# Waves and Their Properties

## Wave Interaction: Superposition and Interference

### Section Key Terms

### Superposition of Waves

Most waves do not look very simple. They look more like the waves in , rather than the simple water wave considered in the previous sections, which has a perfect sinusoidal shape.
Most waves appear complex because they result from two or more simple waves that combine as they come together at the same place at the same time—a phenomenon called superposition.
Waves superimpose by adding their disturbances; each disturbance corresponds to a force, and all the forces add. If the disturbances are along the same line, then the resulting wave is a simple addition of the disturbances of the individual waves, that is, their amplitudes add.

### Wave Interference

The two special cases of superposition that produce the simplest results are pure constructive interference and pure destructive interference.
Pure constructive interference occurs when two identical waves arrive at the same point exactly in phase. When waves are exactly in phase, the crests of the two waves are precisely aligned, as are the troughs. Refer to . Because the disturbances add, the pure constructive interference of two waves with the same amplitude produces a wave that has twice the amplitude of the two individual waves, but has the same wavelength.
 shows two identical waves that arrive exactly out of phase—that is, precisely aligned crest to trough—producing pure destructive interference. Because the disturbances are in opposite directions for this superposition, the resulting amplitude is zero for pure destructive interference; that is, the waves completely cancel out each other.
While pure constructive interference and pure destructive interference can occur, they are not very common because they require precisely aligned identical waves. The superposition of most waves that we see in nature produces a combination of constructive and destructive interferences.
Waves that are not results of pure constructive or destructive interference can vary from place to place and time to time. The sound from a stereo, for example, can be loud in one spot and soft in another. The varying loudness means that the sound waves add partially constructively and partially destructively at different locations. A stereo has at least two speakers that create sound waves, and waves can reflect from walls. All these waves superimpose.
An example of sounds that vary over time from constructive to destructive is found in the combined whine of jet engines heard by a stationary passenger. The volume of the combined sound can fluctuate up and down as the sound from the two engines varies in time from constructive to destructive.
The two previous examples considered waves that are similar—both stereo speakers generate sound waves with the same amplitude and wavelength, as do the jet engines. But what happens when two waves that are not similar, that is, having different amplitudes and wavelengths, are superimposed? An example of the superposition of two dissimilar waves is shown in . Here again, the disturbances add and subtract, but they produce an even more complicated-looking wave. The resultant wave from the combined disturbances of two dissimilar waves looks much different than the idealized sinusoidal shape of a periodic wave.

### Standing Waves

Sometimes waves do not seem to move and they appear to just stand in place, vibrating. Such waves are called standing waves and are formed by the superposition of two or more waves moving in opposite directions. The waves move through each other with their disturbances adding as they go by. If the two waves have the same amplitude and wavelength, then they alternate between constructive and destructive interference. Standing waves created by the superposition of two identical waves moving in opposite directions are illustrated in .
As an example, standing waves can be seen on the surface of a glass of milk in a refrigerator. The vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. The two waves that produce standing waves may be due to the reflections from the side of the glass.
Earthquakes can create standing waves and cause constructive and destructive interferences. As the earthquake waves travel along the surface of Earth and reflect off denser rocks, constructive interference occurs at certain points. As a result, areas closer to the epicenter are not damaged while areas farther from the epicenter are damaged.
Standing waves are also found on the strings of musical instruments and are due to reflections of waves from the ends of the string.  and  show three standing waves that can be created on a string that is fixed at both ends. When the wave reaches the fixed end, it has nowhere else to go but back where it came from, causing the reflection. The nodes are the points where the string does not move; more generally, the nodes are the points where the wave disturbance is zero in a standing wave. The fixed ends of strings must be nodes, too, because the string cannot move there.
The antinode is the location of maximum amplitude in standing waves. The standing waves on a string have a frequency that is related to the propagation speed
   of the disturbance on the string. The wavelength
   is determined by the distance between the points where the string is fixed in place.

### Reflection and Refraction of Waves

As we saw in the case of standing waves on the strings of a musical instrument, reflection is the change in direction of a wave when it bounces off a barrier, such as a fixed end. When the wave hits the fixed end, it changes direction, returning to its source. As it is reflected, the wave experiences an inversion, which means that it flips vertically. If a wave hits the fixed end with a crest, it will return as a trough, and vice versa (Henderson 2015). Refer to .
Rather than encountering a fixed end or barrier, waves sometimes pass from one medium into another, for instance, from air into water. Different types of media have different properties, such as density or depth, that affect how a wave travels through them. At the boundary between media, waves experience refraction—they change their path of propagation. As the wave bends, it also changes its speed and wavelength upon entering the new medium. Refer to .
For example, water waves traveling from the deep end to the shallow end of a swimming pool experience refraction. They bend in a path closer to perpendicular to the surface of the water, propagate slower, and decrease in wavelength as they enter shallower water.

### Check Your Understanding

### Section Summary

1. Superposition is the combination of two waves at the same location.
2. Constructive interference occurs when two identical waves are superimposed exactly in phase.
3. Destructive interference occurs when two identical waves are superimposed exactly out of phase.
4. A standing wave is a wave produced by the superposition of two waves. It varies in amplitude but does not propagate.
5. The nodes are the points where there is no motion in standing waves.
6. An antinode is the location of maximum amplitude of a standing wave.
7. Reflection causes a wave to change direction.
8. Inversion occurs when a wave reflects from a fixed end.
9. Refraction causes a wave’s path to bend and occurs when a wave passes from one medium into another medium with a different density.

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# Sound

## Introduction

If a tree falls in a forest (see ) and no one is there to hear it, does it make a sound? The answer to this old philosophical question depends on how you define sound. If sound only exists when someone is around to perceive it, then the falling tree produced no sound. However, in physics, we know that colliding objects can disturb the air, water or other matter surrounding them. As a result of the collision, the surrounding particles of matter began vibrating in a wave-like fashion. This is a sound wave. Consequently, if a tree collided with another object in space, no one would hear it, because no sound would be produced. This is because, in space, there is no air, water or other matter to be disturbed and produce sound waves. In this chapter, we’ll learn more about the wave properties of sound, and explore hearing, as well as some special uses for sound.


# Sound

## Speed of Sound, Frequency, and Wavelength

### Section Key Terms

### Properties of Sound Waves

Sound is a wave. More specifically, sound is defined to be a disturbance of matter that is transmitted from its source outward. A disturbance is anything that is moved from its state of equilibrium. Some sound waves can be characterized as periodic waves, which means that the atoms that make up the matter experience simple harmonic motion.
A vibrating string produces a sound wave as illustrated in , , and . As the string oscillates back and forth, part of the string’s energy goes into compressing and expanding the surrounding air. This creates slightly higher and lower pressures. The higher pressure... regions are compressions, and the low pressure regions are rarefactions. The pressure disturbance moves through the air as longitudinal waves with the same frequency as the string. Some of the energy is lost in the form of thermal energy transferred to the air. You may recall from the chapter on waves that areas of compression and rarefaction in longitudinal waves (such as sound) are analogous to crests and troughs in transverse waves.
The amplitude of a sound wave decreases with distance from its source, because the energy of the wave is spread over a larger and larger area. But some of the energy is also absorbed by objects, such as the eardrum in , and some of the energy is converted to thermal energy in the air.  shows a graph of gauge pressure versus distance from the vibrating string. From this figure, you can see that the compression of a longitudinal wave is analogous to the peak of a transverse wave, and the rarefaction of a longitudinal wave is analogous to the trough of a transverse wave. Just as a transverse wave alternates between peaks and troughs, a longitudinal wave alternates between compression and rarefaction.

### The Speed of Sound

The speed of sound varies greatly depending upon the medium it is traveling through. The speed of sound in a medium is determined by a combination of the medium’s rigidity (or compressibility in gases) and its density. The more rigid (or less compressible) the medium, the faster the speed of sound. The greater the density of a medium, the slower the speed of sound. The speed of sound in air is low, because air is compressible. Because liquids and solids are relatively rigid and very difficult to compress, the speed of sound in such media is generally greater than in gases.  shows the speed of sound in various media. Since temperature affects density, the speed of sound varies with the temperature of the medium through which it’s traveling to some extent, especially for gases.

