Introduction toWaves and Sound

Principal Authors:Martin Mason, Mt. San Antonio College and Christine Carmichael,

Woodbury University

Based on the work ofJohn Terrell and Roger Edmonds, Middlesex Community College and Tom

O’Kuma, Lee College

This module was developed as a Special Project of the Physics Workshops for the 21st Century Project supported by Joliet Junior College (IL),Lee College (TX), and a grant (#0101589) from the Division of Undergraduate Education (DUE) of the Advanced Technological Education (ATE)

Program of the National Science Foundation (NSF)

Some of these materials contained herein were developed originally for the ICP21 Project under Grant DUE #9553665in the Advanced Technological Education Program of the National Science Foundation.

The National Science Foundation4201 Wilson Boulevard

Arlington, Virginia 22230, USA

The Properties of WavesYou have seen many types of waves in everyday life. Some of these waves

are easily seen, such as the waves in a pond or in an ocean. Others you can feelbut cannot see, such as the winds during a storm or the sensation of riding in a caron a bumpy road. You may have been told that light is a wave, and while you cansee light you can’t see the wave nature of the light. You also may know thatsound is a wave.

In this section you will learn what all of these types of waves have in commonand what distinguishes them from each other. You will also learn the terminologythat applies to all waves, so that a discussion of the specific properties of soundcan be related to waves in general. Finally, you will learn about the sources ofwaves and how the waves travel from one place to another.

What is a wave?

You have experienced waves many times in many different situations. In this introductoryExploration you will be asked to think about what you know about waves, so that you have aninformation base on which to build additional knowledge about waves.

1. The term “wave” has more than on meaning in everyday language. List as many differentmeanings of the word wave as you can. ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

2. What is a wave? Describe a wave in your own words. ____________________________________________________________________________________________________________________________________________________________________________

3. List as many types of waves and/or examples of waves as you can. Try to give at leastfive different examples. _____________________________________________________________________________________________________________________________________________________________________________________________________

4. Look at your answers to the previous question. What, if any, things do all of yourexamples have in common? ______________________________________________________________________________________________________________________________________________________________________________________________

5. Now consider the specific example of a sound wave. List all the ways you can think of tomake a sound wave. Again, try to give at least five different ways to make the soundwave. ___________________________________________________________________________________________________________________________________________________________________________________________________________________

6. What, if anything, do all of your examples of ways to make sound have in common?________________________________________________________________________________________________________________________________________________How do you think sound gets from one place to another? That is, how do you think soundtravels? _________________________________________________________________________________________________________________________________________________________________________________________________________________

7. What types of materials do you think sound travels through? What evidence do you haveto support your answer? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________When a thunderstorm is overhead what have you noticed about the time it takes for you tosee the lightning and to hear the thunder? What does this suggest about the differencesbetween light and sound?______________________________________________________________________________________________________________________________________________

Making Waves

In this Demonstration you will create two different types of waves and compare theirproperties.

Equipment• Slinky®

• Rope or Long Spring• Masking tape

8. Your instructor will ask a student volunteer to helpstretch the rope or spring across the room or hall. The student will generate a wave in therope by moving his or her hand in an up-and-down motion. Observe the wave created inthe rope by this motion. Describe the wave as completely as possible, includinginformation such as direction of motion of the wave, direction of motion of the rope, andanything else you think is important. Draw a sketch of the wave in the space provided.______________________________________________________________________________________________________________________________________________________________________________________

9. A small piece of tape will be attached somewhere near the center of the rope. The studentvolunteer will create a wave pulse in the rope by giving one end of the rope a quick flipwhile your instructor holds the other end stationary. Observe the motion of the pulse downthe rope and compare it to the motion of the piece of tape as the pulse passes the tape.How do these motions compare? Describe your observations. ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

10. The student will generate a continuous wave in the rope as in Step 1. Observe anddescribe the motion of the piece of tape as the wave travels in the rope.________________________________________________________________________________________________________________________________________________

11. Now your instructor will ask the student to generate a wave by moving his or her hand atthe same speed as in Step 1, but with a greater up-and-down motion. What changes doyou see in the wave? ______________________________________________________________________________________________________________________________

12. Your instructor will ask the student to generate the wave by moving his or her hand in afaster up-and-down motion than in Step 3. What changes in the wave do younotice?_______________________________ What if the student moves the rope up anddown slower than in Step 3? ______________________________What property or

properties of the wave seem to change when the speed of the motion of the handgenerating the wave changes? Record your observations for these changes.________________________________________________________________________________________________________________________________________________________________________________________________________________________

13. Now your instructor will ask a student to help stretch a Slinky on a tabletop until it is about1.5 meters long. The student will move one end of the Slinky back and forth, parallel to thetabletop. Observe the wave created in the Slinky by this motion. Describe the wave ascompletely as possible, including information such as direction of motion of the wave,direction of motion of the Slinky, and anything else you think is important. Draw a sketchof the wave in the space provided. ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

