Unit 12 · Waves and Information

Drop a pebble into a still pond. Rings spread out from the splash. A leaf floating nearby bobs up and down, but it does not ride the ring to the shore. It ends up almost where it started. The ring moved. The water and the leaf did not go with it. That is the strange and useful thing about a wave: the pattern travels, the material does not.

A wave is a repeating disturbance that carries energy from one place to another. Stretch a rope along the floor and snap one end up and down. A hump runs down the rope. Each piece of rope only rises and falls, yet the hump reaches the far end and can knock over a cup there. The rope stayed put; the energy arrived.

Sound works the same way. A drum skin pushes the air next to it. That air pushes the air next to it, and so on. The air near the drum never travels to your ear. It just shoves its neighbors. What reaches you is a chain of pushes, and your eardrum feels the energy of the last push in line. Scientists call the material a wave moves through the medium: water for ripples, rope for the hump, air for sound.

Every wave can be measured in three ways. Amplitude is how far the medium moves from its rest position, the height of a water hump above the flat surface or how far a rope swings from the middle. Bigger amplitude means more energy. On the turntable, a loud drum hit cut a wide swing in the groove; a whisper cut a narrow one. Doubling the amplitude of a wave does not just double its energy. It roughly quadruples it, which is why a shout can rattle a window when a normal voice cannot.

Wavelength is the distance from one crest to the next crest, or from one groove wiggle to the next matching wiggle. Long ocean swells might be 100 meters from crest to crest. The ripples in a bathtub are a few centimeters. Frequency is a different measure: it counts how many complete waves pass a point in one second. The unit is the hertz (Hz). A rope shaken twice a second makes a 2 Hz wave.

Wavelength and frequency are linked. If the wave speed stays the same, more waves per second must be packed closer together. So high frequency goes with short wavelength, and low frequency goes with long wavelength. For sound, frequency is what we hear as pitch: a bass note is low frequency, a whistle is high. For the eye, amplitude shows up as brightness and frequency as color, an idea this chapter returns to later.

The site’s oscilloscope draws a wave as a glowing line across a grid. Turn up how fast it wiggles and the crests crowd closer together; turn it down and they spread out, long and lazy. Real water does the same: push a paddle slowly, then quickly, at one end of a long tray, and a marker floating in the tray shows each crest still takes the same time to travel its length. The speed of the ripples did not change. Only the spacing did.

This is a rule you can trust for any wave: wave speed equals frequency times wavelength. Say the paddle makes 2 waves per second and the crests are 0.5 meters apart. Then the crests travel 2 times 0.5, or 1 meter every second. Now double the frequency to 4 Hz. The water has not changed, so the speed is still 1 meter per second. That means the wavelength must shrink to 0.25 meters.

The rule matters because speed depends on the medium, not on the wave. Water of a certain depth carries ripples at one speed. Air at room temperature carries sound at one speed. So when a musician plays a higher note, the sound does not go faster; its wavelength gets shorter. On the turntable, the same idea appears as wiggles packed more tightly in the groove for the high notes, even though the record spins at a fixed rate.

You can use the rule on your own. Count how many ripples pass a mark in ten seconds, divide by ten to get frequency, then measure the crest-to-crest distance with a ruler. Multiply, and you have the wave speed of the bench without ever timing a single ripple.

Around 1660 the English scientist Robert Boyle hung a ringing watch inside a glass jar and pumped the air out. As the air thinned, the ticking faded. With most of the air gone, he could see the watch working but could not hear it. Let the air back in and the sound returned. Boyle had shown that sound needs a medium. Nothing in the jar was able to carry the pushes to the glass.

Sound is a mechanical wave, a wave that moves by shoving particles of matter into each other. A vibrating guitar string squeezes the air on one side, then the other, hundreds of times each second. Each squeeze is a region of slightly higher pressure, and each pull-back is a region of slightly lower pressure. These squeezes travel outward as a compression wave. In a vacuum, a space with no particles, there is nothing to squeeze, and sound simply stops.

This is why films that show roaring explosions in space are wrong. An astronaut outside the International Space Station hears nothing through the vacuum. Inside a suit, a radio carries voices, because radio is not sound. It is a different kind of wave, one that needs no medium, and it is the subject of the next section.