### The Relationship Between the Speed of Sound and the Frequency and Wavelength of a Sound Wave

Sound, like all waves, travels at certain speeds through different media and has the properties of frequency and wavelength. Sound travels much slower than light—you can observe this while watching a fireworks display (see ), since the flash of an explosion is seen before its sound is heard.
The relationship between the speed of sound, its frequency, and wavelength is the same as for all waves:
where v is the speed of sound (in units of m/s), f is its frequency (in units of hertz), and
   is its wavelength (in units of meters). Recall that wavelength is defined as the distance between adjacent identical parts of a wave. The wavelength of a sound, therefore, is the distance between adjacent identical parts of a sound wave. Just as the distance between adjacent crests in a transverse wave is one wavelength, the distance between adjacent compressions in a sound wave is also one wavelength, as shown in . The frequency of a sound wave is the same as that of the source. For example, a tuning fork vibrating at a given frequency would produce sound waves that oscillate at that same frequency. The frequency of a sound is the number of waves that pass a point per unit time.
One of the more important properties of sound is that its speed is nearly independent of frequency. If this were not the case, and high-frequency sounds traveled faster, for example, then the farther you were from a band in a football stadium, the more the sound from the low-pitch instruments would lag behind the high-pitch ones. But the music from all instruments arrives in cadence independent of distance, and so all frequencies must travel at nearly the same speed.
Recall that
 , and in a given medium under fixed temperature and humidity, v is constant. Therefore, the relationship between f and
   is inverse: The higher the frequency, the shorter the wavelength of a sound wave.
The speed of sound can change when sound travels from one medium to another. However, the frequency usually remains the same because it is like a driven oscillation and maintains the frequency of the original source. If v changes and f remains the same, then the wavelength
   must change. Since
 , the higher the speed of a sound, the greater its wavelength for a given frequency.

### Practice Problems

### Check Your Understanding

### Section Summary

1. Sound is one type of wave.
2. Sound is a disturbance of matter that is transmitted from its source outward in the form of longitudinal waves.
3. The relationship of the speed of sound v, its frequency f, and its wavelength
   is given by
 , which is the same relationship given for all waves.
4. The speed of sound depends upon the medium through which the sound wave is travelling.
5. In a given medium at a specific temperature (or density), the speed of sound v is the same for all frequencies and wavelengths.

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# Sound

## Sound Intensity and Sound Level

### Section Key Terms

### Amplitude, Loudness and Energy of a Sound Wave

In a quiet forest, you can sometimes hear a single leaf fall to the ground. But in a traffic jam filled with honking cars, you may have to shout just so the person next to you can hear .The loudness of a sound is related to how energetically its source is vibrating. In cartoons showing a screaming person, the cartoonist often shows an open mouth with a vibrating uvula (the hanging tissue at the back of the mouth) to represent a loud sound coming from the throat.  shows such a cartoon depiction of a bird loudly expressing its opinion.
A useful quantity for describing the loudness of sounds is called sound intensity. In general, the intensity of a wave is the power per unit area carried by the wave. Power is the rate at which energy is transferred by the wave. In equation form, intensity I is
where P is the power through an area A. The SI unit for I is W/m2. The intensity of a sound depends upon its pressure amplitude. The relationship between the intensity of a sound wave and its pressure amplitude (or pressure variation Δp) is
where ρ is the density of the material in which the sound wave travels, in units of kg/m3, and v is the speed of sound in the medium, in units of m/s. Pressure amplitude has units of pascals (Pa) or N/m2. Note that Δp is half the difference between the maximum and minimum pressure in the sound wave.
We can see from the equation that the intensity of a sound is proportional to its amplitude squared. The pressure variation is proportional to the amplitude of the oscillation, and so I varies as (Δp)2. This relationship is consistent with the fact that the sound wave is produced by some vibration; the greater its pressure amplitude, the more the air is compressed during the vibration. Because the power of a sound wave is the rate at which energy is transferred, the energy of a sound wave is also proportional to its amplitude squared.

### The Decibel Scale

You may have noticed that when people talk about the loudness of a sound, they describe it in units of decibels rather than watts per meter squared. While sound intensity (in W/m2) is the SI unit, the sound intensity level in decibels (dB) is more relevant for how humans perceive sounds. The way our ears perceive sound can be more accurately described by the logarithm of the intensity of a sound rather than the intensity of a sound directly. The sound intensity level β is defined to be
where I is sound intensity in watts per meter squared, and I0 = 10–12 W/m2 is a reference intensity. I0 is chosen as the reference point because it is the lowest intensity of sound a person with normal hearing can perceive. The decibel level of a sound having an intensity of 10–12 W/m2 is β = 0 dB, because log10 1 = 0. That is, the threshold of human hearing is 0 decibels.
Each factor of 10 in intensity corresponds to 10 dB. For example, a 90 dB sound compared with a 60 dB sound is 30 dB greater, or three factors of 10 (that is, 103 times) as intense. Another example is that if one sound is 107 as intense as another, it is 70 dB higher.
Since β is defined in terms of a ratio, it is unit-less. The unit called decibel (dB) is used to indicate that this ratio is multiplied by 10. The sound intensity level is not the same as sound intensity—it tells you the level of the sound relative to a reference intensity rather than the actual intensity.

### Solving Sound Wave Intensity Problems

### Practice Problems

### Hearing and Voice

People create sounds by pushing air up through their lungs and through elastic folds in the throat called vocal cords. These folds open and close rhythmically, creating a pressure buildup. As air travels up and past the vocal cords, it causes them to vibrate. This vibration escapes the mouth along with puffs of air as sound. A voice changes in pitch when the muscles of the larynx relax or tighten, changing the tension on the vocal chords. A voice becomes louder when air flow from the lungs increases, making the amplitude of the sound pressure wave greater.
Hearing is the perception of sound. It can give us plenty of information—such as pitch, loudness, and direction. Humans can normally hear frequencies ranging from approximately 20 to 20,000 Hz. Other animals have hearing ranges different from that of humans. Dogs can hear sounds as high as 45,000 Hz, whereas bats and dolphins can hear up to 110,000 Hz sounds. You may have noticed that dogs respond to the sound of a dog whistle which produces sound out of the range of human hearing.
Sounds below 20 Hz are called infrasound, whereas those above 20,000 Hz are ultrasound. The perception of frequency is called pitch, and the perception of intensity is called loudness.
The way we hear involves some interesting physics. The sound wave that hits our ear is a pressure wave. The ear converts sound waves into electrical nerve impulses, similar to a microphone.
 shows the anatomy of the ear with its division into three parts: the outer ear or ear canal; the middle ear, which runs from the eardrum to the cochlea; and the inner ear, which is the cochlea itself. The body part normally referred to as the ear is technically called the pinna.
The outer ear, or ear canal, carries sound to the eardrum protected inside of the ear. The middle ear converts sound into mechanical vibrations and applies these vibrations to the cochlea. The lever system of the middle ear takes the force exerted on the eardrum by sound pressure variations, amplifies it and transmits it to the inner ear via the oval window. Two muscles in the middle ear protect the inner ear from very intense sounds. They react to intense sound in a few milliseconds and reduce the force transmitted to the cochlea. This protective reaction can also be triggered by your own voice, so that humming during a fireworks display, for example, can reduce noise damage.
 shows the middle and inner ear in greater detail. As the middle ear bones vibrate, they vibrate the cochlea, which contains fluid. This creates pressure waves in the fluid that cause the tectorial membrane to vibrate. The motion of the tectorial membrane stimulates tiny cilia on specialized cells called hair cells. These hair cells, and their attached neurons, transform the motion of the tectorial membrane into electrical signals that are sent to the brain.
The tectorial membrane vibrates at different positions based on the frequency of the incoming sound. This allows us to detect the pitch of sound. Additional processing in the brain also allows us to determine which direction the sound is coming from (based on comparison of the sound’s arrival time and intensity between our two ears).

### Check Your Understanding

### Section Summary

1. The intensity of a sound is proportional to its amplitude squared.
2. The energy of a sound wave is also proportional to its amplitude squared.
3. Sound intensity level in decibels (dB) is more relevant for how humans perceive sounds than sound intensity (in W/m2), even though sound intensity is the SI unit.
4. Sound intensity level is not the same as sound intensity—it tells you the level of the sound relative to a reference intensity rather than the actual intensity.
5. Hearing is the perception of sound and involves that transformation of sound waves into vibrations of parts within the ear. These vibrations are then transformed into neural signals that are interpreted by the brain.
6. People create sounds by pushing air up through their lungs and through elastic folds in the throat called vocal cords.

