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Waves and SoundD

292 Unit D

U N I T

09-PHYSICS-11SE-Ch09.indd 29209-PHYSICS-11SE-Ch09.indd 292 7/23/10 1:20:54 PM7/23/10 1:20:54 PM

Unit Contents

Waves transmit energy.

9.1 What Are Waves?

9.2 The Properties of Waves

9.3 Superposition and Interference

Sound is a longitudinal mechanical wave.

10.1 Characteristics of Sound Waves

10.2 Resonance, Standing Waves, and Musical Sound

10.3 Motion Affects Waves

In the Unit Task, you will use a sound frequency generator

to measure the range over which you are able to hear sound.

You will measure both the lowest and highest frequencies

that you can hear. You will gather similar data from some of

your classmates, people in their 30s, and older. You will then

analyze your data to determine whether your acoustic range

changes as you age.

Unit Task

10

DISCOVERING PHYSICSHow does a spider find its lunch? The intricate pattern of the spider web not only acts as a net with which to snare insects, but also as a communication network. The struggling insect sends vibrations racing along the strands of the spider web. The spider can quickly locate its prey by sensing the frequency and strength of these vibrations, as well as determine how big its lunch is going to be. During courtship, a prospective suitor male spider will rapidly vibrate the web of the female. Is he just announcing his presence or preparing to be her next meal?

9


LearningExpectations

294 Unit D ©P

What do bats and dolphins have in common? The phrase “blind as a bat” states a common misconception. Bats have some vision using light, but when placed in pitch-black rooms

crisscrossed with fine wires, they can easily fly around and locate tiny flying insects for food. Dolphins have shown that they can quickly locate and retrieve objects even when they are blindfolded. We usually assume that vision requires light, but both bats and dolphins have evolved the ability to “see” using sound waves (Figure 9.1).

Research in science and technology has developed “eyes” that enable humans to see using sound waves, that is, navigate with senses other than sight. Medicine uses ultrasound (frequencies above the audible range for humans) to look at features such as a fetus or a tumour inside the body. Submarines can circumnavigate the globe without surfacing by using sound waves to explore their underwater environment.

Waves are one of nature’s most common methods to transmit energy between locations. Seismic waves generated during earthquakes can topple buildings or create tsunamis, sound waves carry conversations, and light waves stimulate the cells in the retina of the eye that enable vision. In all cases, waves do this by transmitting energy.

Waves transmit energy.

C H A P T E R

9

By the end of this chapter, you will:

Relating Science to Technology, Society, and the Environment

● analyze how properties of mechanical waves and sound influence the design of structures and technological devices

Developing Skills of Investigation and Communication

● use appropriate terminology related to mechanical waves and sound

● conduct laboratory inquiries or computer simulations involving mechanical waves and their interference

● investigate the relationship between the wavelength, frequency, and speed of a wave, and solve related problems

Understanding Basic Concepts

● distinguish between longitudinal and transverse waves in different media, and provide examples of both types of waves

● explain the components of resonance, and identify the conditions required for resonance to occur in vibrating objects and in various media

● explain and graphically illustrate the principle of superposition with respect to standing waves

● identify the properties of standing waves and explain the conditions required for standing waves to occur

● explain selected natural phenomena with reference to the characteristics and properties of waves

Figure 9.1 Dolphins and bats have evolved sophisticated ways to use the energy transmitted by waves to explore their environments.


Chapter 9 Waves transmit energy. 295©P

What Are Waves?

When a surfer catches a wave, many people assume that the forward motion of the surfer is the result of the forward motion of the water in the wave (Figure 9.2). However, evidence indicates that in a deep-water wave the water does not, in general, move in the direction of the wave motion. Rather, the surfer glides down the surface of the wave just as a skier glides down the surface of a ski hill. Like the skier, the surfer can move across the face of the wave as well as slide down the wave. But, unlike the ski hill, the water in the wave front is constantly rising. So, even though the surfer is sliding down the front of the wave, he or she never seems to get much closer to the bottom of the wave.

It is a common misconception that the water in a wave moves in the direction in which the waves are travelling. This may be because waves arriving at the shoreline move water to and fro across the sand. This motion is a feature of the interaction of the wave with the sloping shoreline rather than the actual motion of the wave itself. In deep water, there is only very limited motion of water when a wave moves past a particular point.

The surfer in Figure 9.2 has ac quired kinetic energy and this energy came from the wave. This example demonstrates one of the most fundamental properties of waves — waves transport energy.

9.1

Section Summary

● Parts of a wave include the crest, trough, amplitude, and wavelength.

● Transverse waves and longitudinal waves are two different types of mechanical waves.

● A longitudinal wave consists of rarefactions and compressions.

Figure 9.2 Surfers use a wave’s energy to speed their boards across the water.


296 Unit D Waves and Sound ©P

Most Waves Need a MediumWhen a stone is thrown into a still pond or lake, a ripple moves outward in ever-enlarging concentric circles (Figure 9.3). The water is the transporting medium for the wave. A medium is a substance through which the wave moves. Most waves require a medium. The elastic property of the medium allows the wave to transport, or propagate, energy as the wave travels through the medium. In the case of the waves shown in Figure 9.3, the undisturbed surface of the water is known as the medium’s equilibrium position. Regions where the water rises above the equilibrium position are called crests and regions where the water is lower than its equilibrium position are called troughs. In the crest or trough, the magnitude of greatest displacement from the equilibrium is defined as the wave’s amplitude (A). The distance between successive crests or troughs is called a wavelength. The symbol for wavelength is the Greek letter lambda, λ (Figure 9.4).

Suggested Activity● D1 Inquiry Activity Overview on

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PHYSICS•SOURCE

AA

λ

λ crest

trough

equilibriumposition

Figure 9.4 Properties of a wavecrestscrestscrests

troughstroughstroughs

wavelengthwavelengthwavelength

Figure 9.3 Many of the terms used to describe wave motions come from the observation of waves on the surface of water.

C D

A

A and B are in phaseC, D, and E are in phase

B

Eλ

λ

λ

Figure 9.5 In-phase points along a wave have identical status relative to the equilibrium position and are separated by one wavelength, λ.

An even simpler wave disturbance is the wave pulse. A wave pulse is a small part of a wave — its crest or trough — that can be produced by a sudden displacement of a medium from its equilibrium position.

The leading edge of a wave is called the wave front. A wave front moving out from the point of origin toward a barrier is called an incident wave. A wave front moving away from the barrier is called a reflected wave. The concept of a wave implies a regular repetition of the motion of the medium through which the wave travels. As a result, many parts of the medium are moving in a motion that is identical to the motion of other points on the wave. These points are said to be in phase (Figure 9.5).