You can feel the medium at work on the site’s drum machine. Tap a drum and hold a hand a few centimeters from the skin. You feel the puffs of air. Rest a fingertip on the skin instead and the sound dies, because you stopped the vibration that was doing the pushing.

Watch a lightning flash and start counting. Thunder arrives a few seconds later. The light reached you almost instantly, but the sound crawled along at about 343 meters per second, the speed of sound in air at room temperature. Every 3 seconds of delay means the strike was roughly 1 kilometer away. Sound is fast by human standards, but the flash shows you it is not instant.

The speed of sound depends on the medium. In water, sound travels about 1,480 meters per second, more than four times faster than in air. In steel, it races along at nearly 6,000 meters per second. Particles in liquids and solids are packed closer and bonded more stiffly, so a push at one end is passed to the far end quickly. Air particles are far apart and must fly across gaps before they bump the next one.

This is why a swimmer hears a clank underwater before a friend on the dock does. It is also why old stories tell of people pressing an ear to a rail to hear a distant train. Temperature matters too. Warm air carries sound a little faster than cold air. Its particles are already moving faster, so they pass the push along sooner.

A sound wave that reaches your head is a train of tiny pressure changes in the air. The outer ear funnels them down a short canal to the eardrum, a thin skin about the size of a fingernail. Each compression pushes the eardrum in; each stretch lets it spring out. The eardrum vibrates at the same frequency as the sound, thousands of times a second for a high note.

Behind the eardrum sit three tiny bones, the smallest in the body. They work like a lever, passing the vibration to a snail-shaped tube called the cochlea, which is filled with fluid. Inside, thousands of hair cells sit along a membrane that is stiff at one end and floppy at the other. High-frequency vibrations shake the stiff end, low ones the far end. Each hair cell fires a nerve signal when its part of the membrane moves, so the brain learns the pitch from which cells are firing.

Loudness is measured in decibels (dB). A whisper is about 30 dB, normal talk about 60 dB, and a loud concert can pass 110 dB. Very loud sound bends the hair cells too far and can break them. Unlike skin, they do not grow back. That is why hearing protection matters near power tools and why doctors warn about turning earbuds all the way up.

Sunlight crosses about 150 million kilometers of nearly empty space to reach Earth. Sound could never make that trip; it needs particles to push. Light does not. It is an electromagnetic wave, a traveling pattern of electric and magnetic fields that carries its own energy along without a medium. In empty space it moves at about 300,000 kilometers per second, the fastest anything can travel.

That speed is so great that light seems instant on Earth. Flip a switch and the room is lit with no delay you can notice. But over big distances the delay shows. Light from the sun takes about 8 minutes to reach us. When you look at the moon, you see it as it was a little over a second ago. The first person to measure this delay was the Danish astronomer Ole Roemer in 1676, who noticed that the eclipses of Jupiter’s moon Io ran late when Earth was far from Jupiter and early when it was close.

Light is still a wave, with amplitude, wavelength and frequency. Amplitude shows up as brightness. Wavelength and frequency show up as color. Red light has the longest wavelength the eye can see, about 700 nanometers, and violet the shortest, about 400. A nanometer is one billionth of a meter, so several hundred waves of visible light fit across the width of a single human hair.

Stand in a sunny room and look around. The mirror shows your face. The black sweater on the chair looks dark and feels warm. The window lets the yard show through. Each object is doing something different with the same sunlight. Light can reflect, bouncing off a surface. It can be absorbed, its energy taken in and usually turned into heat. Or it can be transmitted, passing through. Most objects do a mix of all three.

Reflection follows a simple rule. Light bounces off a smooth surface at the same angle it came in, like a ball off a wall. A mirror is smooth enough to keep the rays in order, so an image forms. A sheet of white paper reflects almost as much light as a mirror. But its surface is bumpy at a tiny scale, so the rays scatter in every direction. That is why paper looks bright but shows no image.

