bridge goes one way, they compensate by stepping the other way, magnifying the amplitude of the wobble. Even engineers admit they don’t know enough about the effects crowds can have on bridges. Fortunately, they know a lot about building skyscrapers that can resist the high winds and earthquakes that threaten to generate resonance frequencies that could destroy their creations. Imagine—the same principles that produced the plaintive sound of our ancestors’ 35,000-year-old flute could threaten the mighty and massive Brooklyn Bridge and the tallest buildings in the world.
CHAPTER 7
The Wonders of Electricity
This works best in the winter, when the air is very dry. Make sure you’re wearing a polyester shirt or sweater, then stand in front of a mirror when it’s dark and start taking your shirt or sweater off. You will have anticipated that you’ll hear crackling noises, just like when you pull laundry out of the dryer (unless you use one of those unromantic dryer sheets designed to reduce all that electricity). But now you will also see the glow of dozens of teeny-weeny little sparks. I love doing this because it r
eminds me just how close physics is to our everyday experience, if only we know how to look for it. And, as I like to point out to my students, the truth is, this little demonstration is even more fun if you do it with a friend.
You know that whenever you walk across a rug in winter and reach for a doorknob—are you wincing?—you may get a shock, and you know that it’s from static electricity. You’ve probably even shocked a friend by shaking her hand, or felt a shock when you’ve handed your overcoat to a coat checker. Frankly, it feels like static electricity is everywhere in wintertime. You can feel your hair separating when you brush it, and sometimes it stands up on its own after you take your hat off. What is it about winter, and why are so many sparks flying?
The answer to all these questions begins with the ancient Greeks, who were the first to name and make a written record of the phenomenon we’ve come to know as electricity. Well over two thousand years ago, the Greeks knew that if you rubbed amber—the fossilized resin that they and the Egyptians made into jewelry—on a cloth, the amber could attract pieces of dry leaves. After enough rubbing, it could even produce a jolt.
I’ve read stories claiming that when Greeks were bored at parties, the women would rub their amber jewelry on their clothing and touch the jewelry to frogs. The frogs would jump, of course, desperately trying to escape the crazy partiers, which apparently made for great fun among the ancients. Nothing about these stories makes any sense. First off, how many parties can you imagine where there are lots of frogs waiting around to be shocked by drunken revelers? Secondly, for reasons I’ll explain in a bit, static electricity doesn’t work so well during the months when you’re more likely to see frogs, and when the air is humid—especially in Greece. Whatever the truth of this story, what is undeniable is that the Greek word for “amber” is electron, so it was really the Greeks who named electricity, along with so much else of the universe, both large and small.
The European physicists of the sixteenth and seventeenth centuries, when physics was known as natural philosophy, didn’t know anything about atoms or their components, but they were terrific observers, experimenters, and inventors, and some were fantastic theorists as well. You had Tycho Brahe, Galileo Galilei, Johannes Kepler, Isaac Newton, René Descartes, Blaise Pascal, Robert Hooke and Robert Boyle, Gottfried Leibniz, Christiaan Huygens—all making discoveries, writing books, disputing one another, and turning medieval scholasticism upside down.
By the 1730s, genuine scientific study of electricity (as opposed to putting on parlor tricks) was well under way in England, France, and, of course, Philadelphia. All of these experimenters had figured out that if they rubbed a glass rod with a piece of silk it would gain a charge of some kind (let’s call it A)—but if they rubbed amber or rubber in the same way it would acquire a different charge (let’s call it B for now). They knew that the charges were different because when they took two glass rods that they’d rubbed with silk, both charged with A, and put them near each other, they would repel each other, by some completely invisible but nevertheless palpable force. Similar objects that had been charged with charge B also repelled each other. And yet differently charged objects, say a charged glass rod (A) and a charged rubber rod (B), would attract rather than repel each other.
Charging objects by rubbing them is a truly intriguing phenomenon, and it even has a wonderful name, the “triboelectric” effect, from the Greek word for “rubbing.” It feels as though the friction between the two objects is what produces the charge, but that’s not the case. It turns out that some materials greedily attract charge B, while other materials can’t wait to lose it, thereby creating charge A. Rubbing works because it increases the number of contact points between substances, facilitating the transfer of charge. There is a ranked list of many materials that make up the “triboelectric series” (you can find it easily online), and the farther apart two materials are on the scale, the more easily they can charge each other.