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### References

Nave, R. Vocal sound production—HyperPhysics. Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/music/voice.html


# Sound

## Doppler Effect and Sonic Booms

### Section Key Terms

### The Doppler Effect of Sound Waves

The Doppler effect is a change in the observed pitch of a sound, due to relative motion between the source and the observer. An example of the Doppler effect due to the motion of a source occurs when you are standing still, and the sound of a siren coming from an ambulance shifts from high-pitch to low-pitch as it passes by. The closer the ambulance is to you, the more sudden the shift. The faster the ambulance moves, the greater the shift. We also hear this shift in frequency for passing race cars, airplanes, and trains. An example of the Doppler effect with a stationary source and moving observer is if you ride a train past a stationary warning bell, you will hear the bell’s frequency shift from high to low as you pass by.
What causes the Doppler effect? Let’s compare three different scenarios: Sound waves emitted by a stationary source (), sound waves emitted by a moving source (), and sound waves emitted by a stationary source but heard by moving observers (). In each case, the sound spreads out from the point where it was emitted.
If the source and observers are stationary, then observers on either side see the same wavelength and frequency as emitted by the source. But if the source is moving and continues to emit sound as it travels, then the air compressions (crests) become closer together in the direction in which it’s traveling and farther apart in the direction it’s traveling away from. Therefore, the wavelength is shorter in the direction the source is moving (on the right in ), and longer in the opposite direction (on the left in ).
Finally, if the observers move, as in , the frequency at which they receive the compressions changes. The observer moving toward the source receives them at a higher frequency (and therefore shorter wavelength), and the person moving away from the source receives them at a lower frequency (and therefore longer wavelength).
We know that wavelength and frequency are related by
   where v is the fixed speed of sound. The sound moves in a medium and has the same speed v in that medium whether the source is moving or not. Therefore, f multiplied by
   is a constant. Because the observer on the right in  receives a shorter wavelength, the frequency she perceives must be higher. Similarly, the observer on the left receives a longer wavelength and therefore perceives a lower frequency.
The same thing happens in . A higher frequency is perceived by the observer moving toward the source, and a lower frequency is perceived by an observer moving away from the source. In general, then, relative motion of source and observer toward one another increases the perceived frequency. Relative motion apart decreases the perceived frequency. The greater the relative speed is, the greater the effect.
For a stationary observer and a moving source of sound, the frequency (fobs) of sound perceived by the observer is
where fs is the frequency of sound from a source, vs is the speed of the source along a line joining the source and observer, and vw is the speed of sound. The minus sign is used for motion toward the observer and the plus sign for motion away from the observer.
Note that the greater the speed of the source, the greater the Doppler effect. Similarly, for a stationary source and moving observer, the frequency perceived by the observer fobs is given by
where vobs is the speed of the observer along a line joining the source and observer. Here the plus sign is for motion toward the source, and the minus sign is for motion away from the source.

### Sonic Booms

What happens to the sound produced by a moving source, such as a jet airplane, that approaches or even exceeds the speed of sound? Suppose a jet airplane is coming nearly straight at you, emitting a sound of frequency fs. The greater the plane’s speed, vs, the greater the Doppler shift and the greater the value of fobs. Now, as vs approaches the speed of sound, vw, fobs approaches infinity, because the denominator in
   approaches zero.
This result means that at the speed of sound, in front of the source, each wave is superimposed on the previous one because the source moves forward at the speed of sound. The observer gets them all at the same instant, and so the frequency is theoretically infinite. If the source exceeds the speed of sound, no sound is received by the observer until the source has passed, so that the sounds from the source when it was approaching are stacked up with those from it when receding, creating a sonic boom. A sonic boom is a constructive interference of sound created by an object moving faster than sound.
An aircraft creates two sonic booms, one from its nose and one from its tail (see ). During television coverage of space shuttle landings, two distinct booms could often be heard. These were separated by exactly the time it would take the shuttle to pass by a point. Observers on the ground often do not observe the aircraft creating the sonic boom, because it has passed by before the shock wave reaches them. If the aircraft flies close by at low altitude, pressures in the sonic boom can be destructive enough to break windows. Because of this, supersonic flights are banned over populated areas of the United States.

### Solving Problems Using the Doppler Shift Formula

### Practice Problems

### Check Your Understanding

### Section Summary

1. The Doppler effect is a shift in the observed frequency of a sound due to motion of either the source or the observer.
2. The observed frequency is greater than the actual source’s frequency when the source and the observer are moving closer together, either by the source moving toward the observer or the observer moving toward the source.
3. A sonic boom is constructive interference of sound created by an object moving faster than sound.

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# Sound

## Sound Interference and Resonance

### Section Key Terms

### Resonance and Beats

Sit in front of a piano sometime and sing a loud brief note at it while pushing down on the sustain pedal. It will sing the same note back at you—the strings that have the same frequencies as your voice, are resonating in response to the forces from the sound waves that you sent to them. This is a good example of the fact that objects—in this case, piano strings—can be forced to oscillate but oscillate best at their natural frequency.
A driving force (such as your voice in the example) puts energy into a system at a certain frequency, which is not necessarily the same as the natural frequency of the system. Over time the energy dissipates, and the amplitude gradually reduces to zero- this is called damping. The natural frequency is the frequency at which a system would oscillate if there were no driving and no damping force. The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance, and a system being driven at its natural frequency is said to resonate.
Most of us have played with toys where an object bobs up and down on an elastic band, something like the paddle ball suspended from a finger in . At first you hold your finger steady, and the ball bounces up and down with a small amount of damping. If you move your finger up and down slowly, the ball will follow along without bouncing much on its own. As you increase the frequency at which you move your finger up and down, the ball will respond by oscillating with increasing amplitude. When you drive the ball at its natural frequency, the ball’s oscillations increase in amplitude with each oscillation for as long as you drive it. As the driving frequency gets progressively higher than the resonant or natural frequency, the amplitude of the oscillations becomes smaller, until the oscillations nearly disappear and your finger simply moves up and down with little effect on the ball.
Another example is that when you tune a radio, you adjust its resonant frequency so that it oscillates only at the desired station’s broadcast (driving) frequency. Also, a child on a swing is driven (pushed) by a parent at the swing’s natural frequency to reach the maximum amplitude (height). In all of these cases, the efficiency of energy transfer from the driving force into the oscillator is best at resonance.
All sound resonances are due to constructive and destructive interference. Only the resonant frequencies interfere constructively to form standing waves, while others interfere destructively and are absent. From the toot made by blowing over a bottle to the recognizability of a great singer’s voice, resonance and standing waves play a vital role in sound.
Interference happens to all types of waves, including sound waves. In fact, one way to support that something is a wave is to observe interference effects.  shows a set of headphones that employs a clever use of sound interference to cancel noise. To get destructive interference, a fast electronic analysis is performed, and a second sound is introduced with its maxima and minima exactly reversed from the incoming noise.
In addition to resonance, superposition of waves can also create beats. Beats are produced by the superposition of two waves with slightly different frequencies but the same amplitude. The waves alternate in time between constructive interference and destructive interference, giving the resultant wave an amplitude that varies over time. (See the resultant wave in ).
This wave fluctuates in amplitude, or beats, with a frequency called the beat frequency. The equation for beat frequency is
where f1 and f2 are the frequencies of the two original waves. If the two frequencies of sound waves are similar, then what we hear is an average frequency that gets louder and softer at the beat frequency.

### Fundamental Frequency and Harmonics

Suppose we hold a tuning fork near the end of a tube that is closed at the other end, as shown in , , and . If the tuning fork has just the right frequency, the air column in the tube resonates loudly, but at most frequencies it vibrates very little. This means that the air column has only certain natural frequencies. The figures show how a resonance at the lowest of these natural frequencies is formed. A disturbance travels down the tube at the speed of sound and bounces off the closed end. If the tube is just the right length, the reflected sound arrives back at the tuning fork exactly half a cycle later, and it interferes constructively with the continuing sound produced by the tuning fork. The incoming and reflected sounds form a standing wave in the tube as shown.
The standing wave formed in the tube has its maximum air displacement (an antinode) at the open end, and no displacement (a node) at the closed end. Recall from the last chapter on waves that motion is unconstrained at the antinode, and halted at the node. The distance from a node to an antinode is one-fourth of a wavelength, and this equals the length of the tube; therefore,
  . This same resonance can be produced by a vibration introduced at or near the closed end of the tube, as shown in .
Since maximum air displacements are possible at the open end and none at the closed end, there are other, shorter wavelengths that can resonate in the tube see ). Here the standing wave has three-fourths of its wavelength in the tube, or
 , so that
  . There is a whole series of shorter-wavelength and higher-frequency sounds that resonate in the tube.
We use specific terms for the resonances in any system. The lowest resonant frequency is called the fundamental, while all higher resonant frequencies are called overtones. All resonant frequencies are multiples of the fundamental, and are called harmonics. The fundamental is the first harmonic, the first overtone is the second harmonic, and so on.  shows the fundamental and the first three overtones (the first four harmonics) in a tube closed at one end.
The fundamental and overtones can be present at the same time in a variety of combinations. For example, the note middle C on a trumpet sounds very different from middle C on a clarinet, even though both instruments are basically modified versions of a tube closed at one end. The fundamental frequency is the same (and usually the most intense), but the overtones and their mix of intensities are different. This mix is what gives musical instruments (and human voices) their distinctive characteristics, whether they have air columns, strings, or drumheads. In fact, much of our speech is determined by shaping the cavity formed by the throat and mouth and positioning the tongue to adjust the fundamental and combination of overtones.

