How to Teach Physics to Your Dog (3 page)

You can already see how waves are different from particles: they don’t have a position. The wavelength and the frequency describe the pattern as a whole, but there’s no single place you can point to and identify as
the
position of the wave. The wave itself is a disturbance spread over space, and not a physical thing with a definite position and velocity. You can assign a velocity to the wave pattern, by looking at how long it takes one crest of the wave to move from one position to another, but again, this is a property of the pattern as a whole.

You also can’t count waves the way you can count particles—you can say how many crests and troughs there are in one particular area, but those are all part of a single wave pattern. Waves are continuous where particles are discrete—you can say that you have one, two, or three particles, but you either have waves, or you don’t. Individual waves may have larger or smaller amplitudes, but they don’t come in chunks like particles do. Waves don’t even add together in the same way that particles do—sometimes, when you put two waves together, you end up with a bigger wave, and sometimes you end up with no wave at all.

Imagine that you have two different sources of waves in the same area—two rocks thrown into still water at the same time, for example. What you get when you add the two waves together depends on how they line up. If you add the two waves together
such that the crests of one wave fall on top of the crests of the other, and the troughs of one wave fall in the troughs of the other (such waves are called “in phase”), you’ll get a larger wave than either of the two you started with. On the other hand, if you add two waves together such that the crests of one wave fall in the troughs of the other and vice versa (“out of phase”), the two will cancel out, and you’ll end up with no wave at all.

This phenomenon is called
interference,
and it’s perhaps the most dramatic difference between waves and particles.

“I don’t know . . . that’s pretty weird. Do you have any other examples of interference? Something more . . . doggy?”

“No, I really don’t. That’s the point—waves are dramatically different than particles. Nothing that dogs deal with on a regular basis is all that wavelike.”

“How about, ‘Interference is like when you put a squirrel in the backyard, and then you put a dog in the backyard, and a minute later, there’s no more squirrel in the backyard.’ ”

“That’s not interference, that’s prey pursuit. Interference is more like putting a squirrel in the backyard, then putting a second squirrel in the backyard one second later, and finding that you have no squirrels at all. But if you wait
two
seconds before putting in the second squirrel, you find four squirrels.”

“Okay, that’s just weird.”

“That’s my point.”

“Oh. Well, good job, then. Anyway, why are we talking about this?”

“Well, you need to know a few things about waves in order to understand quantum physics.”

“Yeah, but this just sounds like math. I don’t like math. When are we going to talk about physics?”

“We are talking about physics. The whole point of physics is to use math to describe the universe.”

“I don’t want to describe the universe, I want to catch squirrels.”

“Well, if you know how to describe the universe with math, that can help you catch squirrels. If you have a mathematical model of where the squirrels are now, and you know the rules governing squirrel behavior, you can use your model to predict where they’ll be later. And if you can predict where they’ll be later . . .”

“I can catch squirrels!”

“Exactly.”

“All right, math is okay. I still don’t see what this wave stuff is for, though.”

“We need it to explain the properties of light and sound waves, which is the next bit.”

We deal with two kinds of waves in everyday life: light and sound. Though these are both examples of wave phenomena, they appear to behave very differently. The reasons for those differences will help shed some light (pardon the pun) on why it is that we don’t see dogs passing around both sides of a tree at the same time.

Sound waves are pressure waves in the air. When a dog barks, she forces air out through her mouth and sets up a vibration that travels through the air in all directions. When it reaches another dog, that sound wave causes vibrations in the second dog’s eardrums, which are turned into signals in the brain that are processed as sound, causing the second dog to bark, producing more waves, until nearby humans get annoyed.

Light is a different kind of wave, an oscillating electric and magnetic field that travels through space—even the emptiness of outer space, which is why we can see distant stars and galaxies. When light waves strike the back of your eye, they get turned into signals in the brain that are processed to form an image of the world around you.

The most striking difference between light and sound in everyday life has to do with what happens when they encounter an obstacle. Light waves travel only in straight lines, while sound waves seem to bend around obstacles. This is why a dog in the dining room can hear a potato chip hitting the kitchen floor, even though she can’t see it.

The apparent bending of sound waves around corners is an example of diffraction, which is a characteristic behavior of waves encountering an obstacle. When a wave reaches a barrier with an opening in it, like the wall containing an open door from the kitchen into the dining room, the waves passing through the opening don’t just keep going straight, but fan out over a range of different directions. How quickly they spread depends on the wavelength of the wave and the size of the opening through

On the left, a wave with a short wavelength encounters an opening much larger than the wavelength, and the waves continue more or less straight through. On the right, a wave with a long wavelength encounters an opening comparable to the wavelength, and the waves diffract through a large range of directions.

which they travel. If the opening is much larger than the wavelength, there will be very little bending, but if the opening is comparable to the wavelength, the waves will fan out over the full available range.

