by Walter Lewin
CHAPTER 6
The Harmonies of Strings and Winds
I took violin lessons as a ten-year-old, but I was a disaster and stopped after a year. Then in my twenties I took piano lessons, and I was a disaster again. I still cannot understand how people can read notes and convert them into music using ten fingers on different hands. I do appreciate music a lot, however, and in addition to having an emotional connection with it, I have come to understand it through physics. In fact, I love the physics of music, which starts, of course, with the physics of sound.
You probably know that sound begins with one or more very rapid vibrations of an object, like a drum surface or a tuning fork or a violin string. These vibrations are pretty obvious, right? What is actually happening when these things vibrate, however, is not so obvious, because it is usually invisible.
The back and forth motion of a tuning fork first compresses the air that is closest to it. Then, when it moves the other way, it decompresses the nearby air. This alternate pushing and pulling creates a wave in the air, a pressure wave, which we call a sound wave. This wave reaches our ears very quickly, at what we call the speed of sound: about 340 meters per second (about a mile in five seconds, or a kilometer in three). This is the speed of sound in air at room temperature. It can differ a great deal, depending on the medium it’s traveling through. The speed of sound is four times faster in water and fifteen times faster in iron than in air.
The speed of light (and all electromagnetic radiation) in vacuum is a famous constant, known as c, about 300,000 kilometers per second (you may have learned it as 186,000 miles per second), but the speed of visible light is about a third slower in water.
Now to get back to the tuning fork. When the wave it produces hits our ears, it pushes our eardrums in and out at exactly the same rate of oscillations as the tuning fork presses on the air. Then, through an almost absurdly complicated process, the eardrum vibrates the bones of the middle ear, known wonderfully as the hammer, anvil, and stirrup, and they in turn produce waves in the fluid in the inner ear. These waves are then converted into electric nerve impulses sent to the brain, and your brain interprets these signals as sound. Quite a process.
Sound waves—in fact all waves—have three fundamental characteristics: frequency, wavelength, and amplitude. Frequency is the number of waves passing a given point in a given period of time. If you are watching waves in the ocean from a boat or a cruise ship, you may notice that, say, ten waves go by in a minute, so we might say they have a frequency of ten per minute. But we actually often measure frequency in oscillations per second, also known as hertz, or Hz; 200 oscillations per second is 200 hertz.
As for wavelength, this is the distance between two wave crests—or also between two wave valleys. One of the fundamental characteristics of waves is that the greater the frequency of a wave, the shorter its wavelength is; and the longer the wavelength, the lower its frequency. Here we’ve reached a terrifically important set of relationships in physics, those between the speed, frequency, and wavelength of waves. The wavelength of a wave is its speed divided by its frequency. This holds for electromagnetic waves (X-rays, visible light, infrared, and radio waves) as well as sound waves in a bathtub and waves in the ocean. As an example, the wavelength in air of a 440 hertz tone (middle A on the piano) is 340 divided by 440, which is 0.77 meters (about 30 inches).
If you think about this for a minute, you’ll see that it makes perfect sense. Since the speed of sound is constant in any given medium (except in gases, where it depends on temperature), the more sound waves there are in a given period of time, the shorter the waves have to be to fit into that time. And the reverse is clearly also true: the fewer the waves in the same time the longer each of them has to be. For wavelength, we have different measurements for different kinds. For example, while we measure wavelengths of sound in meters, we measure the wavelengths of light in nanometers (one nanometer is a billionth of a meter).
Now what about amplitude? Think again about watching the waves out in the ocean from a boat. You will see that some waves are higher than others, even though they may have the same wavelength. This characteristic of the wave is known as its amplitude. The amplitude of a sound wave determines how loud or soft the sound will be: the greater its amplitude, the louder it is, and vice versa. This is because the larger the amplitude, the more energy a wave is carrying. As any surfer can tell you, the taller an ocean wave, the more energy it packs. When you strum guitar strings more vigorously, you are imparting more energy to them and you produce a louder sound. We measure the amplitude of water waves in meters and centimeters. The amplitude of a sound wave in air would be the distance over which the air molecules move back and forth in the pressure wave, but we never express it that way. Instead, we measure the intensity of sound, which is expressed in decibels. The decibel scale turns out to be quite complicated; fortunately, you don’t need to know about it.
The pitch of a sound, meaning how high or low it is on the musical scale, is, on the other hand, determined by the frequency. The higher the frequency, the higher its pitch; the lower the frequency, the lower its pitch. In making music, we change the frequency (thus the pitch) all the time.
The human ear can hear a tremendous range of frequencies, from about 20 hertz (the lowest note on a piano is 27.5 hertz) all the way up to about 20,000 hertz. I have a great demonstration in my classes, in which I run a sound-producing machine, an audiometer, which can broadcast different frequencies and at different intensities. I ask students to hold their hands up as long as they can hear the sound. I gradually increase the frequency. When we get older, most of us lose our ability to hear high frequencies. My own high-frequency cutoff is near 4,000 hertz, four octaves above middle C, at the very end of the piano keyboard. But long after I’m hearing nothing, my students can hear much higher notes. I move the dial upward and still upward, to 10,000 and 15,000 hertz, and some hands start to drop. At 20,000 hertz, only about half of the hands are still up. Then I go more slowly: 21,000, 22,000, 23,000. By the time I get to 24,000 hertz, there are usually just a few hands still raised. At that point, I play a little joke on them; I turn the machine off but pretend to be raising the frequency even higher, to 27,000 hertz. One or two brave souls claim to be hearing these super high notes—until I gently puncture that balloon. It’s all in good fun.