Different Kinds of WavesMost waves travel in a medium. Light waves are an exception — they do not need a medium in which to travel. Waves that require a medium in which to travel are called mechanical waves. For simplicity, we will use the term “wave” to describe mechanical waves in this and the next chapter.

Sound is an example of a mechanical wave that travels through air. Seismic waves travel through rock or water and are produced by earthquakes. A plucked string on a guitar or violin starts as a wave travelling along a tightly stretched string. In all cases, the wave results from displacing an elastic medium from its equilibrium position. Displacing a medium fromits equilibrium position requires a force. This force acts over a distance — the amplitude — so work is being done on the medium when a mechanical wave is produced. The energy of the resulting wave comes from the initial work done.

In general, there are two main ways in which a medium can be displaced from its equilibrium position. Thus, there are also two different ways in which a resulting wave or wave pulse can be created. The resulting two different types of waves are transverse waves and longitudinal waves.

Transverse WavesImagine a long spring resting on a table surface. The spring is in its equilibrium position. A simple way to disturb this equilibrium is to jiggle the spring perpendicular to its length, either side to side or up and down. If you do so, you will see a pulse move along the spring. Since the pulse is directed perpendicular to the spring’s length, it is called a transverse wave pulse (Figure 9.6).

If you continue to jiggle the spring side to side or up and down, you will create a transverse wave. A transverse wave is a wave that moves perpendicular to the medium in which it travels. When a transverse wave travels through a medium, the maximum distance that the medium is displaced from equilibrium is the same as the amplitude of the wave.

Concept Check

1. State a similarity and a difference between an incident wave and a reflected wave.

2. Explain the difference between the motion of a wave and the motion of the medium.

3. What would happen if the medium moved along with the wave? Use waves in a swimming pool as an example.

Hand starts atequilibriumposition of spring.

Front of pulse startsto move along spring.

Hand continuesto move up.

Hand startsto move down.

As the hand moves towardthe equilibrium position,the amplitude of the pulsemoves along the spring.

Hand is at maximumamplitude.

Pulse is completewhen the hand is atequilibrium position.

vhand � 0

vhand � 0

v � 0

vhand

vpulse

A

A

amplitude

lFigure 9.6 The length (l) of the pulse depends on the speed (v)of the pulse and the time (Δt) taken to complete the pulse.



Longitudinal WavesIf you compress a small section of a long spring and then release it, you will notice a disturbance racing along the length of the spring. This is an example of a longitudinal wave pulse. A series of longitudinal pulses creates a longitudinal wave. A longitudinal wave is a wave that moves parallel to the medium in which it travels. If a longitudinal wave travels along a spring, you will see areas in which the spring is stretched and other areas in which the spring is compressed. Areas where the spring is stretched are called rarefactions and areas where the spring is compressed are called compressions. In Figure 9.7, the distance between successive compressions or rarefactions is the wavelength of the longitudinal wave.

Waves in One, Two, and Three DimensionsWaves can travel in one, two, or three dimensions. In fact, the waves you most commonly experience each day are likely sound waves, which travel as three-dimensional wave fronts. Figure 9.8 shows one-dimensional waves along a string, two-dimensional waves in a pool of water, and three-dimensional waves in air. All of the characteristics of waves that you have learned thus far apply. The distance between successive wave crests is the wavelength and the distance from equilibrium to crest (or trough) is the amplitude.

A water wave is an example of a two-dimensional wave. If you watch a toy boat as a water wave passes by, you will see that the boat rises up and down but also moves forward and backward. Both motions combine to produce a circular or rolling motion. This also helps, in part, to explain the formation of breakers and large waves eagerly sought after by surfers. Figure 9.9 illustrates how a water wave works.

Explore More

How can you use a virtual ripple tank to learn about wave basics?

PHYSICS•SOURCE

spring in equilibrium position

compression

rarefaction

wavelength

compression

longitudinal wave in the spring

rarefactionFigure 9.7 Longitudinal waves occur when a medium vibrates in a direction parallel to the direction in which the wave moves.


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PHYSICS•SOURCE

A λλA

λ

Figure 9.8 A comparison of waves in (a) one, (b) two, and (c) three dimensions. In the three-dimensional wave, the thickness of the shells corresponds to the amplitude of the wave.

wave motion

Figure 9.9 When a wave moves across the surface of water, the water moves between crests and troughs by localized circular motions. This local circular motion moves water back and forth between a trough and the adjacent crest.

(a) (b) (c)



No blurringindicatesEk � 0Ep � maximum

Ek � 0Ep � 0

Waves and Rays Describing the direction in which waves in two or three dimensions travel is a bit more complicated than the case for one-dimensional waves. We will consider the case of two-dimensional waves in a ripple tank to develop these ideas.

When waves in a ripple tank are viewed from above, the wave fronts appear as a set of bright and dark bands (crests and troughs) (Figure 9.10). When we draw waves as seen from above, we use a line to represent a wave front along the top of a crest. The point halfway between two lines is the bottom of a trough. A series of concentric circles represents the waves generated by a point source.

Waves are in constant motion. At all points on a wave front, the wave is moving at right angles to the line of the crest. There are two ways to indicate this motion (Figure 9.11). You could draw a series of vector arrows at right angles to the wave front, with their length indicating the speed of the wave (small red arrows). Or, you could draw rays — lines indicating only the direction of motion of the wave front at any point where the ray and the wave front intersect. The rays in Figure 9.11 are called diverging rays since they spread out as they move away from the origin. A ray diagram is a sketch that shows the wave rays for a wave disturbance.

Energy Changes in Pulses and WavesHow does a wave pulse move in a medium such as a spring? A pulse has both elastic potential energy and kinetic energy. Elastic potential energy is the energy stored in an object when it is temporarily forced out of its normal shape. Kinetic energy is the energy possessed by an object due to its motion. Imagine that you have generated a side-to-side transverse wave in a spring. As a section of the spring moves from the equilibrium position to the top of the pulse, that section has both kinetic energy — it is moving sideways relative to the direction of the pulse — and elastic potential energy because it is stretched sideways. At the point on the pulse where the displacement is greatest, the coils of the spring are, for an instant, motionless. Then, the tension in the spring returns the coils to their equilibrium position.

In Figure 9.12, the blurring on the front and back segments of the pulse indicates the transverse motion and the presence of kinetic energy as well as elastic potential energy. At the top, there is no blurring because the coils are temporarily motionless (have zero kinetic energy). At that instant, that segment of the spring has only elastic potential energy. As it returns to its equilibrium position, the segment has, again, both kinetic and potential energies. The energy in a pulse moves along the spring by the sequential transverse motions of the coils.

Figure 9.10 View of a ripple tank from above

Figure 9.11 A point source generates waves that move outward as concentric circles with the source at their centre.

wavesource

crests

rays

velocityvectors

troughs

wavelength

wavefront

Figure 9.12 A transverse pulse is generated when a spring is given a sharp flip to the side. Arrows indicate the direction of motion of the coils.



Concept Check

1. Suppose you generate a small longitudinal pulse in a SlinkyTM stretched out on the floor, with the other end held tightly by a friend. How could you give the next pulse more energy?

2. In what form is the energy just before you release the SlinkyTM?

3. At what points along the SlinkyTM is the energy of the wave pulse kinetic energy only?

One important difference between waves of one, two, and three dimensions is the energy density of a wave front and how it changes as the wave moves. To understand this difference, concentrate on a single pulse or wave front. In a one-dimensional pulse, a wave front travels along a string with essentially constant amplitude. For example, consider a guitar string being plucked. If the wave crest is not losing energy, then it will continue to travel with constant energy. The situation is quite different for waves moving in two or three dimensions. In a two-dimensional wave, the wave front’s energy is being spread over a greater distance along the circumference of the wave front (Figure 9.13(a)). In a three-dimensional wave, the wave front’s energy is being spread over the surface of a wave front (Figure 9.13(b)). In each case, the energy density of the wave front, and hence the amplitude of the wave, decreases the farther the wave front moves from the source. For example, the amplitude of the ripples of water in a pond decrease the farther the wave front moves from the source, such as a stone, that created them.

Take It Further

How did the scorpion find its lunch? Like many insects, scorpions have highly sensitive vibration sensors in their legs. A nearby insect creates vibrations that travel as waves. Scorpions use their wave sensors to sense wave fronts and follow wave rays back to their prey. Research to find out more about how insects use the physics of waves.

PHYSICS•SOURCE

two-dimensional wavesmoving away from a source

(a)

hollow spheres moving away from thesource in a three-dimensional wave

(b)

Figure 9.13 A comparison of the energy density in (a) two-dimensional and (b) three-dimensional waves as the wave fronts move outward. The wave fronts in the three-dimensional wave are hollow spheres.



QuestionWhat are the mechanics by which pulses move through a medium?

Activity OverviewIn this experiment, you will study how a pulse moves through a medium by measuring the amplitude (A) and length (l) of the pulse.

Prelab QuestionsConsider the questions below before beginning this activity.

1. How do you generate a pulse in a medium?

2. How do you determine the amplitude and length of a pulse?

Pulses in an Elastic Medium

Inquiry Activity PHYSICS•SOURCE

REQUIRED SKILLS■ Analyzing patterns■ Reporting results

D1

Creating Two-Dimensional Waves with the Ripple Tank


REQUIRED SKILLS■ Analyzing patterns■ Reporting results

D2

To powersupply

paperscreen

generator support

generator motor

pointsources

light source

Figure 9.14 A transverse pulse

Figure 9.15 A longitudinal wave

QuestionHow do the incident and reflected waves interact when waves reflect from a straight barrier?

Activity OverviewIn this ripple tank experiment, the properties of a two-dimensional wave are analyzed. You are to observe the directions of motion of the incident waves and reflected waves and how these directions are related to each other. Some variables to be observed are the interactions that occur when the incident and reflected waves move in different directions through the same point in the ripple tank. As you observe the wave motions, you should identify which are the dependent and independent variables.


1. How do you distinguish between an incident and a refl ected wave in a medium?

2. How are the wave patterns created by point-source and straight-line generators different?Figure 9.16 The ripple tank and

its parts

DI Key Activity


Check and Reflect


9. The energy of a spherical wave front is spread uniformly over a spherical surface. The surface area of a sphere increases as the square of the radius. So, if you double the radius of a spherical wave front, its surface area increases by a factor of 22 � 4 times. If the total energy of the spherical wave is conserved, what must happen to the energy per square metre for a wave front if you double the radius of the wave front?

10. Why do sounds get fainter the farther you are from their sources?

11. What factor(s) determine how much energy is stored in a wave?

12. You may have seen hockey or football fans do “the wave” in a sports arena. Fans in sequence raise their arms and cheer, and a ripple runs through the crowd. Is this really a wave? If so, what is the medium and what is the wave?

13. Give an example not mentioned in the text of a one-, two-, and three-dimensional wave.

14. You generate a transverse pulse in a spring. (a) At what points along the pulse is the kinetic

energy greatest? How can you tell? (b) At what point along the pulse is the elastic

potential energy the greatest? How can you tell?

15. In each of the following situations, state the source of the wave and the medium through which it moves.

(a) a coach blows her whistle (b) a child stretches a skipping rope, causing it

to vibrate

16. In question 15, how can you tell that only the energy moves through the medium, not the medium itself?

17. How do you create a longitudinal wave in a stretched spring? How would you create a transverse wave in the same spring?

Reflection

18. Identify a concept that you studied in this section that you would like to learn more about.

Key Concept Review

1. If a wave pattern is created by a point source, what is the nature of the ray diagram that would represent the wave fronts?

2. Explain the relationship between the motion of a transverse wave and the motion of the medium through which it moves.

3. Explain how the medium moves when a longitudinal wave passes through it.

4. Explain the difference between motion of a wave and motion of the medium. Why are they not the same?

Connect Your Understanding

5. The figure below shows a ray diagram that represents the motion of a set of wave fronts. If you were observing these wave fronts in a ripple tank, describe what you would see.

6. Give a simple example demonstrating that sound waves carry energy.

7. Imagine dropping a stone into a calm pond. (a) Sketch the wave disturbance that you

would expect would be created.

(b) Explain why the amplitude of this wave must decrease as it moves outward from its source, even if energy is conserved.

8. You are sitting in a boat. Every three seconds, waves cause your boat to rise and fall a total distance of 1 m. What is the amplitude of the wave disturbance causing this up-and-down motion?

9.1

rays

Question 5

For more questions, go to PHYSICS•SOURCE



On a warm summer day in your backyard, you can probably hear bees buzzing around, even if they are a few metres away. That distinctive sound is caused by the very fast, repetitive up-and-down motion of the bees’ wings (Figure 9.17). Waves all share a common set of properties and mathematical description. You have already learned that waves have wavelength and amplitude. Wave motion takes place over time. The period and frequency of waves relate wave motion to time.

Period and FrequencyTake a closer look at the bumblebee. The motion of a bee’s wings repeats at regular intervals. Imagine that you can examine the bee flying through the air. If you start your observation when its wings are at the highest point (Figure 9.18(a)), you see them move down to their lowest point (Figure 9.18(c)), then back up again. When the wings are in the same position as when they started (Figure 9.18(e)), one complete oscillation has occurred. An oscillation is a repetitive back-and-forth motion. One complete oscillation is called a cycle.

The time required for one complete oscillation is the period (T). If the period of each cycle remains constant, then the wings are moving up and down with oscillatory motion. The number of oscillations per second is the frequency (f), measured in hertz (Hz). The equation that relates frequency and period is:

f � 1 _ T

Section Summary

● A cycle is a complete oscillation.

● The period of a wave is the time taken for one complete oscillation.

● The frequency of a wave is the number of oscillations per second.

● The universal wave equation relates speed, frequency, and wavelength of a wave.

● Waves undergo reflection when they meet a rigid barrier.

9.2 The Properties of Waves

Figure 9.17 The wings of a bee in flight make a droning sound because of their motion.

Figure 9.18 The bee’s wings make one full cycle from (a) to (e). The time for this motion is called the period.

(a) (b) (c) (d) (e)

Explore More

How do waves such as tsunamis interact with shorelines?

PHYSICS•SOURCE



Object Period (s) Frequency (Hz)

Bumblebee wings 0.00500 200

Hummingbird wings 0.0128 78.1

Medical ultrasound 1 � 10�8 to 1 � 10�6 106 to 108

Middle C on a piano 0.00382 262

Electrical current in a house 0.0167 60.0

Table 9.1 Periods and Frequencies of Common Items

Table 9.1 shows the period and frequency of a bee’s wings as it hovers, along with other common examples.

Another way to understand period is that it is the time (in seconds) between the passage of successive wave crests. If a wave crest passed you every 0.25 s, you would note the period of the wave as 0.25 s. This would also mean that four wave crests passedby each second, and hence the frequency of the wave is 1 _ 0.25 s or

4 Hz. Frequency and period are inversely related to each other. Figure 9.19 shows two different frequency and period combinations. Note from the graph that as frequency decreases, period increases.

1.00.0 0.5 1.5

period � 0.25 sfrequency � 4 Hz

period � 1.5 sfrequency � 0.67 HzFr

eque

ncy

(Hz)

Period (s)

2

0

4

6

8

2.0

Frequency vs. Period

10

12

Figure 9.19 A graph of the inverse relationship between period and frequency Concept Check

1. If you double the frequency of a wave, what happens to its period?

2. If you halve the frequency of a wave, what happens to its period?

3. If you want to increase the period of a wave, how must you change its frequency?

Example 9.1

You are at the beach and watching waves lap against the shore. Every 3.5 s a new wave arrives. What are the period and frequency for these waves?

GivenWaves arrive every 3.5 s.

Requiredperiod (T)frequency (f)

Analysis and SolutionThe information given is the period of the waves. So you already know that T � 3.5 s.

To find frequency, substitute into the equation f � 1 _ T

.

f � 1 _ 3.5 s

� 0.29 Hz

ParaphraseThe incoming water waves have a period of 3.5 s and frequency of 0.29 Hz.

Practice Problems1. What is the period of a wave

of frequency 10 Hz?

2. Every 5.0 s, a wave front reaches shore. What is the frequency of the wave?

3. How many 20-Hz waves would pass you in 2.0 s?

Answers1. 0.10 s

2. 0.20 Hz

3. 40

Take It Further

You have seen how waves reflect when they meet barriers. Do waves also transmit or transfer some of their energy past barriers? Write a brief summary of your findings.

PHYSICS•SOURCE



The Universal Wave EquationA very simple mathematical relationship combines the ideas of wavelength, frequency, and wave speed. Imagine a water wave of frequency 5 Hz and wavelength 1.2 m. Since the frequency is 5 Hz, in one second, five complete waves pass by your location. Each wave is 1.2 m in length, so the very first wave crest will be five wavelengths, or 5 � 1.2 m � 6 m, beyond you as the last wave crest passes in front of you. Since this all happens in one second, you conclude that the speed of the wave is 6 m/s (Figure 9.20).

This equation can be written as:

v � fλ

where v is the wave speed, f is the frequency of the wave, and λ is the wavelength.

This mathematical formula applies to all waves and is, therefore, called the universal wave equation.


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PHYSICS•SOURCE

A

t � 0.0 s

t � 1.0 s

A

5λ � 5(1.2 m)

f

Figure 9.20 Five waves, each 1.2 m in wavelength, passing by in one second

Explore More

How can you use the universal wave equation in experiments with sound?

PHYSICS•SOURCE

Example 9.2

You generate a series of waves in an 8.5-m-long clothesline by jiggling one end of the clothesline up and down rapidly. You notice that the wave crests are about 0.50 m apart and that the first pulse returns to your hand 2.5 s after you start moving the clothesline. Determine the frequency, wavelength, and speed of the wave that you have created.

GivenThe wave crests are 0.50 m apart. They travel 17 m (8.5 m there and 8.5 m back) in 2.5 s.

Requiredfrequency (f)wavelength (λ)speed of the wave (v)

Analysis and SolutionThe distance between wave crests is the same thing as the wavelength. So, you know that λ � 0.50 m.

The wave travels 17 m in 2.5 s, so the wave speed is

v � 17 m __ 2.5 s

� 6.8 m/s

Now you can use the universal wave equation to determine the frequency.

Since v � fλ, re-arrange this equation to solve for frequency.

f � vλ

� 6.8 m/s __ 0.50 m

� 13.6 Hz

ParaphraseThe wave disturbance that you created has a frequency of 14 Hz, a wavelength of 0.50 m, and a speed of 6.8 m/s.

Practice Problems1. What is the speed of a wave of

frequency 300 Hz and wavelength 1.15 m?

2. What is the wavelength of a 5000-Hz sound in air if the speed of sound is 344 m/s?

3. What is the frequency of a sound wave of wavelength 1.2 cm if the speed is 344 m/s?

Answers1. 345 m/s

2. 0.0688 m

3. 2.9 � 104 Hz



Wave Reflection If you attach a spring to a rigid support on a wall and then create a wave pulse by jiggling the free end, you will discover an important property of waves. The wave pulse will travel along the spring until it reaches the other end, at which point it will be reflected. However, the phase of the reflected wave pulse will be inverted. The phase of a wave refers to the way in which the various parts of the wave (crests and troughs) are positioned with respect to a given point. If the phase of a wave is inverted, then crests become troughs and troughs become crests. When a wave crest reflects from a rigid barrier, the phase of the reflected crest is inverted and the crest becomes a trough (Figure 9.21).

incident pulse

reflected pulse

v

vFigure 9.21 Reflection at a fixed end causes the pulse to be inverted.

Example 9.3

Figure 9.22 shows a wave pulse approaching a rigid barrier. Sketch the reflected wave pulse when it reaches point B. Assume that no energy is lost during reflection.

GivenA wave pulse

RequiredA sketch of the pulse after it has reflected from a rigid barrier

Analysis and SolutionThe wave pulse will have its phase inverted. This means that the wave will reflect with its amplitude flipped. Instead of a wave crest, it will be returning as a trough (Figure 9.23).

ParaphraseAn incident wave pulse with a positive amplitude will reflect with the sign of the amplitude reversed. Hence, a crest will be reflected as a trough.

B

Figure 9.22

B

Figure 9.23

Practice Problems1. A wave pulse approaches a barrier

with an amplitude of 1.0 cm. What is the amplitude of the reflected pulse?

Sketch the reflected pulses for the pulses shown in questions 2 and 3.

2.

3.

Answers1. 1.0 cm

2.

3.

Figure 9.24

Figure 9.25


page 307

PHYSICS•SOURCE



QuestionWhat is the relationship between the amplitude, length, and speed of a pulse?

Activity OverviewIn this activity, you will study the speed, amplitude, and length of pulses and determine the relationship among these variables.


1. How do you measure the speed of a pulse?

2. How can you control the speed of a pulse?

Pulses in a Spring: Speed, Amplitude, and Length


REQUIRED SKILLS■ Analyzing patterns■ Drawing conclusions

D3

Figure 9.26 Pulses in a spring have length, speed, and amplitude.

QuestionWhat happens to the amplitude of a wave pulse when it reflects from a fixed barrier?

Activity OverviewIn this activity, you will study the behaviour of wave pulses when the pulses reflect from a rigid barrier.


1. How does a pulse refl ect from a fi xed barrier?

2. How can you measure the amplitude of a pulse?

Reflection of Wave Pulses



D4

Figure 9.27 A pulse reflecting from a rigid barrier.


Check and Reflect


Key Concept Review

1. A 100-Hz wave and a 200-Hz wave travel through the same medium. Which wave has the longer wavelength?

2. What additional information would you need in question 1 to be able to determine the wavelength of each wave?

3. Sound waves travel through seawater at about 1500 m/s. What frequency would generate a wavelength of 1.25 m in seawater?

4. (a) What is the period of a wave that has a frequency of 55 Hz?

(b) What is the frequency of a wave that has a period of 5.0 s?

5. Three waves, shown below, travel through a medium with the same speed. Which of these waves has the highest frequency? Explain your reasoning.


6. Temperature changes in seawater affect the speed at which sound moves through it. A wave with a length of 2.00 m, travelling at a speed of 1500 m/s, reaches a section of warm water where the speed is 1550 m/s. What would you expect the wavelength in the warmer water to be?

7. The wavelength of a water wave is 0.040 m. If the frequency of the wave is 5.0 Hz, what is its speed?

8. The distance between successive crests in a series of water waves is 8.0 m. The crests travel 4.5 m in 9.0 s. What is the frequency of the waves?

9. A source with a frequency of 10 Hz produces water waves that have a wavelength of 6.0 cm. What is the speed of the waves?

10. A radio station broadcasts radio signals with a frequency of 93.5 MHz. If the radio waves travel at a speed of 3.00 � 108 m/s, what is their wavelength?

11. A wave in a rope travels at a speed of 5.0 m/s. If its wavelength is 2.6 m, what is its period?

12. While sitting in your canoe, you encounter a wave disturbance created by a passing motor boat. Your canoe bobs up and down once every 2.5 s and the average distance between the wave crests moving your boat is 3.6 m. How fast is the wave moving?

13. In question 12, you considered the effect that a passing wave would have on your canoe. How could you use the information given to determine the energy being carried by the wave disturbance? What additional information and measurements would you need to answer this question?

14. One of the most precise ways to measure the speed of light is to measure the wavelength of a known colour or frequency of light. One colour of light emitted by sodium atoms has a frequency of 5.08474 � 1014 Hz and a wavelength of 589.5924 nm. Use this information to determine the speed of light. (Pay attention to the use of significant digits.)

15. In the figure below, a wave pulse is travelling along a spring that is attached to a wall. In your notebook, draw the reflected wave pulse when point A, shown on the pulse, reaches the wall.

Reflection

16. Which concept in this section required the greatest change in your thinking? Explain how your thinking changed.

AQuestion 15


Question 5

9.2



If you have ever been to a concert hall or theatre to hear a symphony or see a musical or theatrical performance, you may have noticed that the musicians, singers, or actors do not play, sing, or speak directly into microphones. How is it possible that we can still hear them at the very back of the theatre or concert hall? The reason is because engineers and architects who design and build theatres and auditoriums take into account sound wave superposition and interference. The shape of the theatre, and the materials used to build it, are chosen based on the interference patterns they can create with the sound waves the performers produce. Since ancient times, architects have designed auditoriums with excellent acoustic properties. Today, architects and engineers use sophisticated computer technology to ensure that good acoustics can be heard in every seat in the concert hall or theatre.

Interference and Superposition of PulsesWhen waves travel through space, it is inevitable that they will cross paths with other waves. In nature, this occurs all the time. Imagine two people who sit facing each other and are speaking at the same time. As each person’s sound waves travel toward the other person, the waves must meet and pass simultaneously through the same point in space (Figure 9.28). Still, both people are able to hear the other person quite plainly. The waves obviously were able to pass through each other so that they reached the other person’s ears unchanged.

How do waves interact when they cross paths without passing through each other? When you observe two waves crossing in the ripple tank, things happen so quickly that it is difficult to see what is happening. Still, it is plain that the waves pass through each other. By sending two pulses toward each other in a spring, it is easier to analyze the events. It is helpful to imagine that the spring in which the pulses are travelling is an ideal, isolated system. The pulses then travel without loss of energy.

First, consider two upright pulses moving through each other. When two pulses pass through the same place in the spring at the same time, they are said to interfere with each other. In the section of the spring where interference occurs, the spring takes on a shape that is different from the shape of either of the pulses individually (Figure 9.29 on the next page). The new shape that the spring takes on is predicted by the principle of superposition:

The displacement of the combined pulse at each point of interference is the algebraic sum of the displacements of the individual pulses.

Section Summary

● The principle of superposition explains constructive and destructive interference of waves.

● Standing waves consist of nodes and antinodes.

● Resonance and resonant frequency have both positive and negative effects on humans and the environment.

9.3 Superposition and Interference

sound wavesfrom girl

sound wavesfrom boy

Figure 9.28 When two people talk simultaneously, each person’s sound waves reach the other person’s ears in their original form.


page 316

PHYSICS•SOURCE



This principle, based on the law of conservation of energy, makes it quite easy to predict the shape of the spring at any instant during which the pulses overlap.

pulse Bactual positionof spring

pulse A

pulse A

pulse A

pulse A

pulse B

pulse B

pulse B

x

x

y

y

z

z

region ofpulse overlap

vA vB

Region of overlapOriginal position of pulse AOriginal position of pulse BResultant

Figure 9.29 When two upright pulses move through each other, the displacement of the resultant pulse is the sum of the displacements of pulse A and pulse B. If at any point in the region of overlap the displacement of one pulse, shown here as x, y, and z, is added to the displacement of the other pulse, the displacement of the resultant pulse is increased.

pulse A

resultant pulse

pulse B

Figure 9.30 Constructive interference

Constructive InterferenceIn Figure 9.29, the two pulses have different sizes and shapes and are moving in opposite directions. The displacement of a pulse is positive for crests and negative for troughs. In Figure 9.29, both displacements are positive. So, at any point where the two pulses overlap, the displacement of the resultant pulse is greater than the displacements of the individual pulses. When pulses overlap to create a pulse of greater amplitude, the result is constructive interference (Figure 9.30).



Destructive InterferenceNow consider the case when an inverted pulse meets an upright pulse. The displacement of the inverted pulse is a negative value. When the displacements of these pulses are added together, the displacement of the resultant pulse is smaller than the displacement of either pulse. When pulses that are inverted with respect to each other overlap to create a pulse of lesser amplitude, the result is destructive interference (Figure 9.31).

pulse A

resultant pulse

pulse BFigure 9.31 Destructive interference

actual position of spring

pulse A

pulse A

pulse B

pulse B

point wherepulses meetpulse A

pulse B

pulse A

pulse B

xx

zz

yy

vA

vB

vB

vA

vB

vApulse A

pulse B

region ofoverlap

Figure 9.32 When identical pulses that are inverted with respect to each other overlap, the displacement of one pulse is reduced by the displacement of the other pulse. At any point in the region of overlap, the displacement of Pulse B, shown here as x, y, and z, reduces the displacement of Pulse A to produce the resultant. This is called destructive interference.

Region of overlapOriginal position of pulse AOriginal position of pulse BResultant

Figure 9.32 shows a special case of destructive interference. Two pulses that have the same shape and size are shown passing through each other. Because the pulses are identical in shape and size, their displacements at any position equidistant from the front of each pulse are equal in magnitude but opposite in sign. At the point where the two pulses meet, the sum of their displacements will always be zero. At the instant when these two pulses exactly overlap, the displacement at all points is zero and the pulses disappear. The resultant is a flat line. Immediately following this instant, the pulses reappear as they move on their way.



Example 9.4

Use the principle of superposition to show the resulting waveform when points A and B on the two waves in Figure 9.33 meet. Use the grid marks to measure wave amplitudes.

GivenWave pulses moving toward each other

RequiredSketch the shape of the pulses when points A and B meet by applying the principle of superposition.

Analysis and SolutionAccording to the principle of superposition, amplitudes add. Remember that the amplitude is positive if the wave is above the equilibrium position (a crest) and negative when it is below the equilibrium position (a trough). To add the waves, reposition the two waves so that points A and B overlap, and carefully add the corresponding parts of the two pulses (Figure 9.34).

Use the grid to add the waves. The leading parts of both wave pulses have amplitudes of �2 and �2 units, whereas at points A and B, the amplitudes are �1 and �2 units (Figure 9.34). Figure 9.35 shows the resulting combined wave at the instant points A and B overlap.

ParaphraseAt the instant the two points overlap, the result will be a wave pulse of amplitude �3 units.

AB

Practice ProblemsUse the principle of superposition to add the following waves when points A and B overlap.

1.

2.

Answers1.

2.

Figure 9.33

A

B

Figure 9.36

A

B

Figure 9.37

B

A

B

A�

A

Both waves have positiveamplitudes here.

These two partsof the wave pulseshave equal andopposite amplitudes.

B

Figure 9.34

A

B

Figure 9.35



Standing WavesWhen two waves with identical wavelengths and amplitudes move through each other, the resulting interference pattern can be explained by using the principle of superposition (Figure 9.38). When crests from the two waves or troughs from the two waves occupy the same point in the medium, the waves are in phase. Waves that are in phase produce constructive interference. When a crest from one wave occupies the same point in the medium as a trough from a second wave, we say that these waves are out of phase. Out-of-phase waves produce destructive interference. As the two waves pass through each other in opposite directions, they continually shift in and out of phase to produce a wave that seems to oscillate between fixed points, rather than move through the medium.

Concept Check

1. The principle of superposition can be used to explain why pulses are inverted when they reflect from the fixed end of a spring. Consider a wave reflecting from the fixed end of a spring.(a) What must the amplitude of a wave always be at the fixed end?(b) If you consider the amplitude at the fixed end to be the result of the

superposition of the incident and reflected waves, why does this necessarily imply that the reflected wave is inverted?

(c) Sketch a graph that shows how the incident and reflected waves would appear at the fixed point.

A

A

λ

A

λ14

14

actual position ofmedium at areawhere waves overlap

A

constructiveinterference

Apoints at whichonly destructiveinterference occurs

A

vA vB

wave Awave Bwave A � B

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9.38 The diagrams show how waves with identical amplitudes and wavelengths travelling in opposite directions interfere as they move through each other.



Looking at Figure 9.38 on the previous page:• Point A is the initial point of contact between the two waves shown in

blue and purple. The crest from the purple wave and the trough from the blue wave arrive at point A at the same instant.

• The two identical waves have moved a distance of 1 _ 4 λ in opposite directions. This movement results in destructive interference and the

spring is flat in the region of overlap. The position of the spring where the two waves overlap, the resultant, is shown in red.

• Each wave has moved a further 1 _ 4 λ. Now the waves are exactly in phase and constructive interference occurs. The regions to the left and right of

point A show a crest and a trough, respectively, with displacement of the resultant being twice that of the blue or purple waves.

• Every time the waves move a further 1 _ 4 λ, the interference changes from constructive to destructive and vice versa.

At point A, only destructive interference occurs. The magnitudes of the displacements of the waves arriving at point A are always equal but opposite in sign. As the waves continue to move in opposite directions, the nature of the interference continually changes. However, at point A and every 1 _ 2 λ from point A, there are points at which only destructive interference occurs. These points are called nodal points or nodes.

Between the nodes, the wave goes into a flip-flop motion as the interference in these areas switches from constructive (crest crossing crest) to destructive (crest crossing trough) and back to constructive interference (trough crossing trough). The midpoints of these regions on the wave are called antinodes. The firstantinode occurs at a distance of 1 _ 4 λ on either side of A, and thenat every 1 _ 2 λ after that point. Because the wave seems to oscillate around stationary nodes along the spring, it is known as a standing wave. Standing waves are also seen in nature. An example is shown in Figure 9.39.

Standing Waves in a Fixed Spring When you generate a wave in a spring that is fixed at one end, the reflected wave must pass back through the incident wave. These two waves have identical wavelengths and nearly identical amplitudes. For incident and reflected waves, the initial point of contact is by definition the fixed point at which reflection occurs. This means that the endpoint of the spring is always a nodal point and, as shown in Figure 9.40, nodesoccur every 1 _ 2 λ from that point with antinodes between them.

Figure 9.39 Standing waves occur in nature. This photograph shows a standing wave in a coffee cup.

antinodeNodes remainmotionless.

Antinodes oscillatebetween shown positions.

λ12

λ12

λ12

λ12

λ12

λ12

Figure 9.40 In a spring with a fixed end, a standing wave must contain a whole number of antinodes. Nodes occur every half-wavelength from the ends.



ResonanceWhen a standing wave is present in a spring, the wave reflects from both ends of the spring. There must be a nodal point at both ends with an integer number of antinodes in between. The spring naturally oscillates at those frequencies that will produce a standing wave pattern. A spring that has a standing wave is in a state of resonance. Resonance is the increase in amplitude of a wave due to the transfer of energy in phase with the natural frequency of the wave. The frequencies at which standing waves form are the resonant frequencies for the spring — the natural frequencies of vibration of an object that produce a standing wave pattern. When the generator is oscillating at a resonant frequency, the energy is added to the spring in phase with existing oscillations. This addition reinforces and enhances the standing wave pattern. The added energy works to construct waves with ever-larger amplitudes. If the generator is not oscillating at a resonant frequency of the medium, the oscillations tend to destroy the standing wave motion.

Amplitude and ResonancePerhaps the most impressive display of a standing wave occurred when resonance set up a standing wave in the bridge across the Tacoma Narrows in the state of Washington in the United States (Figure 9.41). Opened in November 1940, the bridge was in operation only a few months before resonance ripped it apart. More recently, in June 2000, the newly opened Millennium Bridge in London, England, had to be closed for modifications when the footsteps of pedestrians set up resonance patterns.

Anyone who has ever pumped their legs on a swing has used the principle of resonance. To increase the amplitude of its motion, the swing must be given a series of nudges in phase with its natural frequency of motion. Each time the swing begins to move forward, you give it a little push. Since these little pushes are produced in resonance with the swing’s natural motion, they are added to its energy and the amplitude increases. If you pushed out of phase with its natural motion, the swing would soon come to rest.

Concept Check

1. Why does it take so little energy to sustain a standing wave in a spring?

2. A standing wave is created between two fixed points. If the length between the two points is 2λ, where along the wave are the nodes and antinodes located?

3. How many nodes and antinodes does the standing wave in question 2 have?

Explore More

How can you experiment with the frequencies of your voice?

PHYSICS•SOURCE

Figure 9.41 Resonance, caused by wind, set up a standing wave that destroyed the Tacoma Narrows Bridge.

Take It Further

Taipei 101 is a freestanding office tower in Taiwan that is 448 m high. On the 88th floor is a 660-t steel sphere that acts as a tuned mass damper (TMD). Find out more about how TMDs are used in tall buildings to damp out harmful resonant frequencies.

PHYSICS•SOURCE



Concept Check

1. With a partner, think of three examples of resonance in everyday phenomena.

2. You can demonstrate resonance in two dimensions very easily with a StyrofoamTM cup and coffee (or any dark fluid). Fill the cup and slowly push the cup forward or back on a table surface. The friction between the cup and table will cause the cup to stick and slip, and thereby begin to vibrate. Sketch what you observe. Explain how this activity demonstrates resonance.

3. Describe the connection between standing waves and resonance. Can you have one without the other? Explain.

QuestionsPart 1

1. What happens when two pulses pass through the same point in a medium?

2. How can two waves, moving in opposite directions, exist simultaneously in the same space?

Part 2

3. What causes a standing wave?

Activity OverviewIn Part 1 of this lab, you will generate transverse pulses in a spring and observe their superposition and interference. In Part 2, you will generate standing waves at different frequencies in a spring.


1. How can you tell if two pulses have interfered?

2. How can you tell that you have created a standing wave in a spring?

Interference of Waves



D5

Figure 9.42 Two pulses interfering

Figure 9.43 A standing wave in a spring


Chapter 9 Waves transmit energy. 317

Check and Reflect

©P

Key Concept Review

1. Two nodes along a vibrating string are separated by 12 cm. What is the wavelength of the standing wave producing these nodes?

2. What does the term “standing wave” mean? Why do we refer to some waves as standing waves?

3. A standing wave of wavelength 23.6 cm is formed along a vibrating spring. What is the distance between any pair of successive nodes and antinodes?

4. What are resonant frequencies?


5. Two pulses of the same length (l) travel along a spring in opposite directions. The amplitude of the pulse from the right is three units whereas the amplitude of the pulse from the left is four units. Describe the pulse that would appear at the moment when they exactly overlap if

(a) the pulses are on the same side of the spring and

(b) the pulses are on opposite sides of the spring.

6. According to the principle of superposition, what must always occur at a standing wave node?

7. In the figure below, two wave pulses (A and B) are travelling toward each other. At time t � 0.0 s, they are 3 cm apart. (Each grid line represents 1 cm in this diagram.) Sketch the wave pulses at t � 1.0, 2.0, and 3.0 s.

8. A student is investigating the speed of a wave along a string by first finding the wavelength of the wave.

(a) Explain why it would be easier for the student to measure the distance between nodes rather than antinodes.

(b) How can she determine the wavelength from this information?

9. The student in question 8 finds that the distances between five successive nodes produced by a string vibrating at 60.0 Hz are 1.23, 1.20, 1.19, 1.22, and 1.23 cm.

(a) What is the average nodal distance that she should report?

(b) Explain how she can determine the value of the speed of the wave using the information given.

(c) What is the speed of the wave?

10. A clever squirrel that has studied physics finds a large, perfectly round nut in the bottom of a hemispherical bowl. The nut is too big for the squirrel to carry out of the bowl, but if the squirrel pushes it, the nut can be made to roll back and forth a short distance. How can our clever squirrel roll the nut out of the bowl? What important piece of physics will it use?

11. Explain why knowledge of standing waves is an important part of safe building and bridge design.

12. A standing wave is created on a long rope between two fixed ends that are 2.5 m apart. If there are five antinodes in this standing wave, what is the wavelength of this standing wave? Use a sketch to help you.

13. A washing machine sometimes bounces vigorously for a few moments at one or more of its spin cycle speeds. Using the principles you have learned in this section, explain the physics at work.

Reflection

14. Describe to a classmate one misconception youhad about standing waves before reading this section. Explain what you know about this concept now.

9.3

Question 10


A B

v � 1 cm/s

v � �1 cm/st � 0.0 s

Question 7


318 Unit D Waves and Sound

CHAPTER REVIEWC H A P T E R

©P

Key Concept Review

1. How many different examples of wave motion can you identify?

(a) Work with one or two classmates and try to identify at least five different ways in which you see wave motion in everyday things around you. k

(b) In each case, speculate as to whether you can see the actual waves or simply their effects. (Hint: As an analogy, consider whether you can see the wind or only the things it moves.) k

(c) For each case, discuss with a partner how we perceive the energy that the waves transport. k

2. What affects the speed of a wave in water? k

3. What is the nature of the motion of the medium when a longitudinal wave moves through it? k

4. Describe how the speed of a wave affects its wavelength and its amplitude. k

5. If speed is constant, how does wavelength vary with frequency? k

6. Describe the conditions required to produce constructive and destructive interference in waves. k

7. Describe how the principle of superposition applies to what happens when two pulses of identical length and amplitude interfere to produce no apparent pulse. k

8. Define node, antinode, and standing wave. k

9. In terms of the wavelength of the waves that have combined to form a standing wave, describe the position of the nodes and antinodes as you move away from the fixed end of a spring. k

10. When a wave moves on water, what is the nature of the motion of the water within the wave? k

11. What is the relationship between the direction of an incident wave and a reflected wave? k

12. A wave of amplitude 5 cm and frequency 10 Hz and a wave of amplitude 50 cm and frequency 10 Hz pass through the same medium. How do their speeds compare? k

13. While relaxing on a dock, you notice water waves moving past you. If the waves have a frequency of 0.25 Hz, how many waves would pass by you in 10 s? k


14. The speed of a wave in a spring is 15.0 m/s. (a) If the length of the pulse moving in the

spring is 2.00 m, how long did it take to generate the pulse? a

(b) Why don’t we talk about the frequency for a pulse? t

15. In a ripple tank, you measure the speed of a wave to be 12.0 cm/s in the deep section and 9.0 cm/s in the shallow section. If the waves in the deep section that are 11.5 cm long cross over to the shallow section, what would be the wavelength in the shallow section? t

16. The term “ultrasound” means that the frequency is higher than frequencies that human ears can detect (about 20 kHz). Animals can often hear sounds that, to our ears, are ultrasound. For example, a dog whistle has a frequency of 22 kHz. If the speed of sound in air is 350 m/s, what is the wavelength of the sound generated by the whistle? a

17. A spring is stretched to a length of 7.0 m. A frequency of 2.0 Hz generates a standing wave in the spring that has six antinodes.

(a) Sketch the standing wave pattern for the spring. a

(b) Calculate the speed of the wave. a

18. When you create a wave pulse in a stretched spring, it has both kinetic and potential energy. How can the pulse have kinetic and potential energy? (Hint: Think of the role that the medium plays.) k

19. Which of the two wave pulses shown below has the greater amount of energy? How do you know? t

9

A B

Question 19


Chapter 9 Review 319©P

20. Sketch what happens when the two wave pulses shown below overlap. What happens to the energy of the pulses? t

21. The figure below shows two waves that occupy the same point in space. Copy the sketch onto a sheet of paper using the dimensions indicated. Draw the wave that results from the interference of these two waves. k

22. If a frequency of 1.5 Hz generates a standing wave in a spring that has three antinodes, what frequency generates a standing wave with five antinodes in the same spring? t

23. A standing wave interference pattern is produced in a rope. The standing wave has a frequency of 28 Hz. If the wavelength of the wave is 40 cm, what is the distance between successive nodes? t

24. The distance between the second and fifth nodes in a standing wave is 30 cm.

(a) What is the wavelength of the waves? a (b) The frequency of the source is 12.5 Hz.

What is the speed of the waves? a

25. When timing the motion of a pendulum, you notice that it takes 5.2 s to make 10 complete swings.

(a) What is the frequency of the pendulum? a (b) What is its period? a

26. After a rain shower, you notice the curious pattern shown in the figure below, which is produced by water droplets (represented by

the red and green dots) dripping from two parts of a tree branch into a puddle. Use the knowledge that you have acquired in this chapter to comment on what you see. In particular, how does this example illustrate the principles of wave motion and wave superposition? t

27. In the figure below, a wave pulse is travelling with a speed of 1 cm/s toward a rigid barrier. Sketch the appearance of the pulse at t � 1.0 s, 2.0 s, and 5.0 s. The grid spacing is 1 cm in this diagram. k

28. Which wave travels with the higher speed: a wave of amplitude 10 cm, wavelength 8.3 m, and frequency 10 Hz, or a wave of amplitude 1.0 cm, wavelength 0.83 m, and frequency 1000 Hz? Justify your answer. t

Reflection

29. Which concept in this chapter did you find the most challenging? What would help you to understand this concept better? c

10 cm

3.5 cm 5.0 cm

A A

A � 1.5 cm

A A

Question 21

A B

Question 20

ACHIEVEMENT CHART CATEGORIESk Knowledge and understanding t Thinking and investigation

c Communication a Application

Question 26

v � 1 cm/s

t � 0.0 s

Question 27

In the Unit Task, you will be measuring and comparing

the range over which you hear sound. The variable

that you will be measuring is frequency. How is the

frequency of a sound related to its wavelength? Why is

the frequency range being measured in this task rather

than wavelength range?


Unit Task Link


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