Similarly, if sound waves encounter an obstacle like a chair or a tree, they will diffract around it, provided the object is not too much larger than the wavelength. This is why it takes a large wall to muffle the sound of a barking dog—sound waves bend around smaller obstacles, and reach people or dogs behind them.

Sound waves in air have a wavelength of a meter or so, close to the size of typical obstacles—doors, windows, pieces of furniture. As a result, the waves diffract by a large amount, which is why we can hear sounds even around tight corners.

Light waves, on the other hand, have a very short wave-length—less than a thousandth of a millimeter. A hundred wavelengths of visible light will fit in the thickness of a hair. When light waves encounter everyday obstacles, they hardly bend at all, so solid objects cast dark shadows. A tiny bit of diffraction occurs right at the edge of the object, which is why the edges of shadows are always fuzzy, but for the most part, light travels in a straight line, with no visible diffraction.

If we don’t readily see light diffracting like a wave, how do we know it’s a wave? We don’t see diffraction around everyday objects because they’re too large compared to the wavelength of light. If we look at a small enough obstacle, though, we can see unmistakable evidence of wave behavior.

In 1799 an English physicist named Thomas Young did the definitive experiment to demonstrate the wave nature of light. Young took a beam of light and inserted a card with two very narrow slits cut in it. When he looked at the light on the far side of the card, he didn’t just see an image of the two slits, but rather a large pattern of alternating bright and dark spots.

Young’s double-slit experiment is a clear demonstration of the diffraction and interference of light waves. The light passing

An illustration of double-slit diffraction. On the left, the waves from two different slits travel exactly the same distance, and arrive in phase to form a bright spot. In the center, the wave from the lower slit travels an extra half-wavelength (darker line), and arrives out of phase with the wave from the upper slit. The two cancel out, forming a dark spot in the pattern. On the right, the wave from the lower slit travels a full extra wavelength, and again adds to the wave from the upper slit to form a bright spot.

through each of the slits diffracts out into a range of different directions, and the waves from the two slits overlap. At any given point, the waves from the two slits have traveled different distances, and have gone through different numbers of oscillations. At the bright spots, the two waves are in phase, and add together to give light that is brighter than light from either slit by itself. At the dark spots, the waves are out of phase, and cancel each other out.

Prior to Young’s experiment, there had been a lively debate about the nature of light, with some physicists claiming that light was a wave, and others (including Newton) arguing that light was a stream of tiny particles. Interference and diffraction are phenomena that only happen with waves, though, so after Young’s experiment (and subsequent experiments by the French physicist Augustin Fresnel), everybody was convinced that light was a wave. Things stayed that way for about a hundred years.

• • •

“How does this relate to going around both sides of a tree? I’m not interested in going through slits, I want to catch bunnies.”

“The same basic process happens when you put small solid obstacles into the path of a light beam. You can think of the light that goes around to the left and the light that goes around to the right of the obstacle as being like the waves from two different slits. They take different paths to their destination, and thus can be either in phase or out of phase when they arrive. You get a pattern of bright and dark spots, just like when you use slits.”

“Oh. I guess that makes sense. So, I just need to get the bunnies to stand at the spots where I’m in phase with me?”

“No, because of the wavelength thing. We’ll get to that in a minute. I need to talk about particles, first.”

“Okay. I can be patient. As long as it doesn’t take too long.”

The first hint of a problem with the wave model of light came from a German physicist named Max Planck in 1900. Planck was studying the thermal radiation emitted by all objects. The emission of light by hot objects is a very common phenomenon (the best-known example is the red glow of a hot piece of metal), and something so common seems like it ought to be easy to explain. By 1900, though, the problem of explaining how much light of different colors was emitted (the “spectrum” of the light) had thus far defeated the best physicists of the nineteenth century.

Planck knew that the spectrum had a very particular shape, with lots of light emitted at low frequencies and very little at high frequencies, and that the peak of the spectrum—the frequency at which the light emitted is brightest—depends only on the object’s temperature. He had even discovered a formula to
describe the characteristic shape of the spectrum, but was stymied when he tried to find a theoretical justification for the formula. Every method he tried predicted much more light at high frequencies than was observed. In desperation, he resorted to a mathematical trick to get the right answer.

Planck’s trick was to imagine that all objects contained fictitious “oscillators” that emit light only at certain frequencies. Then he said that the amount of energy (
E
) associated with each oscillator was related to the frequency of the oscillation (
f
) by a simple formula:

E = hf

where
h
is a constant. When he first made this odd assumption, Planck thought he would use it just to set up the problem, and then use a common mathematical technique to get rid of the imaginary oscillators and this extra constant
h
. Much to his surprise, though, he found that his results made sense only if he kept the oscillators around—if
h
had a very small but nonzero value.