For the Love of Physics (19 page)

The Wonders of Resonance

The phenomenon we call resonance makes a huge number of things possible that either could not exist at all or would be a whole lot less interesting without it: not only music, but radios, watches, trampolines, playground swings, computers, train whistles, church bells, and the MRI you may have gotten on your knee or shoulder (did you know that the “R” stands for “resonance”?).

What exactly is resonance? You can get a good feeling for this by thinking of pushing a child on a swing. You know, intuitively, that you can produce large amplitudes of the swing with very little effort. Because the swing, which is a pendulum, has a uniquely defined frequency (
chapter 3
), if you accurately time your pushes to be in sync with that frequency, small amounts of additional push have a large cumulative impact on how high the swing goes. You can push your child higher and higher with just light touches of only a couple of fingers.

When you do this, you are taking advantage of resonance. Resonance, in physics, is the tendency of something—whether a pendulum, a tuning fork, a violin string, a wineglass, a drum skin, a steel beam, an atom, an electron, a nucleus, or even a column of air—to vibrate more powerfully at certain frequencies than at others. These we call resonance frequencies (or natural frequencies).

A tuning fork, for instance, is constructed to always vibrate at its resonance frequency. If it does so at 440 hertz, then it makes the note known
as concert A, the A above middle C on the piano. Pretty much no matter how you get it vibrating, its prongs will oscillate, or move back and forth, 440 times per second.

All materials have resonance frequencies, and if you can add energy to a system or an object it may start to vibrate at these frequencies, where it takes relatively little energy input to have a very significant result. When you tap a delicate empty wineglass gently with a spoon, for example, or rub the rim with a wet finger, it will ring with a particular tone—that is a resonance frequency. Resonance is not a free lunch, though at times it looks like one. But at resonance frequencies, objects make the most efficient use of the energy you put into them.

A jump rope works on the same principle. If you’ve ever held one end, you know that it takes a while to get the rope swinging in a nice arc—and while you may have circled your hand around with the end to get that arc, the key part of that motion is that you are going up and down or back and forth, oscillating the rope. At a certain point, the rope starts swinging around happily in a beautiful arc; you barely have to move your hand to keep it going, and your friends can start jumping in the middle of the arc, intuitively timing their jumps to the resonant frequency of the rope.

You may not have known this on the playground, but only one person has to move her hand—the other one can simply hold on to the other end, and it works just fine. The key is that between the two of you, you’ve reached the rope’s lowest resonance frequency, also called the fundamental. If it weren’t for this, the game we know as double-dutch, in which two people swing two ropes in opposite directions, would be just about impossible. What makes it possible for two ropes to be going in opposite directions, and be held by the same people, is that each one requires very little energy to keep it going. Since your hands are the driving force here, the jump rope becomes what we call a driven oscillator. You know, intuitively, once you reach this resonance of the rope, that you want to stay at that frequency, so you don’t move your hand any faster.

If you did, the beautiful rotating arc would break up into rope squiggles, and the jumper would quickly get annoyed. But if you had a long enough rope, and could vibrate your end more quickly, you would find that pretty soon the rope would create two arcs, one going down while the other went up, and the midpoint of the rope would stay stationary. We call that midpoint a node. At that point two of your friends could jump, one in each arc. You may have seen this in circuses. What is going on here? You have achieved a second resonance frequency. Just about everything that can vibrate has multiple resonance frequencies, which I’ll discuss more in just a minute. Your jump rope has higher resonance frequencies too, which I can show you.

I use a jump rope to show multiple resonance frequencies in my class by suspending a single rope, about ten feet long, between two vertical rods. When I move one end of the rope up and down (only an inch or so), oscillating it on a rod, using a little motor whose frequency I can change, soon it will hit its lowest resonance frequency, which we call the first harmonic (it is also called the fundamental), and make one arc like the jump rope. When I oscillate the end of the rope more rapidly, we soon see two arcs, mirror images of each other. We call this the second harmonic, and it will come when we are oscillating the string at twice the rate of the first harmonic. So if the first harmonic is at 2 hertz, two vibrations per second, the second is at 4 hertz. If we oscillate the end still faster, when we reach three times the frequency of the first harmonic, in this case 6 hertz, we’ll reach the third harmonic. We see the string divide equally into thirds with two points of the string (nodes) that do not move, with the arcs alternating up and down as the end goes up and down six times per second.

Remember I said that the lowest note we can hear is about 20 hertz? That’s why you don’t hear any music from a playground jump rope—its frequency is way too low. But if we play with a different kind of string—one on a violin or cello, say—something else entirely happens. Take a violin. You don’t want me to take it, believe me—I haven’t improved in the past sixty years.

In order for you to hear one long, beautiful, mournful note on a violin, there’s an enormous amount of physics that has already happened. The sound of a violin, or cello, or harp, or guitar string—of any string or rope—depends on three factors: its length, its tension, and its weight. The longer the string, the lower the tension, and the heavier the string, the lower the pitch. And, of course, the converse: the shorter the string, the higher the tension, and the lighter the string, the higher the pitch. Whenever string musicians pick up their instruments after a while, they have to adjust the tension of their strings so they will produce the right frequencies, or notes.

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!