Take plastic or hard rubber that combs are typically made of. They are pretty far away from human hair in the triboelectric series, which accounts for how easily your hair can spark and stand up when you comb it in winter—especially my hair. And think about it: not only does it spark, since by vigorously combing my hair I am charging both the comb and my hair; but since the hair all picks up the same charge, whichever it is, each charged hair repels all the other like-charged hairs, and I start to resemble a mad scientist. When you scuff your shoes on a carpet, you charge yourself with A or B, depending on the material of your shoe soles and the carpet. When you get shocked by the nearest doorknob, your hand is either receiving charge from the doorknob or shooting charge to it. It doesn’t matter to you which charge you have; either way, you feel the shock!
It was Benjamin Franklin—diplomat, statesman, editor, political philosopher, inventor of bifocals, swim fins, the odometer, and the Franklin stove—who introduced the idea that all substances are penetrated with what he called “electric fluid,” or “electric fire.” Because it seemed to explain the experimental results of his fellow natural philosophers, this theory proved very persuasive. The Englishman Stephen Gray, for instance, had shown that electricity could be conducted over distances in metal wire, so the idea of a usually invisible fluid or fire (after all, sparks do resemble fire) made good sense.
Franklin argued that if you get too much of the fire then you’re positively charged, and if you have a deficiency of it then you’re negatively charged. He also introduced the convention of using positive and negative signs and decided that if you rub glass with a piece of wool or silk (producing the A charge) you give it an excess of fire, and therefore it should be called positive.
Franklin didn’t know what caused electricity, but his idea of an electrical fluid was brilliant as well as useful, even if not exactly correct. He maintained that if you take the fluid and bring it from one substance to another, the one with an excess becomes positively charged and, at the same time, the one from which you take the fluid becomes negatively charged. Franklin had discovered the law of conservation of electric charge, which states that you cannot truly create or get rid of charge. If you create a certain amount of positive charge, then you automatically create the same amount of negative charge. Electric charge is a zero-sum game—as physicists would say, charge is conserved.
Franklin understood, as we do today, that like charges (positive and positive, negative and negative) repel each other, and that opposite charges (positive and negative) attract. His experiments showed him that the more fire objects had, and the closer they were to each other, the stronger the forces, whether of attraction or repulsion. He also figured out, like Gray and others around the same time, that some substances conduct the fluid or fire—we now call those substances conductors—and others do not, and are therefore called nonconductors, or insulators.
What Franklin had not figured out is what the fire really consists of. If it’s not fire or fluid, what is it? And why does there seem to be so much more of it in the winter—at least where I live, in the northeastern United States, shocking us right and left?
Before we take a look inside the atom to grapple with the nature of electric fire, we need to see that electricity pervades our world far more than Franklin knew—and far more than most of us realize. It not only holds together most of what we experience on a daily basis; it also makes possible everything we see and know and do. We can only think and feel and muse and wonder because electric charges jump between uncountable millions of the roughly 100 billion cells in our brains. At the same time, we can only breathe because electric impulses generated by nerves cause different muscles of our chest to contract and relax in a complicated symphony of movements. For example, and most simply, as your diaphragm contracts and drops in your thorax, it enlarges the chest cavity, drawing air into the lungs. As it relaxes and expands upward again, it pushes air out of the lungs. None of these motions would be possible without countless tiny electric impulses constantly sending messages throughout your body, in th
is case telling muscles to contract and then to stop contracting while others take up the work. Back and forth, back and forth, for as long as you live.
Our eyes see because the tiny cells of our retinas, the rods and cones that pick up black-white and colors, respectively, get stimulated by what they detect and shoot off electric signals through the optic nerves to our brains. Our brains then figure out whether we’re looking at a fruit stand or a skyscraper. Most of our cars run on gasoline, though hybrids use increasing amounts of electricity, but there would be no gasoline used in any engine without the electricity running from the battery through the ignition to the cylinders, where electric sparks ignite controlled explosions, thousands of them per minute. Since molecules form due to electric forces that bind atoms together, chemical reactions—such as gasoline burning—would be impossible without electricity.
Because of electricity, horses run, dogs pant, and cats stretch. Because of electricity, Saran Wrap crumples, packing tape attracts itself, and the cellophane wrapping never seems to want to come off of a box of chocolates. This list is hardly exhaustive, but there’s really nothing that we can imagine existing without electricity; we could not even think without electricity.
That holds true when we turn our focus to things even smaller than the microscopic cells in our bodies. Every bit of matter on Earth consists of atoms, and to really understand electricity we have to go inside the atom and briefly look at its parts: not all of them now, because that gets incredibly complicated, but just the parts we need.
Atoms themselves are so tiny that only the most powerful and ingenious instruments—scanning tunneling microscopes, atomic force microscopes, and transmission electron microscopes—can see them. (There are some astonishing images from these instruments on the web. You can see some at this link: www.almaden.ibm.com/vis/stm/gallery.html.)