### Open-Pipe and Closed-Pipe Resonators

The resonant frequencies of a tube closed at one end (known as a closed-pipe resonator) are

where f1 is the fundamental, f3 is the first overtone, and so on. Note that the resonant frequencies depend on the speed of sound v and on the length of the tube L.
Another type of tube is one that is open at both ends (known as an open-pipe resonator). Examples are some organ pipes, flutes, and oboes. The air columns in tubes open at both ends have maximum air displacements at both ends. (See ). Standing waves form as shown.
The resonant frequencies of an open-pipe resonator are

where f1 is the fundamental, f2 is the first overtone, f3 is the second overtone, and so on. Note that a tube open at both ends has a fundamental frequency twice what it would have if closed at one end. It also has a different spectrum of overtones than a tube closed at one end. So if you had two tubes with the same fundamental frequency but one was open at both ends and the other was closed at one end, they would sound different when played because they have different overtones.
Middle C, for example, would sound richer played on an open tube since it has more overtones. An open-pipe resonator has more overtones than a closed-pipe resonator because it has even multiples of the fundamental as well as odd, whereas a closed tube has only odd multiples.
In this section we have covered resonance and standing waves for wind instruments, but vibrating strings on stringed instruments also resonate and have fundamentals and overtones similar to those for wind instruments.

### Solving Problems Involving Harmonic Series and Beat Frequency

### Practice Problems

### Check Your Understanding

### Section Summary

1. A system’s natural frequency is the frequency at which the system will oscillate if not affected by driving or damping forces.
2. A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
3. Beats occur when waves of slightly different frequencies are superimposed.
4. In air columns, the lowest-frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively, they are called harmonics.
5. The resonant frequencies of a tube closed at one end are
 , where f is the fundamental and L is the length of the tube.
6. The resonant frequencies of a tube open at both ends are

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# Light

## Introduction

The beauty of a coral reef, the warm radiance of sunshine, the sting of sunburn, the X-ray revealing a broken bone, even microwave popcorn—all are brought to us by electromagnetic waves. The list of the various types of electromagnetic waves, ranging from radio transmission waves to nuclear gamma-ray (γ-ray) emissions, is interesting in itself.
Even more intriguing is that all of these different phenomena are manifestations of the same thing—electromagnetic waves (see ). What are electromagnetic waves? How are they created, and how do they travel? How can we understand their widely varying properties? What is the relationship between electric and magnetic effects? These and other questions will be explored.


# Light

## The Electromagnetic Spectrum

### Section Key Terms

### The Electromagnetic Spectrum

We generally take light for granted, but it is a truly amazing and mysterious form of energy. Think about it: Light travels to Earth across millions of kilometers of empty space. When it reaches us, it interacts with matter in various ways to generate almost all the energy needed to support life, provide heat, and cause weather patterns. Light is a form of electromagnetic radiation (EMR). The term light usually refers to visible light, but this is not the only form of EMR. As we will see, visible light occupies a narrow band in a broad range of types of electromagnetic radiation.
Electromagnetic radiation is generated by a moving electric charge, that is, by an electric current. As you will see when you study electricity, an electric current generates both an electric field, E, and a magnetic field, B. These fields are perpendicular to each other. When the moving charge oscillates, as in an alternating current, an EM wave is propagated.  shows how an electromagnetic wave moves away from the source—indicated by the ~ symbol.
From your study of sound waves, recall these features that apply to all types of waves:
1. Wavelength—The distance between two wave crests or two wave troughs, expressed in various metric measures of distance
2. Frequency—The number of wave crests that pass a point per second, expressed in hertz (Hz or s–1)
3. Amplitude: The height of the crest above the null point
As mentioned, electromagnetic radiation takes several forms. These forms are characterized by a range of frequencies. Because frequency is inversely proportional to wavelength, any form of EMR can also be represented by its range of wavelengths.  shows the frequency and wavelength ranges of various types of EMR. With how many of these types are you familiar?
Take a few minutes to study the positions of the various types of radiation on the EM spectrum, above. The narrow band that is visible light extends from lower-frequency red light to higher-frequency violet light. Frequencies just below the visible are called  (below red) and those just above are ultraviolet (beyond violet). Radio waves, which overlap with the frequencies used for media broadcasts of TV and radio signals, occupy frequencies even lower than infrared (IR). The microwave radiation that you see on the diagram is the same radiation that is used in a microwave oven. What we feel as radiant heat is also a form of low-frequency EMR. The high-frequency radiation to the right of ultraviolet (UV) includes X-rays and gamma (γ) rays.

### Characteristics of Electromagnetic Radiation

All the EM waves mentioned above are basically the same form of radiation. They can all travel across empty space, and they all travel at the speed of light in a vacuum. The basic difference between types of radiation is their differing frequencies. Each frequency has an associated wavelength. As frequency increases across the spectrum, wavelength decreases. Energy also increases with frequency. Because of this, higher frequencies penetrate matter more readily. Some of the properties and uses of the various EM spectrum bands are listed in .
The narrow band of visible light is a combination of the colors of the rainbow.  shows the section of the EM spectrum that includes visible light. The frequencies corresponding to these wavelengths are
  at the red end to
  at the violet end. This is a very narrow range, considering that the EM spectrum spans about 20 orders of magnitude.
As a child, you probably learned the color wheel, shown on the left in . It helps if you know what color results when you mix different colors of paint together. Mixing two of the primary  colors—magenta, yellow, or cyan—together results in a secondary color. For example, mixing cyan and yellow makes green. This is called subtractive color mixing. Mixing different colors of light together is quite different. The diagram on the right shows additive color mixing. In this case, the primary colors are red, green, and blue, and the secondary colors are cyan, magenta, and yellow. Mixing pigments and mixing light are different because materials absorb light by a different set of rules than does the perception of light by the eye. Notice that, when all colors are subtracted, the result is no color, or black. When all colors are added, the result is white light. We see the reverse of this when white sunlight is separated into the visible spectrum by a prism or by raindrops when a rainbow appears in the sky.
Humans have found uses for every part of the electromagnetic spectrum. We will take a look at the uses of each range of frequencies, beginning with visible light. Most of our uses of visible light are obvious; without it our interaction with our surroundings would be much different. We might forget that nearly all of our food depends on the photosynthesis process in plants, and that the energy for this process comes from the visible part of the spectrum. Without photosynthesis, we would also have almost no oxygen in the atmosphere.
The low-frequency, infrared region of the spectrum has many applications in media broadcasting. Television, radio, cell phone, and remote-control devices all broadcast and/or receive signals with these wavelengths. AM and FM radio signals are both low-frequency radiation. They are in different regions of the spectrum, but that is not their basic difference. AM and FM are abbreviations for  and . Information in AM signals has the form of changes in amplitude of the radio waves; information in FM signals has the form of changes in wave frequency.
Another application of long-wavelength radiation is found in microwave ovens. These appliances cook or warm food by irradiating it with EM radiation in the microwave frequency range. Most kitchen microwaves use a frequency of
 Hz. These waves have the right amount of energy to cause polar molecules, such as water, to rotate faster. Polar molecules are those that have a partial charge separation. The rotational energy of these molecules is given up to surrounding matter as heat. The first microwave ovens were called Radaranges because they were based on radar technology  developed during World War II.
Radar uses radiation with wavelengths similar to those of microwaves to detect the location and speed of distant objects, such as airplanes, weather formations, and motor vehicles. Radar information is obtained by receiving and analyzing the echoes of microwaves reflected by an object. The speed of the object can be measured using the Doppler shift of the returning waves. This is the same effect you learned about when you studied sound waves. Like sound waves, EM waves are shifted to higher frequencies by an object moving toward an observer, and to lower frequencies by an object moving away from the observer. Astronomers use this same Doppler effect to measure the speed at which distant galaxies are moving away from us. In this case, the shift in frequency is called the red shift, because visible frequencies are shifted toward the lower-frequency, red end of the spectrum.
Exposure to any radiation with frequencies greater than those of visible light carries some health hazards. All types of radiation in this range are known to cause cell damage. The danger is related to the high energy and penetrating ability of these EM waves. The likelihood of being harmed by any of this radiation depends largely on the amount of exposure. Most people try to reduce exposure to UV radiation from sunlight by using sunscreen and protective clothing. Physicians still use X-rays to diagnose medical problems, but the intensity of the radiation used is extremely low.  shows an X-ray image of a patient’s chest cavity.
One medical-imaging technique that involves no danger of exposure is magnetic resonance imaging (MRI). MRI is an important imaging and research tool in medicine, producing highly detailed two- and three-dimensional images. Radio waves are broadcast, absorbed, and reemitted in a resonance process that is sensitive to the density of nuclei, usually hydrogen nuclei—protons.

### Check Your Understanding

### Section Summary

1. The electromagnetic spectrum is made up of a broad range of frequencies of electromagnetic radiation.
2. All frequencies of EM radiation travel at the same speed in a vacuum and consist of an electric field and a magnetic field. The types of EM radiation have different frequencies and wavelengths, and different energies and penetrating ability.