Now think about how a tuning fork works. If you hit a tuning fork harder, the number of vibrations per second of its prongs remains the same—so the frequency of the sound waves it produces stays the same. This is why it always plays the same note. However, the amplitude of the oscillation of its prongs does increase when you hit it harder. You could see this if you were to film the tuning fork as you hit it and then replay the film in slow motion. You would see the prongs of the fork move back and forth, and they would move farther the harder you hit them. Since the amplitude is increased, the note produced will be louder, but since the prongs continue to oscillate at the same frequency, the note stays the same. Isn’t that weird? If you think about it for a bit, you’ll see that it’s exactly like the pendulum (chapter 3), where the period (the time to complete one oscillation) is independent of the amplitude of its swings.
Sound Waves in Space?
Do these relationships of sound hold true beyond Earth? Have you ever heard that there is no sound in space? This would mean that no matter how energetically you play a piano on the surface of the Moon, it wouldn’t produce any sound. Can this be right? Yes, the Moon has no atmosphere; it is basically a vacuum. So you might conclude, perhaps sadly, that yes, even the most spectacular explosions of stars or galaxies colliding with each other occur in utter silence. We might even suppose that the big bang itself, the primordial explosion that created our universe nearly 14 billion years ago, took place entirely in silence. But hold on a minute. Space, like much of life, is considerably messier and more complicated than we thought even a few decades ago.
Even though any of us would quickly perish from a lack of oxyge
n if we tried to breathe in space, the truth is that outer space, even deep space, is not a perfect vacuum. Such terms are all relative. Interstellar and intergalactic space are millions of times closer to a vacuum than the best vacuum we can make on Earth. Still, the fact is that the matter that does float around in space has important and identifiable characteristics.
Most of this matter is called plasma: ionized gases—gases partly or completely made up of charged particles, such as hydrogen nuclei (protons) and electrons—of widely varying density. Plasma is present in our solar system, where we usually call it the solar wind, streaming outward from the Sun (the phenomenon Bruno Rossi did so much to advance our knowledge of). Plasmas are also found in stars, as well as between stars in galaxies (where we call it the interstellar medium), and even between galaxies (where we call it the intergalactic medium). Most astrophysicists believe that more than 99.9 percent of all observable matter in the universe is plasma.
Think about it. Wherever matter exists, pressure waves (thus, sound) can be produced and they will propagate. And because there are plasmas everywhere in space (also in our solar system), there are lots of sound waves out there, even though we can’t possibly hear them. Our poor ears can hear a pretty wide range of frequencies—more than three orders of magnitude, in fact—but we aren’t outfitted to hear the music of the heavenly spheres.
Let me give you one example. Back in 2003 physicists discovered ripples in the superhot gas (plasma) surrounding a supermassive black hole at the center of a galaxy in the Perseus cluster of galaxies, a large group of thousands of galaxies about 250 million light-years from Earth. These ripples clearly indicated sound waves, caused by the release of large amounts of energy when matter was swallowed up by the black hole. (I’ll get into black holes in more detail in chapter 12.) Physicists calculated the frequency of the waves and came up with a pitch of B flat, but a B flat so low that it is 57 octaves (about a factor of 1017) below middle C, whose frequency is about 262 hertz! You can see these cosmic ripples at http://science.nasa.gov/science-news/science-at-nasa/2003/09sep_blackholesounds/.
Now let’s go back to the big bang. If the primordial explosion that birthed our universe created pressure waves in the earliest matter—matter that then expanded and then cooled, creating galaxies, stars, and eventually planets—then we ought to be able to see the remnants of those sound waves. Well, physicists have calculated how far apart the ripples in the early plasma should have been (about 500,000 light-years) and how far apart they should be now, after the universe has been expanding for more than 13 billion years. The distance they came up with is about 500 million light-years.
There are two enormous galaxy-mapping projects going on right now—the Sloan Digital Sky Survey (SDSS) in New Mexico and the Two-degree Field (2dF) Galaxy Redshift Survey in Australia. They have both looked for these ripples in the distribution of galaxies and have independently found… guess what? They found “that galaxies are currently slightly more likely to be 500 million light-years apart than any other distance.” So the big bang produced a bass gong sound that now has a wavelength of about 500 million light-years, a frequency about fifty octaves (a factor of 1015) below anything our ears can hear. The astronomer Mark Whittle has played around a good bit with what he calls big bang acoustics, and you can too, by accessing his website: www.astro.virginia.edu/~dmw8f/BBA_web/index_frames.html. On the site, you can see and hear how he has simultaneously compressed time (turning 100 million years into 10 seconds) and artificially raised the pitch of the early universe fifty octaves, so you can listen to the “music” created by the big bang.
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, wh
ich 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.