Waves
What Travels When Everything Stays Put
Something Crosses, the Water Stays
Drop a stone into a still pond and rings of ripples spread to the shore. A leaf floating on the pond rises and falls as they pass under it. When they have gone, the leaf is still about where it was. Something has crossed the pond, and the water has not. What crosses is a shape, and the energy that keeps it moving.
The shape moves because each bit of water pushes on the bits beside it. Raise a hump and its extra weight presses on the water under it, which is squeezed out sideways and lifts the surface next to it. Gravity pulls the hump back down, and it overshoots into a dip, as a pendulum does, while the surface beside it rises in turn. Each bit hands its motion on to the next a moment later, so the hump runs on while every bit stays near home. Anything that springs back toward rest and overshoots can carry a wave this way: a stretched cord, a spring, the air. A field can too: light is a wave in the electromagnetic field, its electric and magnetic parts rising and falling together, in step.
Fine specks suspended in a tank of water show where the water goes. Seen from the side, as waves run past, each speck traces a small circle.
Each bit of water rises as a crest nears, moves forward at the crest, sinks behind it, and moves back in the trough. Near the surface its circle is as wide as the wave is high. Deeper down the circles shrink fast: to half that size a ninth of a wavelength down, and to about a twentieth of it half a wavelength down. Below that the water hardly moves, whatever the storm above.
On a cord, the cord moves across the way the wave runs. In sound, the air moves to and fro along it, crowding and thinning. Physicists call the first kind of wave transverse and the second longitudinal. Water does both at once, a quarter of a swing apart, and that is why it goes round.
The picture of water that stays put is very nearly right, but not exactly. In theory each circle fails to close by a little, so the water creeps forward with every wave. In a closed tank a slow current back along it cancels the creep. On open water the creep is real, and it carries floating things along with the waves.
Where a wave breaks, the picture fails outright. A wave in deep water cannot grow steeper than about a seventh of its length, and the steepest wave comes to a sharp corner at its crest. Near a beach the water grows shallow, and the wave slows and steepens. Once its height nears the depth of the water under it, its crest runs ahead of its base and topples. Then the water itself is thrown forward.
The Medium Sets the Pace
On a cord or in the air, a wave travels at a set speed, and the speed belongs to what it travels in. Each bit has to be pulled into motion by the bit before it. The harder the medium pulls, the sooner the next bit gets going. The heavier each bit, the slower it answers. On a stretched cord the speed is the square root of the pull divided by the mass of one meter of the cord. Four times the pull doubles the speed, and four times the mass halves it.
The hand that shakes the cord sets only how often the waves come. Shake it twice as fast and the waves come twice as often, but they run no faster, so each one is half as long. Physics teachers have found that many students expect the opposite, that a quicker flick sends a quicker wave. It does not. The speed is the cord's, and the length of each wave is whatever fits: the speed divided by how many waves leave each second.
Sound works the same way. In air at room temperature it runs at 343 meters a second, whatever its pitch, unless it is very loud. Only sound far louder than any voice runs measurably faster. Water is far stiffer than air for its weight, and sound crosses it at about 1,480 meters a second; in steel it runs at nearly 6,000. Light, a wave in the electromagnetic field, crosses empty space at 299,792,458 meters a second.
A wave's height tells how much energy it carries. Each bit of a cord swings to and fro, and at a given rate a swing twice as wide holds four times the energy. So a wave twice as high carries four times as much. On a cord, a wave of the same height at twice the rate makes each bit move twice as fast, and that is four times the energy too.
On the sea, the depth of the water sets the pace of the longest waves. A tsunami can be hundreds of kilometers long and less than a meter high in the open ocean, where over four kilometers of water it runs at about 700 kilometers an hour. As the sea shallows toward a coast the wave slows, shortens and grows. It usually comes ashore not as a curling wall but as a flood many meters high that rises fast and keeps rising for minutes.
When Waves Meet
Throw two stones into a pond side by side, and two sets of rings spread out. Where they meet, they do not bounce off each other. They pass straight through, and each comes out the other side as it went in. While they overlap, the water’s height at each point is the height the first set would give there plus the height the second would give. This adding is called superposition.
Two sources that keep in step make something new. Where a crest from one always arrives with a crest from the other, the water heaves twice as high. Where a crest from one always arrives with a trough from the other, the two cancel and the water stays still. The still places lie along lanes that fan out from between the sources, and they stay put for as long as the sources keep in step. This is interference.
No energy is lost in the still lanes. It is moved out of them into the lanes between, where the water heaves twice as high and so carries four times the energy of one set of waves. Taken over the still lanes and the busy ones together, the energy is just what the two dippers put in. Far apart, that is twice one dipper’s. Closer together, each dipper also pushes against the other’s waves, and what the two put in changes.
Interference shows that light travels as a wave too. The colors of thin films come from two reflections of light that interfere. Let sunlight into a dark room through a pinhole, hold a slip of card less than a millimeter wide in the beam, and inside the card’s shadow lie bright and dark bands. Block the light passing one side of the card, and they vanish.
The pattern is the one water makes. In a ripple tank, a shallow tray of water with a glass bottom and a light below it, two sets of ripples cross with the still lanes between them. Light passing through two narrow slits makes the same bands, and so does one beam of light split into two by a pair of mirrors.
Adding holds while the waves are small. A tall wave changes what it travels in. A high water wave runs a little faster than a low one, and the air in a blast is squeezed so hard that its sound runs faster. Then two waves no longer simply add.
Trapped Waves Have Notes
Tie a cord at both ends and shake it near one end. Each wave runs to the far end, turns upside down there and runs back, so the cord fills with waves going both ways at once. At most rates of shaking, the waves coming back meet the new ones out of step and mostly cancel. The cord hardly stirs.
At certain rates the returning waves arrive in step. Then each trip adds to the last, and the cord swings in loops that stay in place. Between the loops are still points, where the waves going both ways always cancel. The lowest such rate gives a single loop. Twice that rate gives two loops, and three times gives three: a whole number of loops has to fit between the ends.
Those rates are the cord’s notes. A plucked guitar string sounds many of them at once: its lowest note, and the notes at two, three, four and more times that rate, called harmonics. How strongly each one sounds, and how the mix changes from the first instant of the note to its fading, gives a guitar, a violin or a piano its own voice.
A second string can find the still points. Pluck one string, and another tuned an octave lower, at half the rate, trembles in sympathy. It trembles in two halves but not at its middle, as a little bit of paper wrapped loosely round it and slid along it shows. One tuned to a third of the rate trembles in three parts.
The still points are called nodes. Paper riders on a swinging string show them: those on the loops are thrown off, and those on the nodes stay put.
Every trapped wave has notes this way: the air in a flute, a drumhead, a bridge. On a plate, the still points become still lines. Sand sprinkled on a metal plate set ringing with a violin bow gathers along them, in patterns known as Chladni figures. An electron held in an atom is a trapped wave too. Only certain patterns fit in the pull of the nucleus, each a cloud of a set shape round it, and that is why an atom can hold only certain energies.
Real strings are not perfectly flexible, and their stiffness pushes each harmonic a little sharp of its whole-number place. On a piano it is enough to matter. Tuners matching notes by ear end up stretching the octaves, the high notes a little sharp and the low ones a little flat.
Bending Through Gaps
A voice from round a corner can be heard, though its speaker cannot be seen. Part of it bounces off walls, but even in the open, sound bends round the corner, and light, for the most part, does not. Both are waves. The difference is one of size.
Every point a wave reaches sends out its own little ripple, and the next wave front is where they all agree. Add interference, and the idea becomes a tool that gets the patterns right, though not every detail close to an edge. When waves meet a gap, every point of the gap sends out its own ring, and what lies beyond is all of them added up.
Through a gap many wavelengths wide, the rings from across it agree only straight ahead and cancel off to the sides. The waves run on as a beam as wide as the gap, spreading only slowly, and the water behind the walls stays almost calm. Through a gap narrower than a wavelength, the rings from its points cancel in no direction at all, and the waves spread out beyond it in half rings. This spreading is called diffraction.
Through any gap wider than a wavelength, a beam spreads by an angle set by the wavelength and the gap. The first still direction to either side lies where the sine of its angle equals the wavelength divided by the gap’s width. Speech has wavelengths from about three meters down to ten centimeters, the size of doorways and heads, so it spreads widely through a door. Light’s wavelengths are well under a thousandth of a millimeter. A doorway is about two million of them across, so light goes through it nearly straight and leaves a sharp shadow.
The same spreading puts a floor under every lens. A lens is an opening too, and the light through it spreads a little, so it cannot draw a point as a point. An ordinary light microscope cannot separate two things closer together than about half the wavelength of its light. One way past is a shorter wave: electrons, or X-rays. Another uses ordinary light with a trick. Glowing molecules are made to light up a few at a time, and the center of each blur is found far more finely than the blur itself. Such microscopes now place molecules in a cell to within a few billionths of a meter.
One Rule, and Where It Bends
Most of this page comes from one rule. Where a cord is bent, its pull drags the bent part back toward straight, and harder the sharper the bend. So each bit of the cord speeds up in proportion to how sharply the cord is curved there. The rule can be written as one equation. Its solutions are shapes of any form running along the cord without changing, at the speed set by its pull and its mass.
The same equation, with a speed of its own each time, governs sound in air, the waves of an earthquake in rock, and light. Many things that spring back when moved a little, and overshoot, obey it. That is why superposition, interference, notes and diffraction work alike in all of them. It turned up in the laws of electricity and magnetism too, with a speed close to the measured speed of light, and that pointed to light being such a wave. Waves made with sparks, behaving just as light does, settled it.
Deep water breaks the rule. On it a long wave runs faster than a short one. The reason is how deep each wave reaches. On shallow water every wave stirs the whole depth, and every long wave runs at one speed set by that depth, as a tsunami does. On deep water a wave stirs the water only down to about half its length, so a long wave works a deeper layer and runs as if the sea were deeper. Each bit of water goes round its circle in the time a pendulum takes for one swing to and fro, a pendulum as long as the wave’s length divided by about six. A longer pendulum swings more slowly, but only as the square root of its length. A wave four times as long takes twice as long for each swing, so it covers four times the distance in twice the time and runs twice as fast. In meters a second, a wave’s speed is about one and a half times the seconds between its crests. A push that holds waves of many lengths at once, such as the splash of a stone, cannot keep its shape. It spreads out as it goes, with the long waves running ahead. This is called dispersion. Only the finest ripples, under about two centimeters and held by the water’s skin rather than by gravity, sort the other way: among them the shortest run fastest.
The ocean shows it on a grand scale. Swell from storms between New Zealand and Antarctica has been followed across the whole Pacific to Alaska, more than 10,000 kilometers. At every station on the way the long waves arrived first and the shorter ones later. How quickly the waves at a station grow shorter tells how far away the storm is.
A short group of waves on deep water shows dispersion from the inside. A group is a place where waves of slightly different lengths happen to be in step, so that their crests add. Each runs at its own speed, so the place where they agree moves at a speed of its own. On deep water that is half the speed of the crests, so each crest is born at the back of the group, runs through it and dies at its front. It is the group, and the energy in it, that crosses an ocean: in a swell with 20 seconds between its crests, the crests run at about 31 meters a second and the groups at about 16, crossing 10,000 kilometers in about a week. The waves of quantum physics disperse too. An electron’s wave obeys an equation in which short ripples run faster than long ones, and that is why a free electron’s wave spreads out as it travels. That wave is not the electron smeared out: its size at each place sets the chance of catching the whole electron there.
So one equation describes waves wherever they are small and the medium treats every wavelength alike. Where waves grow tall, or the medium sorts them by length, each kind of wave goes its own way, and those departures are as telling as the rule.