### Concept Items

### Critical Thinking

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Light

## The Behavior of Electromagnetic Radiation

### Section Key Terms

### Types of Electromagnetic Wave Behavior

In a vacuum, all electromagnetic radiation travels at the same incredible speed of 3.00 × 108 m/s, which is equal to 671 million miles per hour. This is one of the fundamental physical constants. It is referred to as the speed of light and is given the symbol c. The space between celestial bodies is a near vacuum, so the light we see from the Sun, stars, and other planets has traveled here at the speed of light. Keep in mind that all EM radiation travels at this speed. All the different wavelengths of radiation that leave the Sun make the trip to Earth in the same amount of time. That trip takes 8.3 minutes. Light from the nearest star, besides the Sun, takes 4.2 years to reach Earth, and light from the nearest galaxy—a dwarf galaxy that orbits the Milky Way—travels 25,000 years on its way to Earth. You can see why we call very long distances astronomical.
When light travels through a physical medium, its speed is always less than the speed of light. For example, light travels in water at three-fourths the value of c. In air, light has a speed that is just slightly slower than in empty space: 99.97 percent of c. Diamond slows light down to just 41 percent of c. When light changes speeds at a boundary between media, it also changes direction. The greater the difference in speeds, the more the path of light bends. In other chapters, we look at this bending, called refraction, in greater detail. We introduce refraction here to help explain a phenomenon called thin-film interference.
Have you ever wondered about the rainbow colors you often see on soap bubbles, oil slicks, and compact discs? This occurs when light is both refracted by and reflected from a very thin film. The diagram shows the path of light through such a thin film. The symbols n1, n2, and n3 indicate that light travels at different speeds in each of the three materials. Learn more about this topic in the chapter on diffraction and interference.
 shows the result of thin film interference on the surface of soap bubbles. Because ray 2 travels a greater distance, the two rays become out of phase. That is, the crests of the two emerging waves are no longer moving together. This causes interference, which reinforces the intensity of the wavelengths of light that create the bands of color. The color bands are separated because each color has a different wavelength. Also, the thickness of the film is not uniform, and different thicknesses cause colors of different wavelengths to interfere in different places. Note that the film must be very, very thin—somewhere in the vicinity of the wavelengths of visible light.
You have probably experienced how polarized sunglasses reduce glare from the surface of water or snow. The effect is caused by the wave nature of light. Looking back at , we see that the electric field moves in only one direction perpendicular to the direction of propagation. Light from most sources vibrates in all directions perpendicular to propagation. Light with an electric field that vibrates in only one direction is called polarized. A diagram of polarized light would look like .
Polarized glasses are an example of a polarizing filter. These glasses absorb most of the horizontal light waves and transmit the vertical waves. This cuts down glare, which is caused by horizontal waves.  shows how waves traveling along a rope can be used as a model of how a polarizing filter works. The oscillations in one rope are in a vertical plane and are said to be vertically polarized. Those in the other rope are in a horizontal plane and are horizontally polarized. If a vertical slit is placed on the first rope, the waves pass through. However, a vertical slit blocks the horizontally polarized waves. For EM waves, the direction of the electric field oscillation is analogous to the disturbances on the ropes.
Light can also be polarized by reflection. Most of the light reflected from water, glass, or any highly reflective surface is polarized horizontally.  shows the effect of a polarizing lens on light reflected from the surface of water.

### Quantitative Treatment of Electromagnetic Waves

We can use the speed of light, c, to carry out several simple but interesting calculations. If we know the distance to a celestial object, we can calculate how long it takes its light to reach us. Of course, we can also make the reverse calculation if we know the time it takes for the light to travel to us. For an object at a very great distance from Earth, it takes many years for its light to reach us. This means that we are looking at the object as it existed in the distant past. The object may, in fact, no longer exist. Very large distances in the universe are measured in light years. One light year is the distance that light travels in one year, which is

 kilometers or

 miles (…and 1012 is a trillion!).
A useful equation involving c is
where f is frequency in Hz, and
   is wavelength in meters.
The rate at which light is radiated from a source is called luminous flux, P, and it is measured in lumens (lm). Energy-saving light bulbs, which provide more luminous flux for a given use of electricity, are now available. One of these bulbs is called a compact fluorescent lamp; another is an LED (light-emitting diode) bulb. If you wanted to replace an old incandescent bulb with an energy saving bulb, you would want the new bulb to have the same brightness as the old one. To compare bulbs accurately, you would need to compare the lumens each one puts out. Comparing wattage—that is, the electric power used—would be misleading. Both wattage and lumens are stated on the packaging.
The luminous flux of a bulb might be 2,000 lm. That accounts for all the light radiated in all directions. However, what we really need to know is how much light falls on an object, such as a book, at a specific distance. The number of lumens per square meter is called illuminance, and is given in units of lux (lx). Picture a light bulb in the middle of a sphere with a 1-m radius. The total surface of the sphere equals 4πr2 m2. The illuminance then is given by
What happens if the radius of the sphere is increased 2 m? The illuminance is now only one-fourth as great, because the r2 term in the denominator is 4 instead of 1.  shows how illuminance decreases with the inverse square of the distance.

### Practice Problems

### Check Your Understanding

### Section Summary

1. EM radiation travels at different speeds in different media, produces colors on thin films, and can be polarized to oscillate in only one direction.
2. Calculations can be based on the relationship among the speed, frequency, and wavelength of light, and on the relationship among luminous flux, illuminance, and distance.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Performance Task

1. EYE SAFETY—Chemicals in this lab are poisonous if ingested. If chemicals are ingested, inform your teacher immediately.
2. FUMES—Certain chemicals or chemical reactions in this lab create a vapor that is harmful if inhaled. Follow your teacher's instructions for the use of fume hoods and other safety apparatus designed to prevent fume inhalation. Never smell or otherwise breath in any chemicals or vapors in the lab.
3. FLAMMABLE—Chemicals in this lab are highly flammable and can ignite, especially if exposed to a spark or open flame. Follow your teacher's instructions carefully on how to handle flammable chemicals. Do not expose any chemical to a flame or other heat source unless specifically instructed by your teacher.
4. HAND WASHING—Some materials may be hazardous if in extended contact with the skin. Be sure to wash your hands with soap after handling and disposing of these materials during the lab.
5. WASTE—Some things in this lab are hazardous and need to be disposed of properly. Follow your teacher's instructions for disposal of all items.
1. A large flat tray with raised sides, such as a baking tray
2. Small volumes of motor oil, lighter fluid or a penetrating oil of the type used to loosen rusty bolts, and cooking oil
3. Water
4. A camera
1. Thin-film interference causes colors to appear on the surface of a thin transparent layer. Do you expect to see a pattern to the colors?
2. How could you make a permanent record of your observations?
3. What data would you need to look up to help explain any patterns that you see?
4. What could explain colors failing to appear under some conditions?

### Test Prep Multiple Choice

### Short Answer

### Extended Response


# Mirrors and Lenses

## Introduction

In another moment Alice was through the glass, and had jumped lightly down into the Looking-glass room.
—Through the Looking Glass by Lewis Carol
Through the Looking Glass tells of the adventures of Alice after she steps from the real world, through a mirror, and into the virtual world. In this chapter we examine the optical meanings of real and virtual, as well as other concepts that make up the field of optics.
The light from this page or screen is formed into an image by the lens of your eyes, much as the lens of the camera that made the photograph at the beginning of this chapter. Mirrors, like lenses, can also form images, which in turn are captured by your eyes.
Optics is the branch of physics that deals with the behavior of visible light and other electromagnetic waves. For now, we concentrate on the propagation of light and its interaction with matter.
It is convenient to divide optics into two major parts based on the size of objects that light encounters. When light interacts with an object that is several times as large as the light’s wavelength, its observable behavior is similar to a ray;  it does not display its wave characteristics prominently. We call this part of optics geometric optics. This chapter focuses on situations for which geometric optics is suited.


# Mirrors and Lenses

## Reflection

### Section Key Terms

### Characteristics of Mirrors

There are three ways, as shown in , in which light can travel from a source to another location. It can come directly from the source through empty space, such as from the Sun to Earth. Light can travel to an object through various media, such as air and glass. Light can also arrive at an object after being reflected, such as by a mirror. In all these cases, light is modeled as traveling in a straight line, called a . Light may change direction when it encounters the surface of a different material (such as a mirror) or when it passes from one material to another (such as when passing from air into glass). It then continues in a straight line—that is, as a ray. The word ray comes from mathematics. Here it means a straight line that originates from some point. It is acceptable to visualize light rays as laser rays (or even science fiction depictions of ray guns).
Because light moves in straight lines, that is, as rays, and changes directions when it interacts with matter, it can be described through geometry and trigonometry. This part of optics, described by straight lines and angles, is therefore called . There are two laws that govern how light changes direction when it interacts with matter: the law of reflection, for situations in which light bounces off matter; and the law of refraction, for situations in which light passes through matter. In this section, we consider the geometric optics of reflection.
Whenever we look into a mirror or squint at sunlight glinting from a lake, we are seeing a reflection. How does the reflected light travel from the object to your eyes? The law of reflection states: The angle of reflection,
 , equals the angle of incidence,

. This law governs the behavior of all waves when they interact with a smooth surface, and therefore describe the behavior of light waves as well. The reflection of light is simplified when light is treated as a ray. This concept is illustrated in , which also shows how the angles are measured relative to the line perpendicular to the surface at the point where the light ray strikes it. This perpendicular line is also called the normal line, or just the normal. Light reflected in this way is referred to as specular (from the Latin word for mirror: speculum).
We expect to see reflections from smooth surfaces, but , illustrates how a rough surface reflects light. Because the light is reflected from different parts of the surface at different angles, the rays go in many different directions, so the reflected light is diffused. Diffused light allows you to read a printed page from almost any angle because some of the rays go in different directions. Many objects, such as people, clothing, leaves, and walls, have rough surfaces and can be seen from many angles. A mirror, on the other hand, has a smooth surface and reflects light at specific angles.
When we see ourselves in a mirror, it appears that our image is actually behind the mirror. We see the light coming from a direction determined by the law of reflection. The angles are such that our image is exactly the same distance behind the mirror, di, as the distance we stand away from the mirror, do. Although these mirror images make objects appear to be where they cannot be (such as behind a solid wall), the images are not figments of our imagination. Mirror images can be photographed and videotaped by instruments and look just as they do to our eyes, which are themselves optical instruments. An image in a mirror is said to be a virtual image, as opposed to a real image. A virtual image is formed when light rays appear to diverge from a point without actually doing so.
 helps illustrate how a flat mirror forms an image. Two rays are shown emerging from the same point, striking the mirror, and reflecting into the observer’s eye. The rays can diverge slightly, and both still enter the eye. If the rays are extrapolated backward, they seem to originate from a common point behind the mirror, allowing us to locate the image. The paths of the reflected rays into the eye are the same as if they had come directly from that point behind the mirror. Using the law of reflection—the angle of reflection equals the angle of incidence—we can see that the image and object are the same distance from the mirror. This is a virtual image, as defined earlier.
Some mirrors are curved instead of flat. A mirror that curves inward is called a , whereas one that curves outward is called a . Pick up a well-polished metal spoon and you can see an example of each type of curvature. The side of the spoon that holds the food is a concave mirror; the back of the spoon is a convex mirror. Observe your image on both sides of the spoon.
Ray diagrams can be used to find the point where reflected rays converge or appear to converge, or the point from which rays appear to diverge. This is called the , F. The distance from F to the mirror along the central axis (the line perpendicular to the center of the mirror’s surface) is called the , f.  shows the focal points of concave and convex mirrors.
Images formed by a concave mirror vary, depending on which side of the focal point the object is placed. For any object placed on the far side of the focal point with respect to the mirror, the rays converge in front of the mirror to form a real image, which can be projected onto a surface, such as a screen or sheet of paper However, for an object located inside the focal point with respect to the concave mirror, the image is virtual. For a convex mirror the image is always virtual—that is, it appears to be behind the mirror. The ray diagrams in  show how to determine the nature of the image formed by concave and convex mirrors.
The information in  is summarized in .

You should be able to notice everyday applications of curved mirrors. One common example is the use of security mirrors in stores, as shown in .
Some telescopes also use curved mirrors and no lenses (except in the eyepieces) both to magnify images and to change the path of light.  shows a Schmidt-Cassegrain telescope. This design uses a spherical primary concave mirror and a convex secondary mirror. The image is projected onto the focal plane by light passing through the perforated primary mirror. The effective focal length of such a telescope is the focal length of the primary mirror multiplied by the magnification of the secondary mirror. The result is a telescope with a focal length much greater than the length of the telescope itself.
A parabolic concave mirror has the very useful property that all light from a distant source, on reflection by the mirror surface, is directed to the focal point. Likewise, a light source placed at the focal point directs all the light it emits in parallel lines away from the mirror. This case is illustrated by the ray diagram in . The light source in a car headlight, for example, is located at the focal point of a parabolic mirror.
Parabolic mirrors are also used to collect sunlight and direct it to a focal point, where it is transformed into heat, which in turn can be used to generate electricity. This application is shown in .

### The Application of the Curved Mirror Equations

Curved mirrors and the images they create involve a fairly small number of variables: the mirror’s radius of curvature, R; the focal length, f; the distances of the object and image from the mirror, d and d, respectively; and the heights of the object and image, h and h, respectively. The signs of these values indicate whether the image is inverted, erect (upright), real, or virtual. We now look at the equations that relate these variables and apply them to everyday problems.
 shows the meanings of most of the variables we will use for calculations involving curved mirrors.
The basic equation that describes both lenses and mirrors is the lens/mirror equation
This equation can be rearranged several ways. For example, it may be written to solve for focal length.
Magnification, m, is the ratio of the size of the image, h, to the size of the object, h. The value of m can be calculated in two ways.
This relationship can be written to solve for any of the variables involved. For example, the height of the image is given by
We saved the simplest equation for last. The radius of curvature of a curved mirror, R, is simply twice the focal length.
We can learn important information from the algebraic sign of the result of a calculation using the previous equations:
1. A negative d indicates a virtual image; a positive value indicates a real image
2. A negative h indicates an inverted image; a positive value indicates an erect image
3. For concave mirrors, f is positive; for convex mirrors, f is negative
Now let’s apply these equations to solve some problems.

### Practice Problems

### Check Your Understanding

### Section Summary

1. The angle of reflection equals the angle of incidence.
2. Plane mirrors and convex mirrors reflect virtual, erect images. Concave mirrors reflect light to form real, inverted images or virtual, erect images, depending on the location of the object.
3. Image distance, height, and other characteristics can be calculated using the lens/mirror equation and the magnification equation.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Mirrors and Lenses

## Refraction

### Section Key Terms

### The Law of Refraction

You may have noticed some odd optical phenomena when looking into a fish tank. For example, you may see the same fish appear to be in two different places (). This is because light coming to you from the fish changes direction when it leaves the tank and, in this case, light rays traveling along two different paths both reach our eyes. The changing of a light ray’s direction (loosely called bending) when it passes a boundary between materials of different composition, or between layers in single material where there are changes in temperature and density, is called refraction. Refraction is responsible for a tremendous range of optical phenomena, from the action of lenses to voice transmission through optical fibers.
Why does light change direction when passing from one material (medium) to another? It is because light changes speed when going from one material to another. This behavior is typical of all waves and is especially easy to apply to light because light waves have very small wavelengths, and so they can be treated as rays. Before we study the law of refraction, it is useful to discuss the speed of light and how it varies between different media.
The speed of light is now known to great precision. In fact, the speed of light in a vacuum, c, is so important, and is so precisely known, that it is accepted as one of the basic physical quantities, and has the fixed value
where the approximate value of 3.00
   108 m/s is used whenever three-digit precision is sufficient. The speed of light through matter is less than it is in a vacuum, because light interacts with atoms in a material. The speed of light depends strongly on the type of material, given that its interaction with different atoms, crystal lattices, and other substructures varies. We define the index of refraction, n, of a material to be
where v is the observed speed of light in the material. Because the speed of light is always less than c in matter and equals c only in a vacuum, the index of refraction (plural: indices of refraction) is always greater than or equal to one.
 lists the indices of refraction in various common materials.
 provides an analogy for and a description of how a ray of light changes direction when it passes from one medium to another. As in the previous section, the angles are measured relative to a perpendicular to the surface at the point where the light ray crosses it. The change in direction of the light ray depends on how the speed of light changes. The change in the speed of light is related to the indices of refraction of the media involved. In the situations shown in , medium 2 has a greater index of refraction than medium 1. This difference in index of refraction means that the speed of light is less in medium 2 than in medium 1. Note that, in (a), the path of the ray moves closer to the perpendicular when the ray slows down. Conversely, in (b), the path of the ray moves away from the perpendicular when the ray speeds up. The path is exactly reversible. In both cases, you can imagine what happens by thinking about pushing a lawn mower from a footpath onto grass, and vice versa. Going from the footpath to grass, the right front wheel is slowed and pulled to the side as shown. This is the same change in direction for light when it goes from a fast medium to a slow one. When going from the grass to the footpath, the left front wheel moves faster than the others, and the mower changes direction as shown. This, too, is the same change in direction as light going from slow to fast.
The amount that a light ray changes direction depends both on the incident angle and the amount that the speed changes. For a ray at a given incident angle, a large change in speed causes a large change in direction, and thus a large change in the angle of refraction. The exact mathematical relationship is the law of refraction, or , which is stated in equation form as
In terms of speeds, Snell’s law becomes
Here, n1 and n2 are the indices of refraction for media 1 and 2, respectively, and θ1 and θ2 are the angles between the rays and the perpendicular in the respective media 1 and 2, as shown in . The incoming ray is called the  and the outgoing ray is called the . The associated angles are called the angle of incidence and the angle of refraction. Later, we apply Snell’s law to some practical situations.
Dispersion is defined as the spreading of white light into the wavelengths of which it is composed. This happens because the index of refraction varies slightly with wavelength.  shows how a prism disperses white light into the colors of the rainbow.
Rainbows are produced by a combination of refraction and reflection. You may have noticed that you see a rainbow only when you turn your back to the Sun. Light enters a drop of water and is reflected from the back of the drop, as shown in . The light is refracted both as it enters and as it leaves the drop. Because the index of refraction of water varies with wavelength, the light is dispersed and a rainbow is observed.
A good-quality mirror reflects more than 90 percent of the light that falls on it; the mirror absorbs the rest. But, it would be useful to have a mirror that reflects all the light that falls on it. Interestingly, we can produce total reflection using an aspect of refraction. Consider what happens when a ray of light strikes the surface between two materials, such as is shown in (a). Part of the light crosses the boundary and is refracted; the rest is reflected. If, as shown in the figure, the index of refraction for the second medium is less than the first, the ray bends away from the perpendicular. Because n1 > n2, the angle of refraction is greater than the angle of incidence—that is,
  >
  . Now, imagine what happens as the incident angle is increased. This causes
   to increase as well. The largest the angle of refraction,
 , can be is 90°, as shown in (b). The critical angle,
 , for a combination of two materials is defined to be the incident angle,
 , which produces an angle of refraction of 90°. That is,
   is the incident angle for which
  = 90°. If the incident angle,
 , is greater than the critical angle, as shown in (c), then all the light is reflected back into medium 1, a condition called .
Recall that Snell’s law states the relationship between angles and indices of refraction. It is given by
When the incident angle equals the critical angle (