Color comes from which wavelengths get absorbed. A red apple absorbs most of the blue and green in white light and reflects the red. A black shirt absorbs nearly all colors, which is why it warms up faster in sunlight than a white one. A clear window transmits most light, while tinted glass absorbs some wavelengths and transmits the rest. On a hot Chicago afternoon, that is the difference between a car with a black roof and one with a white roof.

Put a pencil in a glass of water and look from the side. The pencil seems to break at the surface. Nothing happened to the pencil. The light coming from the underwater part bent as it left the water and entered the air. This bending is called refraction, and it happens whenever light crosses from one material into another at an angle.

The cause is speed. Light travels slower in water than in air, and slower still in glass. When a beam hits the surface at a slant, one edge of the beam enters the slow material first and is held back while the other edge is still moving fast. The beam swings around, the way a line of marching students turns if the ones on one end slow down. Light entering a slower material bends toward the line straight into the surface; light leaving it bends away.

Refraction is everywhere. It makes a swimming pool look shallower than it is. It splits white light into a rainbow when it passes through a prism, because each color slows by a slightly different amount and so bends by a slightly different angle. Isaac Newton showed this with a prism in a darkened room in the 1660s. And refraction is the whole trick behind a lens, the subject of the next section.

Your eye is a ball about 2.4 centimeters across with a window in the front. Light enters through the clear cornea, which does most of the bending. Then it passes through the pupil, the dark hole in the middle. The pupil widens in dim light and shrinks in bright light. Behind the pupil sits the lens, a soft, clear disc that fine-tunes the focus. Together the cornea and lens refract the rays so they meet on the back wall of the eye.

That back wall is the retina, a thin layer of light-sensing cells. The image that lands there is upside down and backward, just as in a camera, but the brain flips it without your noticing. Two kinds of cells do the sensing. Rods work in dim light and see only shades of gray. Cones need brighter light and come in three types tuned to different wavelengths, roughly red, green and blue. Mixing their signals gives every color you see.

The lens can change shape. Small muscles around it squeeze it thicker to focus on a near page and let it flatten to focus on a distant sign. That is why your eyes feel tired after a long time reading up close: the muscles have been working the whole time. With age the lens stiffens, which is why many adults past 40 start holding menus at arm’s length.

Some eyes are a little too long from front to back. Light from distant objects comes to a focus before it reaches the retina, then spreads out again, so the far-away sign is a blur while a nearby book is sharp. This is nearsightedness, and it is common in teenagers. The fix is a concave lens, thinner in the middle than at the edges. It spreads the rays slightly apart before they enter the eye, so they meet a little farther back, right on the retina.

Other eyes are a little too short, or their lens has stiffened with age. Light from a near object would come to a focus behind the retina if it could, so the page is blurry while the far sign is clear. This is farsightedness. The fix is a convex lens, thicker in the middle, which bends the rays inward so they meet sooner. Reading glasses sold at a drugstore are simple convex lenses.

An eye doctor finds the right lens by trial. You look at letters through lens after lens until the blur snaps into focus, and the strength of that lens is written on your prescription. Contact lenses do the same job by sitting directly on the cornea. Laser surgery reshapes the cornea itself to change how strongly it refracts. All three are the same idea: move the focus onto the retina.

Hold a magnifying glass over a sidewalk on a sunny day and slide it up and down. At one height the sunlight collapses into a tiny, painfully bright spot. That spot is the focal point, and its distance from the lens is the focal length. The lens is convex, thicker in the middle, so every part of it bends the incoming light toward the center. Parallel rays from the sun meet at one place.

Now hold the same lens over a printed page, closer than its focal length. The letters swell. You are looking at a magnified image, larger than the object. The rays leaving each letter are bent by the lens so that, when they reach your eye, they seem to come from a much bigger letter farther away. The eye and brain cannot tell the difference between real rays and bent ones, so you see a giant letter.

A fatter lens, with more curve, has a shorter focal length and magnifies more. But it also blurs and colors the edges of the image. Early lens makers fought these flaws for centuries. Wider lenses gather more light. That is why a telescope lens is much bigger than a magnifying glass. The point is not just to enlarge. It is to collect enough dim light from a distant object to see it at all.

The site’s microscope has two sets of lenses. Nearest the specimen is the objective lens, a small, strongly curved lens that sits a few millimeters above the glass slide. It forms a magnified image of the specimen inside the tube. At the top is the eyepiece, which you look through. It acts like a magnifying glass on the image the objective already made. Two enlargements in a row multiply: a 10x objective and a 10x eyepiece give 100x.

Light matters as much as glass. A mirror or lamp under the stage sends light up through the thin specimen. What you see is light transmitted through the sample, with dark parts where cells absorb it. That is why a slice must be very thin. It is also why stains are used. A dye that a cell nucleus absorbs turns the nucleus dark and visible.

In 1665 Robert Hooke published a book of drawings made with a two-lens microscope, including a thin slice of cork that looked like rows of tiny boxes. He called them cells. The name stuck to the units of every living thing. A few years later, Antonie van Leeuwenhoek in the Netherlands used single, tiny, very powerful lenses to see living things in pond water that nobody had known were there.

Light microscopes have a hard limit. Visible light has a wavelength of around 500 nanometers, and anything much smaller than that wavelength simply cannot be resolved by light, no matter how good the glass. Seeing viruses or atoms took a different kind of wave, and electron microscopes in the twentieth century supplied it.

Galileo’s telescope was a refractor: a large lens at the front gathered light and a small lens at the back magnified it. Refractors improved for nearly three centuries. The largest ever put into regular use is at Yerkes Observatory in Williams Bay, Wisconsin, a two-hour drive from Chicago. Its lens, finished in 1897, is 40 inches, about 1 meter, across. Nobody built a bigger one, because a glass lens that large sags under its own weight and blurs.

The answer was a mirror. In 1668 Isaac Newton built a small telescope that used a curved mirror instead of a front lens. A mirror can be supported from behind, so it can be made huge, and it reflects every color to the same focus, avoiding the color fringes that lenses add. Nearly every large telescope since has been a reflector. The Hale telescope on Palomar Mountain, California, finished in 1948, has a mirror about 5 meters across.

Two more problems remained: air and daylight. Earth’s atmosphere wobbles, making stars twinkle and images shimmer. The Hubble Space Telescope, launched in 1990 with a 2.4-meter mirror, solved both by orbiting above the air. The James Webb Space Telescope, launched on December 25, 2021, carries a 6.5-meter mirror built from 18 gold-coated hexagons that unfolded in space. It sees in infrared, a kind of light explained in the next section.

Chicago has its own place in this story. The Adler Planetarium on the lakefront, opened in 1930, was the first planetarium in the Western Hemisphere. And every telescope, from Galileo’s tube to Webb, works on the same two ideas from the last chapter: gather as much light as possible, and bend it to a focus.

In 1800 the astronomer William Herschel spread sunlight into a rainbow with a prism and set a thermometer in each color. Red warmed the thermometer more than violet. Then he tried something odd: he moved the thermometer past the red end, into what looked like empty shadow. It got warmer still. Something invisible was arriving there, carrying energy. He had found infrared light, light with a wavelength too long for the eye.

A year later Johann Ritter tried the other end. Paper soaked in silver chloride darkens in light. Ritter found it darkened fastest just beyond the violet, where nothing could be seen. That was ultraviolet. Over the next century the family kept growing. In 1887 Heinrich Hertz made and detected radio waves in his lab. In 1895 Wilhelm Röntgen found X-rays, which passed through flesh and cast shadows of bone.

All of these are the same kind of wave as visible light. They are electromagnetic waves traveling at the same speed, differing only in wavelength and frequency. Lined up from longest wavelength to shortest, they make the electromagnetic spectrum: radio, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays. Visible light, from about 700 down to 400 nanometers, is a thin sliver near the middle. Shorter wavelength means higher frequency and more energy per wave. That is why ultraviolet burns skin and radio waves do not.

Look around a kitchen and count the spectrum. The radio on the counter picks up FM signals near 100 megahertz, waves about 3 meters long that bend around buildings. The microwave oven fills its box with waves about 12 centimeters long that water absorbs strongly, so the food heats from the water inside it while the dry plate stays cool. The toaster glows with infrared. The overhead lamp gives visible light. Each job uses the wavelength that behaves the right way.