14. A small piece of tape will be attached to one of the coils somewhere near the center of theSlinky. The student will generate a wave pulse in the Slinky by giving the end of the Slinkya quick push with his or her hand while your instructor holds the other end stationary.Observe the motion of the pulse down the Slinky and compare it to the motion of the pieceof tape as the pulse passes the tape. How do these motions compare? Describe yourobservations. ____________________________________________________________________________________________________________________________________________________________________________________________________________

15. Now the student will generate a continuous wave in the Slinky by moving his or her handback and forth, as in Step 6. Observe and describe the motion of the piece of tape as thewave travels in the Slinky. __________________________________________________________________________________________________________________________________________________________________________________________________

16. Your instructor will now ask the volunteer to move his/her hand back and forth faster thanin Step 8. What changes in the wave do you notice? What if the hand moves back andforth slower than in Step 8? What property or properties of the wave seem to changewhen the speed hand generating the waves changes? Record your observations for thesechanges. ______________________________________________________________________________________________________________________________________________________________________________________________________________

17. Summarize what you have observed about the similarities and differences between thetype of wave generated in the rope and the type of wave generated in the Slinky.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Sound: Waves in Air

Previously you looked at waves in a string. Sound waves behave the same way, except thatsince they travel in air, we can’t see them. To study and identify the basic properties of soundwaves we will have to use our ears and the computer as our measuring instrument.

Sound waves are a common form of mechanical waves. Just like any other mechanical waves,they need a medium in which to move. They are longitudinal waves carried by the vibrations ofparticles in the medium. In a gas, like air, sound waves may also be thought of as a pressuredisturbance (variation from the equilibrium condition) which moves from one place to another.Sound waves moving through air may be captured by a microphone and displayed.

Equipment• Sound Analysis Center Software

(www.mtsacphysics.org)• Microphone• Computer• various sources of sound waves.

In this activity you will see how to collect and displayvarious sound waves on the screen.

1. Plug the microphone into the input of your soundcard(This is often identified with a smallpicture of a microphone). Open the Sound Analysis Center software. You should see Recordand Play buttons as well as a Sound level-time axes on the screen.

2. First, capture an "ooo" sound by clicking on the Record button with the mouse.Make a long, drawn out "ooo" sound into the microphone (as in "moo"). Try to keep theloudness of the sound constant. The graph you see represents the sound wave (in volts). Ifyou can’t see the pattern, use the Zoom In button to get a closer look at the signal.

a. Is the microphone collecting the wave disturbance at a fixed time or a fixed location?

________________________________________________________________________________________________________________________________________________

What would you need to be able to display the wave the other way? Explain.

________________________________________________________________________________________________________________________________________________

3. Now that you have captured sound waves, let's take a closer look at two sound waves on thescreen and compare them (one being music and the other noise). Try to get graphs on thescreen with sound waves large enough to be seen clearly but not touching the top and bottom ofthe display box. Get good, clear sound waves captured in the box so that you can study themcarefully.

a. First make a sound wave of a long, drawn out "ooo" sound, as in "moo".

b. Save the “Data” as “Store Latest Run” and make a graph of the sound you get when youcrumple a piece of paper. Crumple the paper close enough to the microphone to get a fairlylarge wave.

In what ways are the sound waves similar?________________________________________________________________________________________________________________________________________________

In what ways are the two waves different?

________________________________________________________________________________________________________________________________________________

4. Sketch both of your sound waves on the graph below.

In the next activity you will examine several ways in which sound waves can differ.

Differences in sound waves

1. Compare two samples of the same sound. Capture the same sound twice, save the firstdisplay and capture the second. Try to keep the sound at the same pitch and volume.Repeat for many kinds of sounds. Are the pairs of sound waves exactly identical or onlysimilar? If only similar, describe in what ways they differ.

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2. Compare loud and soft sounds. Capture pairs of loud and soft examples of the same kind of

sound. Repeat for voice, instruments, hand claps and other types of sounds. Whatcharacteristic of the waves seems to represent the difference between loud and soft in all ofthese pairs? What seems to stay the same?

3. Compare high and low sounds. Capture pairs of high-pitched and low-pitched sounds. Forexample, sing "ah" as a high note once and then as a low note. Repeat for many kinds ofsounds. Try to make the pairs of sounds about just as loud. What characteristic of thewaves seems to represent the difference between high and low pitch in all of these pairs?

Have you wondered what makes each sound wave so different? Can you control the shape of asound wave? Can you find any similarities in sound waves?

In this activity, you will be making several sounds. Try to decide whether each sound waveshows a repeating pattern.

1. Clap your hands close to the microphone to trigger a sound wave. Try different distancesand loudness as you clap your hands until you get a sound wave that almost fills the box butdoes not touch the top or bottom lines. When you get a good representation of a clap, clickon Stop.

2. Draw a sketch to show the approximate shape of the sound wave you captured.

3. Record other sounds.

a.A plastic whistle.

b. Hum with an even sound AND Clear your throat.

Which of your recordings had patterns which repeated over time?

Indicate with vertical lines on your sketches the distance between repeats on your repeatingpatterns. The time the wave takes to repeat is called the period of the wave. How dorepeating and non-repeating waves sound differently?

Amplitude is the name given to the height of a wave. The amplitude of the wave below isindicated by the double arrow and labeled A.

In the next activity you will compare loud sounds and soft sounds. You will examine therelationship between amplitude and the loudness of a wave.

Loud and soft sounds -- amplitude of sound waves

1. See what happens when you move the microphone away from a source of sound. Hold themicrophone close to your mouth. Hum, and adjust the position of the microphone until thewave almost fills the height of the box. Wait several seconds in each position of themicrophone so that the graph you see definitely represents that position. Move themicrophone away from your mouth while you continue to hum just as loud. Again waitseveral seconds in each position.

For this activity,For this activity,a.in what way(s) does the wave change as you move the microphone away from yourmouth?

b.does the amplitude change?

c.does the number of peaks of the wave change? (That is, does the period change?)

2. Capture a loud note from the whistle. Blow moderately hard, but not so hard that you hear ashill higher pitched tone. Position the microphone close to the flute. Save the wave andsketch the screen below.

3. Capture a softer sound wave. Now play the same note until volume sounds about half asloud as before. Capture the wave and sketch the result above.

How is the wave for the soft sound different from the wave for the loud sound of the samefrequency?

How is the amplitude of a wave related to the quality of a sound which you hear calledloudness? What types of sounds have large amplitudes? Small amplitudes?

Is loudness a very precisely defined physical concept? Explain.

In the next activity, you will compare a high-pitched sound (soprano) with a low-pitched sound(bass). You may make the sounds by singing, or by playing instruments. The period of a wave,T, is the time the wave pattern takes to repeat. It is the time for one complete cycle (repeat) ofthe wave. Be sure that you understand how to determine the period, as indicated in thefollowing diagram.

In the diagram, since two and a half cycles fit into 25 ms, the period of this wave is 25 ms/2.5= 10 ms. You also know that the frequency of a wave is the number of complete cycles(repeats) per second. You can calculate the frequency of a wave by determining how manycycles fit into one second. This is equal to by one divided by the period of the wave in seconds.For the wave above, the frequency is 1/10 ms or 1/.01 s = 100 cycles per second. Anothername for cycles per second is hertz (Hz). The frequency is 100 Hz.

High pitch, low pitch -- period and frequency of sound waves

1. Capture a high-pitched repeating sound wave. Practice making regular pattern for this soundbefore you click on Stop to capture the picture. Get a large wave, but do not let it touch thetop or bottom of the box. Click quietly so its sound is not picked up by the microphone.Sketch the sound wave of your high-pitched sound and save:

2. Capture a low-pitched repeating sound wave. Try to get a large wave - the same height asyour high-pitched sound. Sketch the sound wave of your low-pitched sound.

3. Measure the periods of your high-pitched and low pitched sound. Note how many completecycles are displayed in the box for your high-pitched sound. Read the total time for all ofthese cycles in milliseconds using the cursors on the graph. Notice that as you move theyellow or red cursor, the values displayed in the amplitude and time graph change.Calculate the time for one cycle in seconds. Show your calculations. High-Pitched Sound Low-Pitched Sound

4. Calculate the frequencies of your high-pitched sound and low-pitched sound. Show yourcalculations.High-Pitched Sound Low-Pitched Sound

How is the period of a wave related to the quality of the sound which you hear called pitch?What types of sounds have long periods? Short periods?

How is the frequency of a wave related to the quality of the sound which you hear calledpitch? What types of sounds have high frequencies? Low frequencies?

Is pitch a very precisely defined physical concept? Explain.

Transverse and Longitudinal Waves

Previously waves were created in a rope and a Slinky. The volunteer held the end of therope or Slinky and moved his or her hand in a regular manner, either up and down or back andforth. This caused a vibratory motion, or disturbance, in the rope or the Slinky. The motion of adisturbance in a medium is called a wave. (By the word medium we mean some sort of matteror substance that must be present to carry the wave.) In the demonstration the medium was therope in one case and the Slinky coils in the other. In the sound experiment, the medium was theair. Waves in the ocean are produced by a disturbance in the water, so water is the medium forocean waves. These examples all represent mechanical waves. A mechanical wave is awave that travels in a material medium such as a rope, a spring, air or water. An additionalproperty of mechanical waves is that Newton’s laws govern their motion and that of the materialcarrying the waves.

When a wave moves through a medium, it is said to propagate through the medium. Botha vibrating source and a medium are necessary to generate a mechanical wave. The wave iscreated by a disturbance of the medium. After the disturbance creates the wave, the wavepropagates through the medium. These two events are independent of each other: a change inthe way the wave is created does not change the way in which it propagates. The propagationof the wave is determined by the medium, not by the source. In particular, the speed at whichthe wave travels is determined by the properties of the medium that carries the wave.

In Demonstration 1, what was the source of the waves? That is, what created thedisturbance? ________________________ What was the medium that carried each of thewaves you have seen so far? ____________________________________________________________________________________________________________________________

The disturbance, or wave, travels through the medium by way of interactions between“nearest neighbors” in the medium. For example, as a wave travels through a solid object, themolecules that make up the material are densely packed andinteract with each other through intermolecular forces thatcause them to act like small masses connected by tiny springs.A disturbance in the material disturbs molecules, which stretchagainst their bonds. These bonds act like springs, stretching orcompressing, and affecting the surrounding molecules. Inliquids the bonds between molecules are not as strong as theyare for solids. Because of this, surface tension and otherfactors play an important role in the propagation of waves through liquids. In gases such as airintermolecular bonds are virtually non-existent, yet waves can and do travel through gases. Inthis case, nearest neighbor molecules play a role in the propagation of the wave; as a wavepasses through, the molecules of the gas are crowded together. The crowded, or compressed,molecules tend to move to adjoining portions of the air where there are not so many airmolecules. This rearrangement of the air molecules causes the wave to propagate through thegas.

You can probably think of many examples of waves traveling, or propagating, through amedium: sound waves in air or in solids, waves in a string, vibrations in a metal rod clamped atone end, earthquake waves, water waves, waves in the head of a drum, and vibrations inmachinery. Even “the wave” performed by fans at sporting events has many of the same

properties scientists attribute to waves. In Demonstration 1 you looked at examples oftransverse and longitudinal waves.

A transverse wave is a wave that is produced when the particles of amedium vibrate in a direction perpendicular to the direction of propagation of thewave itself.

A longitudinal wave is a wave that is produced when the vibration of theparticles of the medium is parallel to the direction of propagation of the wave.

In Demonstration 1, which wave was a transverse wave? __________________________Which wave was a longitudinal wave? _____________________________________________

When you move your hand up and down to create a wave in a rope, you can see theheight of the wave change. If you imagine a line that represents the undisturbed position of therope, you can see that moving your hand higher or lower creates a wave that is higher or lowerwith respect to the undisturbed position of the rope. The displacement, y, is the distance fromthe undisturbed position. The maximum distance that the rope moves, either above or below itsundisturbed position, is called the amplitude(A) of the wave. This is illustrated in the diagrambelow. For a wave in a rope or water, displacement and amplitude are measured in distanceunits such as meters or centimeters. The amplitude thus represents the maximum displacementof a particle in the medium carrying the wave as thewave passes by. The maximum upwarddisplacement point on a transverse wave is called acrest, while the maximum downward displacementpoint is called a trough. For sound waves theamplitude can also be measured in distance units,but is usually measured in pressure units.

Another property of waves is the wavelength. The wavelength of a wave is the distancebetween one complete cycle of wave disturbances; that is, the distance from one point of thedisturbance to another similar point. In other words, the wavelength is the distance along thewave before the wave pattern starts to repeat itself. To measure the wavelength on a graph of

the wave, the x-axis must show position along the wave. This is like taking a photograph of thewave at a point in time; that is, the time is held constant while you measure wavelength. For atransverse wave this distance can be measured from one crest to the next, from one trough tothe next, or from one undisturbed position to the second one following. For a longitudinal wave,the easiest way to measure the wavelength isfrom the center of one compression to the centerof the next compression. This is illustrated in thediagram below. Wavelength is measured indistance units, such as meters or centimeters,and is symbolized by the Greek letter λ (lambda).

When you move your hand up and down tocreate waves in a rope, you will notice that the faster you move your hand the faster the wavesyou generate move through the rope. By moving your hand fast you will also create a largernumber of waves in a given amount of time than if you moved your hand slowly. The number ofcycles, or complete oscillations, generated in a unit of time is called the frequency of the wave.Frequency is given the symbol f, and has units of cycles/second or Hertz, abbreviated Hz. InDemonstration 1, the movement of the student’s hand was the source of the wave, and the ropeor Slinky was the medium through which the wave propagated. The frequency at which thehand moved corresponded to the frequency of the wave. This is the case with all wave motion:the source, not the medium, determines the frequency of the wave.

Another property of a wave is the time it takes fora complete wave to be formed. This quantity is calledthe period of the wave, symbolized by T, and ismeasured in time units such as seconds. The periodof a wave is the time between one crest of the waveand the next. To measure this on a graph of a wave,the x-axis must show time. That is, the position of thewave is held constant while measuring the period.This can be thought of as standing in one position witha stopwatch and measuring the time between crests as they pass.

You have already read that frequency is the number of waves in a second, and that theperiod is how many seconds in a wave. This suggests a relationship between frequency andperiod. The period of a wave is just the inverse of the frequency:1Tf=

.

Example

A person standing on a pier notes that 3 waves come across the end of the pier every second.What are the frequency and period of the water waves?

Solution

Since the frequency of a wave is the number of waves per second, the frequency of thewater waves is 3 Hz. The period is the inverse of the frequency:

ssf

T 33.03

11=== .

Properties of Waves

Previously you observed examples of transverse and longitudinal waves and saw that thefrequency of the wave was determined by the source of the wave, not the medium throughwhich the wave traveled. In this Exploration you will use transverse waves to investigate howthe frequency of a wave is related to another property – the wavelength.

Equipment• Wave generator• String• Pulley• Masses and mass hanger• Meter stick

Section I

1. Open the Sound Analysis CenterWave Generator software. Be sure that the cable from the Wave Generator box isplugged into the speaker output of your computer and the Wave Generator box is receivingpower.

2. Set up the wave generator as seen in the photo. Connect a string to the paperclip vibratoron the wave generator and run it to a pulley on a support pole a distance away as in thephoto. The string should be horizontal and at a convenient height for viewing. Afterpassing the string over the pulley, connect a mass hanger to the end of the string and adda small amount of mass (about 20 grams) to the hanger.

3. Set the frequency on the generator to a value between 60 and 100 Hz and record thevalue you chose. _________ Do not change the frequency until told to do so.

4. In Dialog 1 you learned that the speed of a wave was determined by the medium throughwhich the wave traveled. For the wave to be created by the wave generator, what is themedium through which the wave will travel? ________________________ For a wave in astring, the speed of the wave is determined by the tension in the string; the greater thetension the faster the wave will travel. The speed of the wave set up in the string can be

calculated from

µ=SV

, where S is the tension in the string and µ is the mass per unit

length of the string. Your instructor will provide you with the value of µ. Record that valuehere. __________________ How will you determine the tension in the string?________________________________________________________________________________________________________________________________________________When you do this Exploration, take your time and be sure to measure a completewave. Change to metric units before doing the calculations.

5. Turn on the wave generator and observe the wave in the string. If you do not have a“clean” wave adjust the tension slightly by add or remove a few grams of mass until aclean wave is observed in the string. If you are not sure if your wave is “clean” enough,

check with your instructor. Measure the wavelength of the waves set up in the string.Record the number of loops in the string, the wavelength and the tension in the string inthe table below. (DO NOT USE MORE THEN 200 GRAMS ON THE STRING!)

Numberof Loops

Total Mass onString (kg)

Tension(N)

Wave Speed(m/s)

Wavelength,λ (m)

6. Change the speed of the wave by adjusting the tension. You can adjust the tension byadding or removing mass from the hanger until another clean wave goes through thestring. Be sure that the number of loops is different from the number of loops you had inStep 4. Repeat the measurements in Step 4 and record your data in the table.

7. Repeat Step 5 for two other values of tension.

8. For each tension, calculate the speed of the wave in the string and record your results inthe table.

9. Using the data from the Section I of this Exploration, plot a graph of speed vs. wavelength.What type of a relationship is suggested by your graph? _________________________Find the best-fit equation for your graph. Sketch the shape of the graph in the spacebelow.

10. What are the units of your slope? What physical quantity do you think is represented bythe slope of your graph? ____________________________________________________________________________________________________________________________

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

Wavelength (m)

Wavesp

eed

(m

/s)

Section II

11. Place an amount of mass on the mass hanger that is about the middle value of the rangeof masses used in the first part of this Exploration. Record the amount of mass used._________________ Record the speed of the wave with this amount of mass._____________________________

12. Adjust the frequency output of the wave generator until you see a clean wave is seen inthe string. Measure the wavelength and the frequency and record the values in the tablebelow.

Frequency(Hz)

Wavelength,λ (m)

13. Repeat Step 9 for four other values of frequency, leaving the tension and thus the velocityconstant.

14. Using the data from the second part of the Exploration, plot a graph of wavelength vs.frequency. What type of relationship is suggested by your graph? ___________________Sketch your graph and the best-fit equation for your graph in the space below.

15. What physical quantity is represented by the constant term in your best-fit equation?Again, if you are having trouble with this, try looking at the units of the quantities in yourequation. ________________________________________________________________________________________________________________________________

16. Summarize what you have discovered about the relationship between the speed of awave, its frequency and its wavelength. ________________________________________________________________________________________________________________________________________________________________________________________

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 20 40 60 80 100

Frequency (Hz)

Wavele

ng

th (

m)

The Speed of a Wave

Previously you saw that as the speed of the wave increased at a fixed frequency, thewavelength also increased. Likewise, you saw that as the frequency increased at a fixed speed,the wavelength decreased. Is there a way to put these two findings together into a relationshipthat will help describe the wave?

The key to the relationship between wavelength and frequency is the speed of the wave.You learned in the Motion module that the average speed of an object is the distance traveleddivided by the time required for the object to travel that distance. In our case the “object”traveling is the wave. The average speed of the wave is the distance traveled in one cycle ofthe wave divided by the time required for one complete cycle of the wave. Since a wave travelsat a constant speed in a given medium, the average speed of the wave is the only speed in thatmedium.

In Dialog 1 you found that the distance of one cycle of a wave is called the wavelength,and the time required for one cycle is called the period of the wave. Thus, the speed of a waveis given by

Tt

dv

λ==

Since the period of the wave is the inverse of the frequency, this can also be written as

.1

fT

vf

λλλ===

This final relationship is the mathematical model for the relationship between wave speed,wavelength and frequency. Look back at your results from Exploration 2. Your first graphshould have been a straight line. Based on the best-fit line and the discussion in this Dialog,what physical quantity is represented by the slope of this graph? ___________________ Howdoes the value of your slope compare to the true value? __________________________

The second graph from the previous activity should have shown an inverse relationship. An

inverse relationship is one that has the mathematical form x

Ay = , where A is a constant. Based

on your results and the discussion in this dialog, what physical quantity is represented by theconstant term in your graph? ______________________ How does the value of your constantcompare to the true value? ______________________________________________________

Do your graphs correspond to the mathematical relationship between wave speed, wavelengthand frequency? ____________________________________

Example

A sound wave travels in air with a speed of 345 m/s. If the wavelength of the sound wave is 5m, what is the frequency of the wave?

Solution

Steps 1 and 2: Real world and physical representation.

Step 3: Knowns and unknowns.

v = 345 m/s λ = 5 m f = ?

Step 4: Mathematical representation.

The relationship we need is vfl=

, which we will need

to rearrange to solve for f:

vfl=.

Step 5: Solution.

Substitute the known quantities into the relationship for the frequency of the wave.345/169695vmsfHzmsl====

Notice the units in the answer. The m in the numerator cancels with the m in the denominator,leaving units of 1/s. These units are equivalent to cycles/s, or Hertz (Hz).

What is it that the wave carries as it travels? Consider your results from Exploration 2. Did thestring move from one place to another as the wave was carried in it? Or did the string onlyvibrate in place? ______________________________________________________________

A wave was previously defined as being a disturbance traveling through a medium. Thisdefinition tells us the answer to the questions above. The disturbance travels through themedium; the medium that the wave travels in is not moved from one place to another. Theparticles that make up the medium oscillate in place, but once the wave passes they move backto their original positions. Think about Exploration 2: once you finished the experiment andturned off the generator, the string was in the same position it was originally. Likewise, if youhave ever seen wind blowing across a field of grain, you can see the waves traveling across thefield, but the grain remains in place. Even “the wave,” often done by sports fans by moving backand forth in their seats, demonstrates this property of waves. The fans move back and forth inplace, but observers on the other side of the stadium see a wave move across the spectators.The spectators don’t move from their seats, but the wave travels.

So what is it that the wave carries? Ultimately, it is energy that moves from place to placein the medium. It takes energy to create the original disturbance that generates the wave, andas the disturbance moves through the medium, the energy of the disturbance moves with it. Ina way, a wave can be considered as an energy transport system.

Working with Wave Relationships

1. A student connects a wave generator to a string that has a fixed amount of tension appliedto it. Describe what happens to the wave in the string as the frequency of the generator isincreased.

2. A student connects a wave generator that has a fixed frequency to a string. What happensto the wave in the string as the student pulls the string tighter?

3. What is it that actually moves from one place to another as a wave travels through amaterial? .

4. A despondent young man sits at the end of a pier, contemplating his lost love. As hewatches the waves come across the end of the pier, he notes that the wave crests are 3 mapart, and that the crests pass the end of the pier one every 2 seconds.

a) What is the frequency of the water waves?

b) What is the wavelength of the water waves?

c) What is the speed of the waves?

5. Strong winds can actually make skyscrapers oscillate back and forth. In fact, the buildingsare designed to have a certain amount of “sway” in windy conditions. If the oscillationfrequency of the building is 0.1 Hz, what is the period of the motion?

6. A woman competing in a fishing tournament sits in a 6 meter long fishing boat that is 0.70m high. The fish aren’t biting, and in her boredom the lady notices that the verticaldistance between a crest and a trough of a water wave passing the boat is one-quarter theheight of the boat. At any time there are 10 crests from the front of the boat to the back ofthe boat. Find the amplitude and the wavelength of the waves.

7. A wave travels through the air at 3 x 108 m/s. If the wavelength of the wave is 5 cm, findthe frequency of the wave. (Note: this is a microwave.)

8. The diagram below represents a wave. The time covered by the picture is 10 seconds.Using this information and the information on the diagram find a) the period of the wave; b)the frequency of the wave; c) the wavelength; d) the amplitude; and e) the speed of thewave.

Wave Interference

Previously you may have noticed that the waves you were working with formed fixedpatterns in the string. That is, the wave pattern in the string did not seem to change. Therewere certain positions in the string where there did not appear to be any motion of the string,while other portions of the string seemed to be in constant motion. An example of the type ofpattern you saw is seen in the diagram below. This type of a wave pattern is called a standingwave. In this Exploration you will discover some of the conditions that cause a standing wave tooccur.

Equipment• Wave worksheet (found at end of this Exploration)• Colored pencils (three different colors)

1. On the top graph of the wave worksheet, draw a wave that has an amplitude of 1 cm and awavelength of 8 cm. Start the wave with a vertical displacement y = 0 cm at x = 0 cm.This will be wave 1 and it will be considered to be moving to the right. Using the samecolored pencil as you used to draw the wave, indicate the direction of motion of the waveby an arrow on the graph. How far would this wave travel in one period of the wave?__________ How do you know? ______________________________________________________________________________________________________________________

2. Using a second color, draw another wave on the same graph that is identical to wave 1and in phase with wave 1. This second wave will be wave 2, and it will be considered tobe moving to the left. Using the same colored pencil you used to draw wave 2, indicatethe direction of motion of wave 2 by an arrow on the graph. How far would this wave travelin one period of the wave? __________ How do you know? _______________________________________________________________________________________________

3. When two or more waves are traveling together in the same medium, each wave will try todisturb the medium by the same amount it would if it was the only wave in that medium.That is, if one wave would cause a displacement of 5 cm at some point, and a secondwave would cause a displacement of 2 cm at the same point, the total displacement of themedium at that point will be 7 cm. If the two waves you have drawn are traveling togetherin the same medium, what would be the maximum displacement of the medium at anypoint? ___________How do you know? ________________________________________________________________________________________________________________

4. Using the third colored pencil, draw the wave that represents the overall displacement ofthe medium when the two original waves are traveling in the same medium at the sametime. What is the amplitude of this resultant wave? ____________ How far would theresultant wave travel in one period of the resultant wave? __________________________

5. On the second blank grid on the worksheet, draw the original two waves as they wouldappear a time equal to one-quarter of a period after the first graph. Use the same color for

each wave as you used previously. Remember that wave 1 is moving to the right andwave 2 is moving to the left. On the same graph, use the third colored pencil to draw theresultant wave for this situation. What is the amplitude of the resultant wave in this case?______________________________________________________________________

6. Repeat Step 5, this time showing the original two waves at a time equal to one-half aperiod later than in the first graph. What is the amplitude of the resultant wave in thiscase? ________________________________________________________________

7. Repeat Step 5 one more time, this time showing the original two waves at a time equal tothree-quarters of a period later than in the first graph. What is the amplitude of theresultant wave in this case? ______________________________________________

8. Which of the situations would you categorize as having a “constructive” effect takingplace? _____________________ Why do you think this is so?________________________________________________________________________________________

9. Which of the situations would you categorize as having a “destructive” effect taking place?_____________________ Why do you think this is so?____________________________________________________________________________________________________

10. Look at the resultant waves on the series of graphs. Are there any locations where themedium carrying the wave would have a vertical displacement of zero at all times? ______Locations where there is no vertical motion of the medium carrying a wave are callednodes. Where are the nodes of the resultant wave located?________________________________________________________________________

11. Again observe the resultant waves on your graphs. Are there any locations where themedium carrying the wave would oscillate in a vertical direction between maximum positivedisplacement and maximum negative displacement? _______ Locations where themedium carrying a wave oscillates between maximum positive and negative displacementare called antinodes. Where are the antinodes of the resultant wave located?________________________________________________________________________

12. Summarize what you have learned in this Exploration by making a general statementabout the effect on the medium when two waves are in the same location at the sametime. ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Wave Worksheet

y (cm) 2

t = 0 s1

0 2 4 6 8 10 12 14 16 18 x (cm)

-1

-2

y (cm) 2

t = (1/4)Ts

1

0 2 4 6 8 10 12 14 16 18 x (cm)

-1

-2

y (cm) 2

t = (1/2)Ts

1

0 2 4 6 8 10 12 14 16 18 x (cm)

-1

-2

y (cm) 2

t = (3/4)Ts

1

0 2 4 6 8 10 12 14 16 18 x (cm)

-1

-2

The Superposition of Waves

Previously you looked at several situations in which two identical waves overlapped. Ifthese two waves were traveling together in the same medium, the medium would react to bothwaves together, resulting in a single wave that was the superposition of the two original waves.When two waves interact in this manner, they are said to be interfering with each other.

Consider what happens when you combine two waves that have the same wavelength andthe same amplitude, oriented in such a way that the crests and troughs of the two waves arealigned with each other. Such waves are said to be in phase. The resultant wave that occurswhen the two waves overlap and interfere with each other is a wave with twice the amplitude ofthe original waves. This is an example of constructive interference, and is illustrated in thediagram below. The overall displacement of any point in the medium is the algebraic sum of thedisplacements provided by each of the two waves that are interfering with each other.

Now imagine that the two waves are oriented in such a way that the peaks of one wavealign with troughs of the second wave. Such waves are said to be 180o, or π radians, out ofphase. This corresponds to a shift equivalent in time to one-half the period of the waves. Thetwo waves now in effect cancel each other out, so that the resultant wave has an amplitude ofzero. This is an example of destructive interference, as illustrated in the diagram below.

What if the two waves that are interfering with each other are not perfectly in phase or 180o

out of phase, but rather something in between? In situations such as these the waves stillinterfere with each other, but the result is not perfectly constructive or perfectly destructive. Theoverall displacement of the medium is still the algebraic sum of the individual displacementsprovided by the original wave. An example of such a situation is seen in the diagram below.

In situations where the waves that are interfering constructively are continually created, theresulting amplitude can become quite large. This type of situation can be beneficial, or it can bedisastrous. For example, for a musical instrument to create the sounds we generally regard aspleasant, a number of different sound waves must interfere constructively. On the other hand,the unrestrained constructive interference of waves in a machine might result in the catastrophicfailure of that machine as the vibrations created by the interfering waves become larger than the

ability of the structure of the machine to support them. This rather dramatic situation hasoccurred many times in the past, resulting in the collapse of bridges and the crashing ofairliners.

In your first experiment yousaw a wave pattern that seemed tobe stationary; certain places on thestring appeared to never move,while others were in constantmotion. This resulted in a stationary pattern similar to that in the diagram above.

This type of a wave is called a standing wave. The locations along the wave where nodisplacement of the medium appears to take place are called nodes, while those places thatoscillate between maximum positive and negative displacement are called antinodes. Thestanding wave is the result of the interference of the original wave traveling in one direction andits reflection traveling in the opposite direction. In the case of the string, the wave generatorcreated a wave in the string, which traveled away from the generatortowards the opposite end. The wave reflected off of the pulley at the otherend of the string and traveled back towards the wave generator. Becausethe generator was continually generating a wave, there was continualreflection, and this reflected wave interfered with the wave from thegenerator.

To understand the process by which the standing wave occurs, wecan break the process down into small steps, looking at what is happeningat various times during the period of the wave and its reflection. In thefirst diagram at right, the wave is traveling to the left, while the reflectiontravels to the right. At this point in the motion, the wave and its reflectionare perfectly aligned, resulting in constructive interference.

The wave and its reflection continue to travel in their respectivedirections. At a time equivalent to one-quarter of a period later, the twowaves are now 180o out of phase, resulting in complete destructiveinterference.

After a time equal to one-half a period, the waves are again aligned, this time in theopposite orientation as before. Again, the overall result is constructive interference.

After three-quarters of a period have passed, the wave and its reflection are once again180o out of phase and destructive interference occurs. The cycle is complete after a time equalto one period of the original waves, when constructive interference again occurs.

There are a couple of points to be made about this situation. First, notice that there arecertain points in the resultant wave pattern that always have a zero displacement. These arethe nodes of the standing wave. These points always occur a distance of one-half a wavelength(λ/2) apart. Also notice that the constructive interference condition occurs at times that are one-half a period apart. The second constructive interference pattern is inverted with respect to thenext one, and that after a time equal to one period of the wave, the constructive interferencepattern is exactly the same as the first one. In other words, the period of the standing wave isthe same as the period of the wave that caused the standing wave.

Not all frequencies of waves will set up standing waves in a given situation. The object ormaterial in which the wave is traveling determines what frequencies will make standing waves.We have seen that a node is a point on the standing wave where the displacement is alwayszero, and that an antinode is a point on the standing wave where the displacement oscillateswith maximum amplitude. Nodes on a standing wave in a string occur at distances one-half awavelength apart, and antinodes always occur midway between nodes. If we consider thevibrations of a guitar string, or any taut string clamped at two points, the points where thevibrating string is clamped are nodes. (Why?) