Chapter 14: Waves and Energy Transfer - Denton · PDF fileTo find out more about waves and energy transfer, visit the Glencoe Science Web site at science.glencoe.com WHAT YOU’LL

Surf’s UpFrom where does the

surfer’s kinetic energy

come? Trace the energy

source back as far as

you can.

➥ Look at the text on page 329 for the answer.

Waves and Energy Transfer14

CHAPTER

PHYSICSTo find out more about waves and energy transfer, visit theGlencoe Science Web site at science.glencoe.com

WHAT YOU’LL LEARN• You will determine how

waves transfer energy.• You will describe wave

reflection and discuss itspractical significance.

WHY IT’S IMPORTANT• Waves enable the sun’s

energy to reach Earth andmake possible all communi-cation through sound.

• Because light waves can bereflected, you are able to seethe world around you andeven read these very words.

• Knowledge of the behaviorof waves is essential to thedesigning of bridges andmany other structures.

327

Have you ever been in a wave pool and caught a wave withyour body, experiencing a push from the wave? Perhapsyou’ve splashed about in the tub, creating a miniwave and

letting the water lift your body upward. Or maybe you have had the exhilarating experience of riding a

surfboard near the ocean shore. As the wave pushes you toward theshore, you gain speed and stay just ahead of the breaking surf. Asyou surf, you may try to ride almost parallel to the wave at a veryhigh speed, as this surfer does. Surfing, however, can be dangerous.Unless you are skilled, the energy carried by the wave can cause youto wipe out, throwing you into the ocean or up onto the sand.

In each case, your body’s kinetic energy increases. The questionis, where does this extra energy come from, and how does it getfrom one place to another?

In the wave pool, you can see the waves move from one end ofthe pool to the other. Even in the tub, you can see the miniwavemove from your head to your toes. At the ocean shore, surfing awave allows you to come rapidly toward the shore. In each case,you may think the water is moving from one location to another.However, that is not what occurs. In water waves, the wave carriesenergy from one location to another, while the water itself movesin circles. In the coming chapters, you will study the wave motionsof light and sound. You will find out how light waves and soundwaves are similar and how they are different.

http://science.glencoe.com

Both particles and waves carry energy, butthere is an important difference in how they

do this. Think of a ball as a particle. If you tossthe ball to a friend, the ball moves from you to your friend and carriesenergy. However, if you and your friend hold the ends of a rope and yougive your end a quick shake, the rope remains in your hand, and eventhough no matter is transferred, the rope still carries energy. The wavescarry energy through matter.

Mechanical WavesYou have learned how Newton’s laws of motion and conservation of

energy principles govern the behavior of particles. These laws also gov-ern the motion of waves. There are many kinds of waves. All kinds ofwaves transmit energy, including the waves you cannot see, such as thesound waves you create when you speak and the light waves that reflectfrom the leaves on the trees.

Transverse waves A wave is a rhythmic disturbance that carriesenergy through matter or space. Water waves, sound waves, and thewaves that travel down a rope or spring are types of mechanical waves.Mechanical waves require a medium. Water, air, ropes, or springs are thematerials that carry the energy of mechanical waves. Other kinds ofwaves, including electromagnetic waves and matter waves, will bedescribed in later chapters. Because many of these waves cannot bedirectly observed, mechanical waves can serve as models for their study.

The two disturbances that go down the rope shown in Figure 14–1are called wave pulses. A wave pulse is a single bump or disturbancethat travels through a medium. If the person continues to move the ropeup and down, a continuous wave is generated. Notice that the rope isdisturbed in the vertical direction, but the pulse travels horizontally.This wave motion is called a transverse wave. A transverse wave is awave that vibrates perpendicular to the direction of wave motion.

Longitudinal and surface waves In a coiled spring such as a Slinkytoy, you can create a wave pulse in a different way. If you squeezetogether several turns of the coiled spring and then suddenly releasethem, pulses of closely spaced turns will move away in both directions,as in Figure 14–2. In this case, the disturbance is in the same directionas, or parallel to, the direction of wave motion. Such a wave is called alongitudinal wave. Sound waves are longitudinal waves. Fluids such asliquids and gases usually transmit only longitudinal waves.

Although waves deep in a lake or ocean are longitudinal, at the sur-face of the water, the particles move in a direction that is both parallel

OBJ ECTIVES• Identify how waves transfer

energy without transferringmatter.

• Contrast transverse andlongitudinal waves.

• Relate wave speed, wave-length, and frequency.

14.1 Wave Properties

328 Waves and Energy Transfer

FIGURE 14–1 A quick shake ofa rope sends out wave pulses inboth directions.

and perpendicular to the direction of wave motion, as shown in Figure 14–3. These are surface waves, which have characteristics ofboth transverse and longitudinal waves. The energy of water waves usu-ally comes from storms far away. The energy of the storms initially camefrom the heating of Earth by solar energy. This energy, in turn, was car-ried to Earth by transverse electromagnetic waves.

Measuring a WaveThere are many ways to describe or measure a wave. Some methods

depend on how the wave is produced, whereas others depend on themedium through which the wave travels.

14.1 Wave Properties 329

FIGURE 14–3 Surface waveshave properties of both longitu-dinal and transverse waves (a).The paths of the particles are circular (b).

FIGURE 14–2 The squeeze andrelease of a coiled spring toysends out wave pulses in bothdirections.

Crest

Trough

Wave motion

a

b

Surf’s Up➥ Answers question from

page 326.

Waves on a Coiled SpringProblem

How can you model the properties of transverse waves?

HypothesisA coiled spring toy can be used to modeltransverse waves and to investigate waveproperties such as speed, frequency, amplitude, and wavelength.

Possible Materials a long coiled spring toystopwatchmeterstick

Plan the Experiment1. Work in pairs or groups, and clear a path

of about 6 meters for this activity.

2. One member of the team should grip theSlinky firmly with one hand. Another mem-ber of the team should stretch the springto the length suggested by your teacher.Team members should take turns holdingthe end of the spring. CAUTION: Coiledsprings easily get out of control. Do notallow them to get tangled or overstretched.

3. The second team member should thenmake a quick sideways snap of the wristto produce transverse wave pulses. Otherteam members can assist in measuring,timing, and recording data. It is easier tosee the motion from one end of the Slinky,rather than from the side.

4. Design experiments to answer the questions under Analyze and Conclude.

5. Check the Plan Make sure your teacherhas approved your final plan before youproceed with your experiments.

Analyze and Conclude1. Interpreting Data What happens to

the amplitude of the transverse wave as it travels?

2. Recognizing Cause and Effect Does the transverse wave’s speed depend uponits amplitude?

3. Observing and Interpreting If you puttwo quick transverse wave pulses into thespring and consider the wavelength to bethe distance between the pulses, does thewavelength change as the pulses move?

4. Applying How can you decrease thewavelength of a transverse wave?

5. Interpreting As transverse wave pulsestravel back and forth on the spring, do theybounce off each other or pass through each other?

Apply1. How do the speeds of high frequency

(short wavelength) transverse waves com-pare with the speeds of low frequency(long wavelength) transverse waves?

2. Suppose you designed the experiment usinglongitudinal waves. How would the proce-dure for longitudinal waves be differentfrom the procedure for transverse waves?

3. Would you expect the results of an experi-ment with longitudinal waves to be similarto the results of the transverse waveexperiment? Explain why or why not.



Speed and amplitude How fast does a wave move? The speed of thepulse shown in Figure 14–4 can be found in the same way in which youwould determine the speed of a moving car. First, you measure the dis-placement of the wave peak, �d; then you divide this by the time inter-val, �t, to find the speed, as shown by v � v� � �d/�t. The speed of acontinuous wave, can be found the same way. For most mechanicalwaves, both transverse and longitudinal, the speed depends only on themedium through which the waves move.

How does the pulse generated by gently shaking a rope differ from thepulse produced by a violent shake? The difference is similar to the dif-ference between a ripple in a pond and a tidal wave. They have differentamplitudes. The amplitude of a wave is its maximum displacement fromits position of rest, or equilibrium. Two similar waves having differentamplitudes are shown in Figure 14–5. A wave’s amplitude depends onhow the wave is generated, but not on its speed. More work has to bedone to generate a wave with a larger amplitude. For example, strongwinds produce larger water waves than those formed by gentle breezes.Waves with larger amplitudes transfer more energy. Thus, although asmall wave might move sand on a beach a few centimeters, a giant wavecan uproot and move a tree. For waves that move at the same speed, therate at which energy is transferred is proportional to the square of the

FIGURE 14–4 These two photo-graphs were taken 0.20 s apart.During that time, the crest moved0.80 m. The velocity of the waveis 4.0 m/s.

FIGURE 14–5 The amplitude ofwave A is larger than that ofwave B.

Dis

pla

cem

ent

Wave A

Trough

Crest

Wave B

Amplitude

Distance

λ

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amplitude. Thus, doubling the amplitude of a wave increases theamount of energy it transfers each second by a factor of four.

Wavelength Rather than focusing on one point on a wave, imaginetaking a snapshot of a wave, so that you can see the whole wave at oneinstant in time. Figure 14–5 shows the low points, or troughs, and thehigh points, or crests, of a wave. The shortest distance between pointswhere the wave pattern repeats itself is called the wavelength. Crests arespaced by one wavelength. Each trough is also one wavelength from thenext. The Greek letter lambda, �, represents wavelength.

Period and frequency Although wave speed and amplitude candescribe both pulses and continuous waves, period (T) and frequency(f) apply only to continuous waves. You learned in Chapter 6 that theperiod of a simple harmonic oscillator, such as a pendulum, is the timeit takes for the motion of the oscillator to repeat itself. Such an oscilla-tor is usually the source, or cause, of a continuous wave. The period ofa wave is equal to the period of the source. In Figure 14–6, the period,T, equals 0.04 s, which is the time it takes the source to return to thesame point in its oscillation. The same time is taken by point P, a pointon the rope, to return to its initial position.

The frequency of a wave, f, is the number of complete oscillations itmakes each second. Frequency is measured in hertz. One hertz (Hz) is one oscillation per second. The frequency (f) and period (T) of a waveare related by the following equation.

Frequency of a Wave f � �1

T�

Both the period and the frequency of a wave depend only on its source.They do not depend on the wave’s speed or the medium.

Although you can measure a wavelength directly, the wavelengthdepends on both the frequency of the oscillator and the speed of the

FIGURE 14–6 One end of astring, with a piece of tape atpoint P, is attached to a vibratingblade. Note the change in positionof point P over time. P

Vibratingblade

P

P

P

a

c

b

d


wave. In the time interval of one period, a wave moves one wavelength.Therefore, the speed of a wave is the wavelength divided by the period,v � �/T. Because the frequency is more easily found than the period,this equation is more often written as

Speed of a Wave v � �f.

14.1 Wave Properties

Speed of a WaveA sound wave has a frequency of 262 Hz and a wavelength

measured at 1.29 m.

a. What is the speed of the wave?

b. How long will it take the wave to travel the length of a football field, 91.4 m?

c. What is the period of the wave?

Sketch the Problem• Draw a model of one wavelength.• Diagram a velocity vector.

Calculate Your AnswerKnown: Unknown:

f � 262 Hz v � ?

� � 1.29 m t � ?

d � 91.4 m T � ?

Check Your Answer• Hz has the units s–1 and so m�Hz equals m/s, which is correct.• Are the magnitudes realistic? A typical sound wave travels

approximately 343 m/s. You can notice the delay in sound across a football field, so a few tenths of a second is reasonable.

Strategy:

a. Find the speed of sound from the frequency and wavelength.

b. Find the time required from speed and distance.

c. Find the period from the frequency.

Calculations:

v � �f � (1.29 m)(262 Hz) � 338 m/s

v � �d

t�, so t � �

d

v� � �

3

9

3

1

8

.4

m

m

/s� � 0.270 s

T � �1

f� � �

262

1

Hz� � 0.00382 s

λ

v

F.Y.I.Because radio waves travel at 3.00 � 108 m/sand sound waves areslower, 3.4 � 102 m/s, abroadcast voice can beheard sooner 10 000 milesaway than it can be heardat the back of the room inwhich it originated.

Example Problem

334 Waves and Energy TransferWaves and Energy Transfer

Predicting EarthquakesEarthquakes are produced by the suddenmotion of rock masses within Earth’s crust.Although rocks can bend and twist to someextent, they break if they are exposed toforces that exceed their strength. The frictionand crushing motions of breaking rock createseismic waves of energy that radiate outward.These seismic waves cause the shaking andtrembling called an earthquake. The breaks in the rock are known as earthquake faults.

The subsurface region where the rock rup-tures is called the focus of an earthquake. Thepoint on the surface directly above the focusis known as the epicenter. Several types ofseismic waves travel outward from the focus.Waves that travel along the surface are calledsurface waves. They have characteristics ofboth transverse and longitudinal waves andcause most major earthquake damage tohomes and cities.

How much damage?Many earthquakes are quite small and arehardly felt. However, catastrophic earth-quakes have been recorded in many regionsof the world. One example is the deadlyearthquake that hit the densely populatedregion of Izmir, Turkey, in 1999. Eighteenthousand people were killed, and damageestimates reached $40 billion.

Can earthquakes be predicted?Most geologists agree that regions along por-tions of California’s famous San Andreas Faultwill experience a major earthquake before themiddle of the 21st century. The broad timeframe of a prediction such as this doesn’tenable people to evacuate an area in antici-pation of a quake on any particular date.

Efforts are under way to develop more precise earthquake predictions. Scientists

constantly use seismographs to monitor majorfaults such as the San Andreas Fault for thesmallest Earth tremors. A seismograph recordsthe magnitude of seismic waves by suspendinga pen on a pendulum over a paper-covereddrum. The stronger the motion, the largerthe arc of the pen’s motion on the drum.

Lasers and creep meters measure differ-ences in land movement on the two sides of a fault. A creep meter consists of wiresstretched across a fault, and a laser beam is timed as it returns to its source. Scientistsmonitor radon and hydrogen concentrationsin groundwater to determine how theychange prior to an earthquake. Antennaemonitor changes in electromagnetic wavescoming from deep beneath Earth’s surface.

Investigating the Issue1. Acquiring Information Use your library

skills to find out more about the Parkfieldexperiment near the San Andreas Fault inCalifornia. What kinds of instruments arebeing used by seismologists to monitorthe fault? Have any earthquakes predictedfor Parkfield actually occurred?

2. Debating the Issue Evaluate the impact ofearthquake research on society and theenvironment. How can studying waveproperties and behaviors impact the build-ing industry? What is the responsibility ofscientists, government, and citizens withrespect to earthquake-resistant structures?

PHYSICSTo find out more about earthquakes, visit the GlencoeScience Web site atscience.glencoe.com


14.1 Wave Properties 33514.1 Wave Properties

EARTH SCIENCECONNECTION

Earth’s Core An earth-quake produces bothtransverse and longitudi-nal waves that travelthrough Earth. Geologistsstudying these waveswith seismographs foundthat only the longitudinalwaves could passthrough Earth’s core. Only longitudinal wavescan move through a liquid or a gas. From thisobservation, what can be deduced about thenature of Earth’s core?

Section Review1. Suppose you and your lab partner are asked

to measure the speed of a transverse wavein a giant, coiled spring toy. How could youdo it? List the equipment you would need.

2. Describe longitudinal waves. What typesof media transmit longitudinal waves?

3. You are creating transverse waves in a ropeby shaking your hand from side to side.Without changing the distance your handmoves, you begin to shake it faster andfaster. What happens to the amplitude, frequency, period, and velocity of the wave?

4. If you pull on one end of a coiled springtoy, does the pulse reach the other endinstantaneously? What if you pull on arope? What if you hit the end of a metalrod? Compare the responses of thesethree materials.

5. Critical Thinking If a raindrop falls into a pool, small-amplitude waves result. If a swimmer jumps into a pool, a large-amplitude wave is produced. Why doesn’tthe heavy rain in a thunderstorm producelarge waves?

14.1

Practice Problems

1. A sound wave produced by a clock chime is heard 515 m away1.50 s later.a. What is the speed of sound of the clock’s chime in air?b. The sound wave has a frequency of 436 Hz. What is its period?c. What is its wavelength?

2. A hiker shouts toward a vertical cliff 685 m away. The echo isheard 4.00 s later.a. What is the speed of sound of the hiker’s voice in air?b. The wavelength of the sound is 0.750 m. What is its frequency?c. What is the period of the wave?

3. If you want to increase the wavelength of waves in a rope,should you shake it at a higher or lower frequency?

4. What is the speed of a periodic wave disturbance that has a frequency of 2.50 Hz and a wavelength of 0.600 m?

5. The speed of a transverse wave in a string is 15.0 m/s. If a sourceproduces a disturbance that has a frequency of 5.00 Hz, what isits wavelength?

6. Five pulses are generated every 0.100 s in a tank of water. Whatis the speed of propagation of the wave if the wavelength of thesurface wave is 1.20 cm?

7. A periodic longitudinal wave that has a frequency of 20.0 Hztravels along a coil spring. If the distance between successivecompressions is 0.400 m, what is the speed of the wave?

When a wave encounters the boundary ofthe medium in which it is traveling, it

sometimes reflects back into the medium. Inother instances, some or all of the wave passesthrough the boundary into another medium, often changing direction atthe boundary. In addition, many properties of wave behavior result fromthe fact that two or more waves can exist in the same medium at the sametime—quite unlike particles, which consist of matter and take up space.

Waves at BoundariesRecall that the speed of a mechanical wave depends only on the prop-

erties of the medium it passes through, not on the wave’s amplitude orfrequency. For water waves, the depth of the water affects wave speed.For sound waves in air, the temperature affects wave speed. For waves ona spring, the speed depends upon the spring’s rigidity and its mass perunit length.

Examine what happens when a wave moves across a boundary fromone medium into another, as in two springs of different thicknessesjoined end to end. Figure 14–7 shows a wave pulse moving from a largespring into a smaller one. The wave that strikes the boundary is calledthe incident wave. One pulse from the larger spring continues in thesmaller spring, at the speed of waves on the smaller spring. Note thatthis transmitted wave pulse remains upward.

Some of the energy of the incident wave’s pulse is reflected backwardin the larger spring. This returning wave is called the reflected wave.Whether or not the reflected wave is upward (erect) or downward (inverted)depends on the comparative thicknesses of the two springs. If waves in thesmaller spring have a higher speed because the spring is heavier or stiffer,then the reflected wave will be inverted.

OBJ ECTIVES• Relate a wave’s speed to

the medium in which thewave travels.

• Describe how waves arereflected and refracted atboundaries between media,and explain how waves diffract.

• Apply the principle ofsuperposition to the phe-nomenon of interference.

14.2 Wave Behavior


FIGURE 14–7 The junction ofthe two springs is a boundarybetween two media. A pulsereaching the boundary (a) is partially reflected and partiallytransmitted (b).

a b

14.2 Wave Behavior 337

What happens if the boundary is a wall rather than another spring?When a wave pulse is sent down a spring connected to a rigid wall, the energy transmitted is reflected back from the wall, as shown in Figure 14–8. The wall is the boundary of a new medium through whichthe wave attempts to pass. The pulse is reflected from the wall withalmost exactly the same amplitude as the pulse of the incident wave.Thus, almost all the wave’s energy is reflected back. Very little energy istransmitted into the wall. Note also that the pulse is inverted.

FIGURE 14–8 A pulseapproaches a rigid wall (a) andis reflected back (b). Note thatthe amplitude of the reflectedpulse is nearly equal to theamplitude of the incident pulse,but it is inverted and appears asa downward curve.

BA

FIGURE 14–10

a b

Pocket LabWave Reflections

Waves lose amplitude andtransfer energy when theyreflect from a barrier. Whathappens to the speed of thewaves? Use a wave tank with aprojection system. Half-fill thetank with water. Dip your fingerinto the water near one end ofthe tank and notice how fastthe wave that you make movesto the other end.Analyze Does the wave slowdown as it travels? Use a stop-watch to measure the time for awave to cover two lengths, thenfour lengths, of the wave tank.

Practice Problems

8. A pulse is sent along a spring. The spring is attached to a light-weight thread that is tied to a wall, as shown in Figure 14–9.a. What happens when the pulse reaches point A?b. Is the pulse reflected from point A erect or inverted?c. What happens when the transmitted pulse reaches point B?d. Is the pulse reflected from point B erect or inverted?

9. A long spring runs across the floor of a room and out the door.A pulse is sent along the spring. After a few seconds, an invertedpulse returns. Is the spring attached to the wall in the next roomor is it lying loose on the floor?

10. A pulse is sent along a thin rope that is attached to a thick rope,which is tied to a wall, as shown in Figure 14–10.a. What happens when the pulse reaches point A? Point B?b. Is the pulse reflected from point A displaced in the same

direction as the incident pulse, or is it inverted? What about the pulse reflected from point B?

FIGURE 14–9

A B

FIGURE 14–11 When two equalpulses meet there is a point,called the node, (N), where themedium remains undisturbed (a).Constructive interference resultsin maximum displacement at theantinode, (A), (b). If the oppositepulses have unequal amplitudes,cancellation is incomplete (c).

N

N

N

N

N

1

2

3

4

5

A

A

A

1

A5

2

A

4

3


Superposition of WavesSuppose a pulse traveling down a spring meets a reflected pulse com-

ing back. In this case, two waves exist in the same place in the mediumat the same time. Each wave affects the medium independently. The dis-placement of a medium caused by two or more waves is the algebraicsum of the displacements caused by the individual waves. This is calledthe principle of superposition. In other words, two or more waves cancombine to form a new wave. If the waves are in opposite directions,they can cancel or form a new wave of less or greater amplitude. Theresult of the superposition of two or more waves is called interference.

Wave interference Wave interference can be either constructive ordestructive. When two pulses with equal but opposite amplitudes meet,the displacement of the medium at each point in the overlap region isreduced. The superposition of waves with equal but opposite ampli-tudes causes destructive interference, as shown in Figure 14–11a.When the pulses meet and are in the same location, the displacement iszero. Point N, which doesn’t move at all, is called a node. The pulsescontinue to move and eventually resume their original form. As in con-structive interference, the waves pass through each other unchanged.

Constructive interference occurs when the wave displacements arein the same direction. The result is a wave that has an amplitude largerthan any of the individual waves. Figure 14–11b shows the constructiveinterference of two equal pulses. A larger pulse appears at point A whenthe two waves meet. Point A has the largest displacement and is calledthe antinode. The two pulses pass through each other without

a b

N1

2

4

3

N

N

N

c

changing their shapes or sizes. If the pulses have opposite and unequalamplitudes, the resultant pulse at the overlap is the algebraic sum of thetwo pulses, as shown in Figure 14–11c.

Continuous waves You have read how wave pulses move througheach other and are reflected from boundaries. What happens when con-tinuous waves meet a boundary? Figure 14–12a shows a continuouswave moving from a region with higher speed to a region with lowerspeed. On the left-hand side of the boundary, the amplitude of thereflected wave has been added to that of the incident wave because someof the wave energy has been reflected.

Recall that velocity of a wave is the product of wavelength and frequency, v � �f. Thus, the transmitted wave on the other side of theboundary has a shorter wavelength because of its slower speed. Thetransmitted wave also has a smaller amplitude because less wave energyis available. Notice in Figure 14–12b how the relative amplitudes andwavelengths change when a continuous wave moves from a region withlower speed to one with higher speed.

Standing waves You can use the concept of superimposed waves tocontrol the frequency and formation of waves. If you attach one end ofa rope or coiled spring to a fixed point such as a doorknob, and thenstart to vibrate the other end, the wave leaves your hand, is reflected atthe fixed end, is inverted, and returns to your hand. When it reachesyour hand, it is inverted and reflected again. Thus, when the wave leavesyour hand the second time, its displacement is in the same direction itwas when it left your hand the first time.

But what if you want to magnify the amplitude of the wave you cre-ate? Suppose you adjust the motion of your hand so that the period ofthe rope’s vibration equals the time needed for the wave to make oneround-trip from your hand to the door and back. Then, the displacement


FIGURE 14–12 In each of thetwo examples below, continuouswaves pass from the medium onthe left to the medium on theright. Wave speed depends uponthe medium. For example, lightwaves travel faster in air thanthey do in water.

Incident + Reflected wave Transmitted wave

Higher speedLonger wavelength

Lower speedShorter wavelength

Boundary

v1 v2–v1

Incident + Reflected wave Transmitted wave

Lower speedShorter wavelength

Higher speedLonger wavelength

Boundary

v1 v2–v1

Dis

pla

cem

ent

a b

Pocket LabWave Interaction

What happens to the wavescoming from different direc-tions when they meet? Dothey slow down, bounce offeach other, or go througheach other?Design an ExperimentUse a coiled spring toy to testthese questions. Record yourprocedures and observations.

FIGURE 14–13 Interferenceproduces standing waves in arope. As the frequency of vibra-tion is increased, as shown fromtop to bottom, the numbers ofnodes and antinodes increase.


given by your hand to the rope each time will add to the displacementof the reflected wave. As a result, the oscillation of the rope in one seg-ment will be much larger than the motion of your hand, as expectedfrom your knowledge of constructive interference. This large-amplitudeoscillation is an example of mechanical resonance. The nodes are at theends of the rope and an antinode is in the middle, as shown in Figure14–13 top. Thus, the wave appears to be standing still and is called astanding wave. If you double the frequency of vibration, you can pro-duce one more node and one more antinode in the rope. Now itappears to vibrate in two segments, as in Figure 14–13 center. Furtherincreases in frequency produce even more nodes and antinodes, asshown in Figure 14–13 bottom.

Waves in Two DimensionsYou have studied waves on a rope or spring reflecting from a rigid

support, where the amplitude of the wave is forced to be zero bydestructive interference. These mechanical waves move in only onedimension. However, waves on the surface of water move in two dimen-sions, and sound waves and electromagnetic waves will later be shownto move in three dimensions. How can two-dimensional waves bedemonstrated?

Reflection of waves in two dimensions A ripple tank can be usedto show the properties of two-dimensional waves. A ripple tank containsa thin layer of water. Vibrating boards produce wave pulses or, in thiscase, traveling waves of water with constant frequency. A lamp above thetank produces shadows below the tank that show the locations of thecrests of the waves. Figure 14–14a shows a wave pulse traveling towarda rigid barrier that reflects the wave. The incident wave moves upward.The reflected wave moves to the right.

Pocket LabBent Out of Shape

What happens to a water wave’sspeed when the depth of thewater changes? How does achange in speed affect the shapeof the waves? Try this activity tofind out. Use a wave tank witha projection system. Adjust thetank so that the water is shallowon one side and deep on theopposite side. Dip your finger or pencil eraser into the middleof the tank, and gently tap thewater to make a circular wave.Watch closely. Do the waves hit the sides of the tank at thesame time? What happens tothe wavelength in differentdirections?Modeling Make a sketch ofthe shape of the waves. Labelthe deep and shallow ends onyour drawing. Describe therelationship between the depthof the water and the wave’sspeed (inverse or direct). Whatdid you notice about the wave-length in different directions?

The direction of waves moving in two dimensions can be modeled byray diagrams. Ray diagrams model the movement of waves. A ray is aline drawn at a right angles to the crests of the waves. Figure 14–14bshows the ray diagram for the wave in the ripple tank. The ray repre-senting the incident ray is the arrow pointing upward. The ray repre-senting the reflected ray points to the right.

The direction of the barrier is also shown by a line, which is drawn at a right angle to the barrier. This line is called the normal. The anglebetween the incident ray and the normal is called the angle of incidence. The angle between the normal and the reflected ray is calledthe angle of reflection. The law of reflection states that the angle ofincidence is equal to the angle of reflection.

Refraction of waves in two dimensions A ripple tank also can beused to model the behavior of waves as they move from one mediuminto another. Figure 14–15a shows a glass plate placed in a ripple tank.The water above the plate is shallower than the water in the rest of thetank, and the water there acts like a different medium. As the wavesmove from deep to shallow water, their wavelength decreases, and thedirection of the waves changes. Because the waves in the shallow waterare generated by the waves in deep water, their frequency is not changed.Based on the equation v � �f, the decrease in the wavelength of thewaves means that the velocity is lower in the shallower water. This issimilar to what happens to a sound wave in the air that collides withand then travels through another medium such as a wall.

This same phenomenon can be seen at the coast when the land gen-tly slopes down into the sea. In Figure 14–15b, the waves approach theshore. The ray direction is not parallel to the normal. Not only does thewavelength decrease over the shallower bottom, but also the directionof the waves changes. The change in the direction of waves at the bound-ary between two different media is known as refraction.


Barrier

Incident ray

Reflected ray

Normal

r

iθ

θ

FIGURE 14–14 A wave pulsein a ripple tank is reflected bya barrier (a). The ray diagrammodels the wave in timesequence as it approaches thebarrier and then is reflected tothe right (b).

a b

FIGURE 14–15 As the waterwaves move over a shallowerregion of the ripple tank where atriangular glass plate is placed,they slow down and their wave-length decreases (a). Whenwaves approach the shore, theyare refracted by the change indepth of the water (b).


Diffraction and Interference of WavesIf particles are thrown at a barrier with holes in it, the particles will

either reflect off the barrier or pass straight through the holes. Whenwaves encounter a small hole in a barrier, however, they do not passstraight through. Rather, they bend around the edges of the barrier,forming circular waves that radiate out, as shown in Figure 14–16. Thespreading of waves around the edge of a barrier, such as a small barriercoral reef, is called diffraction. Diffraction also occurs when wavesmeet a small obstacle. They can bend around the obstacle, producingwaves behind it. The smaller the wavelength in comparison to the sizeof the obstacle, the less the diffraction.

If a barrier has two closely spaced holes, the waves are diffracted byeach hole and form circular waves, as shown in Figure 14–17a. Thesetwo sets of circular waves interfere with each other. There are regions ofconstructive interference where the resulting waves are large, and bandsof destructive interference where the water remains almost undisturbed.Constructive interference occurs where two crests or two troughs of thecircular waves meet. The antinodes formed lie on antinodal lines that

FIGURE 14–16 Waves thatbend around the edges of barriers demonstrate diffraction.

a

b

radiate outward from the barrier, Figure 14–17b. Between these anti-nodal lines are areas where a crest of one wave meets a trough fromanother wave. Destructive interference produces nodes where the wateris undisturbed. The lines of nodes, or nodal lines, lie between adjacentantinodal lines. Thus, it is interference of the water waves that producesthe series of light and dark lines observed in Figure 14–17a.


Section Review1. Which of the following wave

characteristics remain unchangedwhen a wave crosses a boundary into a different medium: frequency,amplitude, wavelength, velocity, or direction?

2. A rope vibrates with the two wavesshown in Figure 14–18. Sketch theresulting wave.

3. Describe diffraction. How can diffrac-tion lead to interference?

4. Would you expect high-frequency orlow-frequency sound waves to bemore diffracted when they passthrough an open door? Explain.

5. Critical Thinking As another way tounderstand wave reflection, cover theright-hand side of each drawing inFigure 14–11a with a piece of paper.The edge of the paper should be atpoint N, the node. Now, concentrateon the resultant wave, shown in blue.Note that it acts like a wave reflectedfrom a boundary. Is the boundary arigid wall or open ended? Repeat thisexercise for Figure 14–11b.

14.2

FIGURE 14–17 Waves are diffracted at two openings in thebarrier. At each opening, circularwaves are formed that interferewith each other. Points of con-structive interference appear asbright spots in the photograph(a). Lines of constructive inter-ference, called antinodal lines,occur where crest meets crest (b).

ba

FIGURE 14–18

CHAPTER 14 REVIEW


14.1 Wave Properties• Waves transfer energy without transfer-

ring matter.• Mechanical waves require a medium.• A continuous wave is a regularly repeat-

ing sequence of wave pulses.• In transverse waves, the displacement

of the medium is perpendicular to thedirection of wave motion. In longitudi-nal waves, the displacement is parallelto the wave direction. In surface waves,matter is displaced in both directions.

• The wave source determines the fre-quency of the wave, f, which is thenumber of vibrations per second.

• The wavelength of a wave, �, is theshortest distance between points wherethe wave pattern repeats itself.

• The medium determines wave speed,which can be calculated for continuouswaves using the equation v � �f.

14.2 Wave Behavior• When a wave crosses a boundary

between two media, it is partially trans-mitted and partially reflected, depend-ing on how much the wave velocities inthe two media differ.

• When a wave moves to a medium withhigher wave speed, the reflected wave isinverted. When moving to a mediumwith lower wave speed, the displace-ment of the reflected wave is in thesame direction as the incident wave.

• The principleof superposi-tion statesthat the dis-placement of a medium resulting fromtwo or more waves is the algebraic sum of the displacements of the indi-vidual waves.

• Interference occurs when two or morewaves move through a medium at thesame time.

• Destructive interference results indecreased wave displacement with itsleast amplitude at the node.

• Constructive interference results inincreased wave displacement with itsgreatest amplitude at the antinode.

• A standing wave has stationary nodesand antinodes.

• When two-dimensional waves arereflected from boundaries, the angles ofincidence and reflection are equal.

• The change in direction of waves at theboundary between two different mediais called refraction.

• The spreading of waves around a barrieris called diffraction.

Key Terms

14.1• wave• wave pulse• continuous

wave• transverse

wave• longitudinal

wave• surface wave• trough• crest• wavelength• frequency

14.2• incident wave• reflected wave• principle of

superposition• interference• destructive

interference• node• constructive

interference• antinode• standing wave• law of reflection• refraction• diffraction

Summary

Key Equations

14.1

Reviewing Concepts

f � �1

T� v � �f

Section 14.11. How many general methods of energy

transfer are there? Give two examplesof each.

2. What is the primary differencebetween a mechanical wave and anelectromagnetic wave?

3. What are the differences among trans-verse, longitudinal, and surface waves?

4. Suppose you send a pulse along arope. How does the position of apoint on the rope before the pulsearrives compare to the point’s posi-tion after the pulse has passed?

CHAPTER 14 REVIEW

Chapter 14 Review 345

5. What is the difference between a wave pulseand a continuous wave?

6. Describe the difference between wave frequencyand wave velocity.

7. Suppose you produce a transverse wave byshaking one end of a spring from side to side.How does the frequency of your hand comparewith the frequency of the wave?

8. Waves are sent along a spring of fixed length.a. Can the speed of the waves in the spring be

changed? Explain.b. Can the frequency of a wave in the spring be

changed? Explain.9. What is the difference between the speed of a

transverse wave pulse down a spring and themotion of a point on the spring?

10. Suppose you are lying on a raft in a wave pool.Describe, in terms of the waves you are riding,each of the following: amplitude, period, wave-length, speed, and frequency.

11. What is the amplitude of a wave and what doesit represent?

12. Describe the relationship between the ampli-tude of a wave and the energy it carries.

Section 14.213. When a wave reaches the boundary of a new

medium, part of the wave is reflected and partis transmitted. What determines the amount ofreflection?

14. A pulse reaches the boundary of a medium inwhich the speed of the pulse becomes higher. Isthe reflection of the pulse the same as for theincident pulse or is it inverted?

15. A pulse reaches the boundary of a medium inwhich the speed is lower than the speed of themedium from which it came. Is the reflectedpulse erect or inverted?

16. When a wave crosses a boundary between athin and a thick rope, its wavelength and speedchange, but its frequency does not. Explain whythe frequency is constant.

17. When two waves interfere, is there a loss ofenergy in the system? Explain.

18. What happens to a spring at the nodes of astanding wave?

19. A metal plate is held fixed in the center andsprinkled with sugar. With a violin bow, the

plate is stroked along one edge and made tovibrate. The sugar begins to collect in certainareas and move away from others. Describethese regions in terms of standing waves.

20. If a string is vibrating in four parts, there arepoints where it can be touched without dis-turbing its motion. Explain. How many ofthese points exist?

21. How does a spring pulse reflected from a rigidwall differ from the incident pulse?

22. Describe interference. Is interference a propertyof only some types of waves or all types of waves?

Applying Concepts23. Suppose you hold a 1-m metal bar in your

hand and hit its end with a hammer, first, in adirection parallel to its length, second, in adirection at right angles to its length. Describethe waves you produce in the two cases.

24. Suppose you repeatedly dip your finger into asink full of water to make circular waves. Whathappens to the wavelength as you move yourfinger faster?

25. What happens to the period of a wave as thefrequency increases?

26. What happens to the wavelength of a wave asthe frequency increases?

27. Suppose you make a single pulse on a stretchedspring. How much energy is required to make apulse with twice the amplitude?

28. Sonar is the detection of sound waves reflectedoff boundaries in water. A region of warmwater in a cold lake can produce a reflection, ascan the bottom of the lake. Which would youexpect to produce the stronger echo? Explain.

29. You can make water slosh back and forth in ashallow pan only if you shake the pan with thecorrect frequency. Explain.

30. AM-radio signals have wavelengths between600 m and 200 m, whereas FM signals havewavelengths of about 3 m, Figure 14–19.Explain why AM signals can often be heardbehind hills whereas FM signals cannot.

FIGURE 14–19

346

31. In each of the four waves in Figure 14–20, thepulse on the left is the original pulse movingtoward the right. The center pulse is a reflectedpulse; the pulse on the right is a transmittedpulse. Describe the boundaries at A, B, C, and D.

ProblemsSection 14.132. The Sears Building in Chicago sways back and

forth in the wind with a frequency of about0.10 Hz. What is its period of vibration?

33. An ocean wave has a length of 10.0 m. A wavepasses a fixed location every 2.0 s. What is thespeed of the wave?

34. Water waves in a shallow dish are 6.0 cm long.At one point, the water oscillates up and downat a rate of 4.8 oscillations per second.a. What is the speed of the water waves?b. What is the period of the water waves?

35. Water waves in a lake travel 4.4 m in 1.8 s. Theperiod of oscillation is 1.2 s.a. What is the speed of the water waves?b. What is their wavelength?

36. The frequency of yellow light is 5.0 � 1014 Hz.Find the wavelength of yellow light. The speedof light is 300 000 km/s.

37. AM-radio signals are broadcast at frequenciesbetween 550 kHz and 1600 kHz (kilohertz)and travel 3.0 � 108 m/s.

a. What is the range of wavelengths for thesesignals?

b. FM frequencies range between 88 MHz and108 MHz (megahertz) and travel at the samespeed. What is the range of FM wavelengths?

38. A sonar signal of frequency 1.00 � 106 Hz hasa wavelength of 1.50 mm in water.a. What is the speed of the signal in water?b. What is its period in water?c. What is its period in air?

39. A sound wave of wavelength 0.70 m and velo-city 330 m/s is produced for 0.50 s.a. What is the frequency of the wave?b. How many complete waves are emitted in

this time interval?c. After 0.50 s, how far is the front of the wave

from the source of the sound?40. The speed of sound in water is 1498 m/s. A

sonar signal is sent straight down from a shipat a point just below the water surface, and1.80 s later the reflected signal is detected. Howdeep is the ocean beneath the ship?

41. The time needed for a water wave to changefrom the equilibrium level to the crest is 0.18 s.a. What fraction of a wavelength is this?b. What is the period of the wave?c. What is the frequency of the wave?

42. Pepe and Alfredo are resting on an offshore raftafter a swim. They estimate that 3.0 m sepa-rates a trough and an adjacent crest of surfacewaves on the lake. They count 14 crests thatpass by the raft in 20.0 s. Calculate how fast thewaves are moving.

43. The velocity of the transverse waves producedby an earthquake is 8.9 km/s, and that of thelongitudinal waves is 5.1 km/s. A seismographrecords the arrival of the transverse waves 73 sbefore the arrival of the longitudinal waves.How far away was the earthquake?

44. The velocity of a wave on a string depends onhow hard the string is stretched, and on themass per unit length of the string. If FT is thetension in the string, and � is the mass/unitlength, then the velocity, v, can be determined.

v � ��F

�T��

CHAPTER 14 REVIEW

Waves and Energy Transfer

A

B

C

D

FIGURE 14–20

Chapter 14 Review 347

A piece of string 5.30 m long has a mass of15.0 g. What must the tension in the string be to make the wavelength of a 125-Hz wave 120.0 cm?

Section 14.245. Sketch the result for each of the three cases

shown in Figure 14–21, when centers of thetwo wave pulses lie on the dashed line so thatthe pulses exactly overlap.

46. If you slosh the water back and forth in a bath-tub at the correct frequency, the water rises first at one end and then at the other.Suppose you can make a standing wave in a150-cm-long tub with a frequency of 0.30 Hz.What is the velocity of the water wave?

47. The wave speed in a guitar string is 265 m/s.The length of the string is 63 cm. You pluck thecenter of the string by pulling it up and lettinggo. Pulses move in both directions and arereflected off the ends of the string.a. How long does it take for the pulse to move

to the string end and return to the center?b. When the pulses return, is the string above

or below its resting location?c. If you plucked the string 15 cm from one

end of the string, where would the two pulses meet?

Critical Thinking Problems48. Gravel roads often develop regularly spaced

ridges that are perpendicular to the road. Thiseffect, called washboarding occurs becausemost cars travel at about the same speed andthe springs that connect the wheels to the cars oscillate at about the same frequency. Ifthe ridges are 1.5 m apart and cars travel atabout 5 m/s, what is the frequency of thesprings’ oscillation?

Going FurtherApplying Calculators or Computers Use agraphing calculator or computer program to plotthe following equation that describes a snapshotof a wave at a fixed time: y � A sin (2� x/�),where y is the displacement, � the wavelength,x the distance along the wave, and A the amplitude. Evaluate this equation for radians,not degrees. Start with A � 10, � � 6.28, and let x vary from 0 to 1. Repeat plotting forshorter and longer wavelengths and larger and smaller amplitudes. Display your printedgraphs that describe the wave that each set of data represents.

CHAPTER 14 REVIEW

1

2

3

FIGURE 14–21

Extra Practice For more practice solving problems, go to Extra Practice Problems, Appendix B.

PHYSICSTo review content, do the interactive quizzes on theGlencoe Science Web site atscience.glencoe.com


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