Radio waves, the longest, pass through walls and travel far. That makes them ideal for broadcasting and for cell phones, whose signals sit between a few hundred megahertz and a few gigahertz. Microwaves are shorter and can be aimed in tight beams, useful for radar and for Wi-Fi. Infrared cameras see warm bodies in the dark, because everything warm gives off infrared. Ultraviolet from the sun tans and burns. Sunscreen absorbs it before the skin does.

X-rays pass easily through soft tissue but are absorbed by bone. So a bone casts a shadow on the film or sensor. Gamma rays, the most energetic, come from radioactive atoms and are used in some cancer treatments. Higher-frequency waves carry more energy, so X-rays and gamma rays can damage cells. That is why a dentist steps out of the room and a lead apron covers your chest.

The universe does not shine only in colors we can see. Clouds of dust that block visible light are nearly transparent to infrared. So an infrared telescope can look straight through them at the stars forming inside. The James Webb Space Telescope was built for exactly this. Its mirror is coated in a thin layer of gold, because gold reflects infrared very well. The whole telescope hides behind a sunshield the size of a tennis court, so its own warmth does not blind it.

There is a second reason Webb sees in infrared. The universe is expanding. Light from the most distant galaxies has been stretched during its long trip, shifting visible light into the infrared. Catching that stretched light lets Webb see galaxies as they were more than 13 billion years ago, when the universe was young. A telescope is a kind of time machine. The farther away it looks, the further back in time it sees.

Radio telescopes, huge dish antennas, map cold hydrogen gas and the leftover glow of the early universe. X-ray telescopes in orbit catch the light from gas heated to millions of degrees near black holes. Each window in the spectrum shows a different universe. Astronomers combine them into one picture. A doctor does the same with an X-ray, an ultrasound and a photograph of one patient.

The groove on a vinyl record is an analog signal. Its wiggle is a direct copy of the sound wave’s shape: twice as loud, twice as wide. That is simple and elegant, but it has a weakness. Every scratch, every speck of dust, every bit of wear adds its own wiggle, and the player cannot tell the added wiggle from the music. Copy an analog tape to another tape and the hiss of the first copy is copied too, plus new hiss. After a few generations the music is buried.

A digital signal takes a different route. Thousands of times a second, it measures the wave’s height and writes the measurement down as a number. Each number is stored as a string of bits, ones and zeros. A CD samples the wave 44,100 times each second. Instead of a shape, the recording is a long list of numbers.

Numbers can be protected in a way shapes cannot. A one is sent as a high voltage or a bright flash, a zero as a low voltage or darkness. Noise may nudge a high signal a little lower or a low signal a little higher, but as long as the receiver can still tell high from low, it reads the bit correctly and the noise vanishes. Extra check bits let the receiver notice and fix bits that did get flipped. A digital file copied a million times is still exactly the same file.

This is why phone calls, streaming music, photographs and this web page are all digital. The trade-off is that a digital recording only keeps what it measured. Sample too rarely or store too few bits per sample and the sound gets coarse. Engineers choose sample rates high enough that the ear cannot tell the difference.

You take a photo of the Bean in Millennium Park and tap send. Inside the camera, a grid of millions of tiny light sensors, one per pixel, has measured how much red, green and blue light landed on it. Each color is stored as a number from 0 to 255, so each pixel is three numbers, and each number is 8 bits. A 12-million-pixel picture is nearly 300 million bits before anything else happens.

That is too much to send quickly, so the phone compresses the image. It looks for patches where neighboring pixels are nearly the same color, such as the sky, and stores a short description instead of every value. A typical photo shrinks to a few million bits with almost no visible loss. The file is then chopped into packets, each labeled with where it belongs.

The phone’s antenna sends the packets as radio waves to a cell tower that may be a kilometer away. The wave is switched on and off, or shifted slightly, to mean one or zero. From the tower the bits usually travel by glass fiber, as pulses of light. They pass through switching centers, possibly across an ocean, to the tower nearest your friend. Her phone receives the radio packets and checks each one for errors. It asks again for any that were damaged, then rebuilds the numbers.

Finally her screen does the reverse of your camera: at each pixel it lights three tiny red, green and blue elements to the strength the numbers say. The whole trip takes about a second, and nothing physical made the journey. Only a pattern did, exactly as with the ripple on the pond in the last chapter.

Aim a flashlight up from under water at a shallow angle and the light does not leave the pool. It bounces off the surface as if the surface were a mirror. This is total internal reflection. Light moving from a slower material such as water or glass toward a faster one such as air is bent away from the surface, and past a certain angle it cannot escape at all. It reflects back in, losing almost nothing.

An optical fiber uses this trick on purpose. It is a strand of extremely pure glass thinner than a human hair. It is wrapped in a second glass layer that light travels through faster. Light pulses entering one end glance off the boundary again and again, tens of thousands of times per kilometer. They stay trapped in the core until they come out the far end. A laser flashes on for a one and off for a zero, billions of times each second.

Fiber beats copper wire in every way that matters for signals. The glass is so pure that a pulse can travel 100 kilometers before it needs a boost. A window pane that pure would let you see clearly through a stack a kilometer thick. Light pulses do not interfere with each other the way electric currents in nearby wires do. One fiber can carry many colors at once, each color a separate stream of bits. Undersea fiber cables now carry nearly all international internet traffic. The first was laid across the Atlantic in 1988.

The idea reaches back to 1880. That year Alexander Graham Bell sent a voice a few hundred meters on a beam of sunlight, with a device he called the photophone. Clouds blocked it, so it was never practical. Glass gave the light a protected road. In 1966 the engineer Charles Kao showed that glass pure enough could make that road span the world.

Shout across a canyon and a moment later the canyon shouts back. The delay is the round trip of your sound to the far wall and back. If you know the speed of sound, you can measure the distance without a tape: distance equals speed times time, divided by two because the sound went there and back. A 2-second echo in air means the wall is about 343 meters away.

Sonar does this on purpose, under water. A ship sends a sharp click downward and listens. In seawater, sound travels about 1,500 meters per second, so an echo that returns after 4 seconds came from a bottom about 3,000 meters down. Sweeping the click across the seafloor maps mountains and trenches that no one can see. Sonar was developed after the sinking of the Titanic in 1912 and during the First World War to find icebergs and submarines, and it still does both.

Bats invented the method millions of years earlier. A hunting bat squeaks at frequencies up to about 100,000 hertz, far above human hearing, and listens for the echo from a moth. Its calls are high-pitched because short wavelengths bounce off small objects that long waves would pass around. Dolphins do the same with clicks under water. The site’s drum machine lets you try a slow version: tap once and listen for the return from the far wall of the room.

Ultrasound in a doctor’s office is sonar at very small scale. A probe sends pulses into the body and times the echoes from organs and from a baby’s heartbeat, building a picture out of nothing but delay. Sound is a safer choice than X-rays for this job because it carries far less energy per wave.

Radar is sonar with radio waves. A dish sends a pulse of microwaves and listens for the reflection. Because the waves travel at the speed of light, the timing is fierce: an echo from an airplane 15 kilometers away returns in one ten-thousandth of a second. Electronics measure that easily, and a rotating dish turns the delays into dots on a screen. Radar was developed in Britain and the United States during the 1930s, and by 1940 it was warning of approaching aircraft.

Raindrops reflect microwaves too. Weather radar sends pulses into the sky and paints a map of where rain and hail are, colored by how strong the echo is. Watch a Chicago forecast and the green, yellow and red blotches sliding across Illinois are radar echoes, updated every few minutes.

Modern weather radar does something more. When a wave reflects from an object moving toward the dish, the returning wave is squeezed to a slightly higher frequency. From an object moving away, it is stretched lower. This is the Doppler effect, the same reason a passing siren drops in pitch. By measuring the shift, Doppler radar sees which way the rain is moving inside a storm. A patch of echoes rushing toward the dish right beside a patch rushing away means a rotating column. That is a possible tornado, and the warning goes out before the funnel touches down.

Every technology in this unit is the same idea wearing different clothes. Send a wave, or catch one that is already there. Measure its amplitude, its timing, its frequency. Turn the measurements into information. The turntable does it with a groove, the eye with a lens, the bat with a squeak and the weather service with a spinning dish.