=

), the angle of refraction is 90° (

= 90°). Noting that sin 90° = 1, Snell’s law in this case becomes
The critical angle,
 , for a given combination of materials is thus
for n1 > n2.
Total internal reflection occurs for any incident angle greater than the critical angle,
 , and it can only occur when the second medium has an index of refraction less than the first. Note that the previous equation is written for a light ray that travels in medium 1 and reflects from medium 2, as shown in .
There are several important applications of total internal reflection. Total internal reflection, coupled with a large index of refraction, explains why diamonds sparkle more than other materials. The critical angle for a diamond-to-air surface is only 24.4°; so, when light enters a diamond, it has trouble getting back out (). Although light freely enters the diamond at different angles, it can exit only if it makes an angle less than 24.4° with the normal to a given surface. Facets on diamonds are specifically intended to make this unlikely, so that the light can exit only in certain places. Diamonds with very few impurities are very clear, so the light makes many internal reflections and is concentrated at the few places it can exit—hence the sparkle.
A light ray that strikes an object that consists of two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came. This parallel reflection is true whenever the reflecting surfaces are perpendicular, and it is independent of the angle of incidence. Such an object is called a  because the light bounces from its inside corner. Many inexpensive reflector buttons on bicycles, cars, and warning signs have corner reflectors designed to return light in the direction from which it originates. Corner reflectors are perfectly efficient when the conditions for total internal reflection are satisfied. With common materials, it is easy to obtain a critical angle that is less than 45°. One use of these perfect mirrors is in binoculars, as shown in . Another application is for periscopes used in submarines.
Fiber optics are one common application of total internal reflection. In communications, fiber optics are used to transmit telephone, internet, and cable TV signals, and they use the transmission of light down fibers of plastic or glass. Because the fibers are thin, light entering one is likely to strike the inside surface at an angle greater than the critical angle and, thus, be totally reflected (). The index of refraction outside the fiber must be smaller than inside, a condition that is satisfied easily by coating the outside of the fiber with a material that has an appropriate refractive index. In fact, most fibers have a varying refractive index to allow more light to be guided along the fiber through total internal reflection. Rays are reflected around corners as shown in the figure, making the fibers into tiny light pipes.

### Calculations with the Law of Refraction

The calculation problems that follow require application of the following equations:
and
These are the equations for refractive index, the mathematical statement of the law of refraction (Snell’s law), and the equation for the critical angle.

### Practice Problems

### Check Your Understanding

### Section Summary

1. The index of refraction for a material is given by the speed of light in a vacuum divided by the speed of light in that material.
2. Snell’s law states the relationship between indices of refraction, the incident angle, and the angle of refraction.
3. The critical angle,
 , determines whether total internal refraction can take place, and can be calculated according to
  .

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Mirrors and Lenses

## Lenses

### Section Key Terms

### Characteristics of Lenses

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom lens. In this section, we use the law of refraction to explore the properties of lenses and how they form images.
Some of what we learned in the earlier discussion of curved mirrors also applies to the study of lenses. Concave, convex, focal point F, and focal length f have the same meanings as before, except each measurement is made from the center of the lens instead of the surface of the mirror. The convex lens shown in  has been shaped so that all light rays that enter it parallel to its central axis cross one another at a single point on the opposite side of the lens. The central axis, or axis, is defined to be a line normal to the lens at its center. Such a lens is called a converging lens because of the converging effect it has on light rays. An expanded view of the path of one ray through the lens is shown in  to illustrate how the ray changes direction both as it enters and as it leaves the lens. Because the index of refraction of the lens is greater than that of air, the ray moves toward the perpendicular as it enters and away from the perpendicular as it leaves. (This is in accordance with the law of refraction.) As a result of the shape of the lens, light is thus bent toward the axis at both surfaces.
Note that rays from a light source placed at the focal point of a converging lens emerge parallel from the other side of the lens. You may have heard of the trick of using a converging lens to focus rays of sunlight to a point. Such a concentration of light energy can produce enough heat to ignite paper.
 shows a concave lens and the effect it has on rays of light that enter it parallel to its axis (the path taken by ray 2 in the figure is the axis of the lens). The concave lens is a diverging lens because it causes the light rays to bend away (diverge) from its axis. In this case, the lens has been shaped so all light rays entering it parallel to its axis appear to originate from the same point, F, defined to be the focal point of a diverging lens. The distance from the center of the lens to the focal point is again called the focal length, or “ƒ,” of the lens. Note that the focal length of a diverging lens is defined to be negative. An expanded view of the path of one ray through the lens is shown in  to illustrate how the shape of the lens, together with the law of refraction, causes the ray to follow its particular path and diverge.
The power, P, of a lens is very easy to calculate. It is simply the reciprocal of the focal length, expressed in meters
The units of power are diopters, D, which are expressed in reciprocal meters. If the focal length is negative, as it is for the diverging lens in , then the power is also negative.
In some circumstances, a lens forms an image at an obvious location, such as when a movie projector casts an image onto a screen. In other cases, the image location is less obvious. Where, for example, is the image formed by eyeglasses? We use ray tracing for thin lenses to illustrate how they form images, and we develop equations to describe the image-formation quantitatively. These are the rules for ray tracing:
1. A ray entering a converging lens parallel to its axis passes through the focal point, F, of the lens on the other side
2. A ray entering a diverging lens parallel to its axis seems to come from the focal point, F, on the side of the entering ray
3. A ray passing through the center of either a converging or a diverging lens does not change direction
4. A ray entering a converging lens through its focal point exits parallel to its axis
5. A ray that enters a diverging lens by heading toward the focal point on the opposite side exits parallel to the axis
Consider an object some distance away from a converging lens, as shown in . To find the location and size of the image formed, we trace the paths of select light rays originating from one point on the object. In this example, the originating point is the top of a woman’s head.  shows three rays from the top of the object that can be traced using the ray-tracing rules just listed. Rays leave this point traveling in many directions, but we concentrate on only a few, which have paths that are easy to trace. The first ray is one that enters the lens parallel to its axis and passes through the focal point on the other side (rule 1). The second ray passes through the center of the lens without changing direction (rule 3). The third ray passes through the nearer focal point on its way into the lens and leaves the lens parallel to its axis (rule 4). All rays that come from the same point on the top of the person’s head are refracted in such a way as to cross at the same point on the other side of the lens. The image of the top of the person’s head is located at this point. Rays from another point on the object, such as the belt buckle, also cross at another common point, forming a complete image, as shown. Although three rays are traced in , only two are necessary to locate the image. It is best to trace rays for which there are simple ray-tracing rules. Before applying ray tracing to other situations, let us consider the example shown in  in more detail.
The image formed in  is a real image—meaning, it can be projected. That is, light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye.
In , the object distance, d, is greater than f. Now we consider a ray diagram for a convex lens where d f, and another diagram for a concave lens.
The examples in  and  represent the three possible cases—case 1, case 2, and case 3—summarized in . In the table, m is magnification; the other symbols have the same meaning as they did for curved mirrors.
Image formation by lenses can also be calculated from simple equations. We learn how these calculations are carried out near the end of this section.
Some common applications of lenses with which we are all familiar are magnifying glasses, eyeglasses, cameras, microscopes, and telescopes. We take a look at the latter two examples, which are the most complex. We have already seen the design of a telescope that uses only mirrors in .  shows the design of a telescope that uses two lenses. Part (a) of the figure shows the design of the telescope used by Galileo. It produces an upright image, which is more convenient for many applications. Part (b) shows an arrangement of lenses used in many astronomical telescopes. This design produces an inverted image, which is less of a problem when viewing celestial objects.
 shows the path of light through a typical microscope. Microscopes were first developed during the early 1600s by eyeglass makers in the Netherlands and Denmark. The simplest compound microscope is constructed from two convex lenses, as shown schematically in . The first lens is called the objective lens; it has typical magnification values from 5
   to 100
  . In standard microscopes, the objectives are mounted such that when you switch between them, the sample remains in focus. Objectives arranged in this way are described as . The second lens, the eyepiece, also referred to as the , has several lenses that slide inside a cylindrical barrel. The focusing ability is provided by the movement of both the objective lens and the eyepiece. The purpose of a microscope is to magnify small objects, and both lenses contribute to the final magnification. In addition, the final enlarged image is produced in a location far enough from the observer to be viewed easily because the eye cannot focus on objects or images that are too close.
Real lenses behave somewhat differently from how they are modeled using rays diagrams or the thin-lens equations. Real lenses produce aberrations. An aberration is a distortion in an image. There are a variety of aberrations that result from lens size, material, thickness, and the position of the object. One common type of aberration is chromatic aberration, which is related to color. Because the index of refraction of lenses depends on color, or wavelength, images are produced at different places and with different magnifications for different colors. The law of reflection is independent of wavelength, so mirrors do not have this problem. This result is another advantage for the use of mirrors in optical systems such as telescopes.
(a) shows chromatic aberration for a single convex lens, and its partial correction with a two-lens system. The index of refraction of the lens increases with decreasing wavelength, so violet rays are refracted more than red rays, and are thus focused closer to the lens. The diverging lens corrects this in part, although it is usually not possible to do so completely. Lenses made of different materials and with different dispersions may be used. For example, an achromatic doublet consisting of a converging lens made of crown glass in contact with a diverging lens made of flint glass can reduce chromatic aberration dramatically ((b)).

### Physics of the Eye

The eye is perhaps the most interesting of all optical instruments. It is remarkable in how it forms images and in the richness of detail and color they eye can detect. However, our eyes commonly need some correction to reach what is called normal vision, but should be called ideal vision instead. Image formation by our eyes and common vision correction are easy to analyze using geometric optics.  shows the basic anatomy of the eye. The cornea and lens form a system that, to a good approximation, acts as a single thin lens. For clear vision, a real image must be projected onto the light-sensitive retina, which lies at a fixed distance from the lens. The lens of the eye adjusts its power to produce an image on the retina for objects at different distances. The center of the image falls on the fovea, which has the greatest density of light receptors and the greatest acuity (sharpness) in the visual field. There are no receptors at the place where the optic nerve meets the eye, which is called the blind spot. An image falling on this spot cannot be seen. The variable opening (or pupil) of the eye along with chemical adaptation allows the eye to detect light intensities from the lowest observable to 1010 times greater (without damage). Ten orders of magnitude is an incredible range of detection. Our eyes perform a vast number of functions, such as sense direction, movement, sophisticated colors, and distance. Processing of visual nerve impulses begins with interconnections in the retina and continues in the brain. The optic nerve conveys signals received by the eye to the brain.
Refractive indices are crucial to image formation using lenses.  shows refractive indices relevant to the eye. The biggest change in the refractive index—and the one that causes the greatest bending of rays—occurs at the cornea rather than the lens. The ray diagram in  shows image formation by the cornea and lens of the eye. The rays bend according to the refractive indices provided in . The cornea provides about two-thirds of the magnification of the eye because the speed of light changes considerably while traveling from air into the cornea. The lens provides the remaining magnification needed to produce an image on the retina. The cornea and lens can be treated as a single thin lens, although the light rays pass through several layers of material (such as the cornea, aqueous humor, several layers in the lens, and vitreous humor), changing direction at each interface. The image formed is much like the one produced by a single convex lens. This result is a case 1 image. Images formed in the eye are inverted, but the brain inverts them once more to make them seem upright.
As noted, the image must fall precisely on the retina to produce clear vision—that is, the image distance, di, must equal the lens-to-retina distance. Because the lens-to-retina distance does not change, di must be the same for objects at all distances. The eye manages to vary the distance by varying the power (and focal length) of the lens to accommodate for objects at various distances. In , you can see the small ciliary muscles above and below the lens that change the shape of the lens and, thus, the focal length.
The need for some type of vision correction is very common. Common vision defects are easy to understand, and some are simple to correct.  illustrates two common vision defects. Nearsightedness, or myopia, is the inability to see distant objects clearly while close objects are in focus. The nearsighted eye overconverges the nearly parallel rays from a distant object, and the rays cross in front of the retina. More divergent rays from a close object are converged on the retina, producing a clear image. Farsightedness, or hyperopia, is the inability to see close objects clearly whereas distant objects may be in focus. A farsighted eye does not converge rays from a close object sufficiently to make the rays meet on the retina. Less divergent rays from a distant object can be converged for a clear image.
Because the nearsighted eye overconverges light rays, the correction for nearsightedness involves placing a diverging spectacle lens in front of the eye. This lens reduces the power of an eye that has too short a focal length ((a)). Because the farsighted eye underconverges light rays, the correction for farsightedness is to place a converging spectacle lens in front of the eye. This lens increases the power of an eye that has too long a focal length ((b)).

### Calculations Using Lens Equations

As promised, there are no new equations to memorize. We can use equations already presented for solving problems involving curved mirrors. Careful analysis allows you to apply these equations to lenses. Here are the equations you need
where P is power, expressed in reciprocal meters (m–1) rather than diopters (D), and f is focal length, expressed in meters (m). You also need
where, as before, do and di are object distance and image distance, respectively. Remember, this equation is usually more useful if rearranged to solve for one of the variables. For example,
The equations for magnification, m, are also the same as for mirrors
where hi and ho are the image height and object height, respectively. Remember, also, that a negative di value indicates a virtual image and a negative hi value indicates an inverted image.
These are the steps to follow when solving a lens problem:
1. Step 1. Examine the situation to determine that image formation by a lens is involved.
2. Step 2. Determine whether ray tracing, the thin-lens equations, or both should be used. A sketch is very helpful even if ray tracing is not specifically required by the problem. Write useful symbols and values on the sketch.
3. Step 3. Identify exactly what needs to be determined in the problem (identify the unknowns).
4. Step 4. Make a list of what is given or can be inferred from the problem as stated (identify the knowns). It is helpful to determine whether the situation involves a case 1, 2, or 3 image. Although these are just names for types of images, they have certain characteristics (given in ) that can be of great use in solving problems.
5. Step 5. If ray tracing is required, use the ray-tracing rules listed earlier in this section.
6. Step 6. Most quantitative problems require the use of the thin-lens equations. These equations are solved in the usual manner by substituting knowns and solving for unknowns. Several worked examples were included earlier and can serve as guides.
7. Step 7. Check whether the answer is reasonable. Does it make sense? If you identified the type of image (case 1, 2, or 3) correctly, you should assess whether your answer is consistent with the type of image, magnification, and so on.
All problems will be solved by one or more of the equations just presented, with ray tracing used only for general analysis of the problem. The steps then simplify to the following:
1. Identify the unknown.
2. Identify the knowns.
3. Choose an equation, plug in the knowns, and solve for the unknown.
Here are some worked examples:

### Practice Problems

### Check Your Understanding

### Section Summary

1. The characteristics of images formed by concave and convex lenses can be predicted using ray tracing. Characteristics include real versus virtual, inverted versus upright, and size.
2. The human eye and corrective lenses can be explained using geometric optics.
3. Characteristics of images formed by lenses can be calculated using the mirror/lens equation.

### Key Equations

### Concept Items

### Critical Thinking

### Problems

### Performance Task

### Test Prep Multiple Choice

### Test Prep Short Answer

### Test Prep Extended Response


# Diffraction and Interference

## Introduction

Examine a compact disc under white light, noting the colors observed and their locations on the disc. Using the CD, explore the spectra of a few light sources, such as a candle flame, an incandescent bulb, and fluorescent light. If you have ever looked at the reds, blues, and greens in a sunlit soap bubble and wondered how straw-colored soapy water could produce them, you have hit upon one of the many phenomena that can only be explained by the wave character of light. That and other interesting phenomena, such as the dispersion of white light into a rainbow of colors when passed through a narrow slit, cannot be explained fully by geometric optics. In such cases, light interacts with small objects and exhibits its wave characteristics. The topic of this chapter is the branch of optics that considers the behavior of light when it exhibits wave characteristics.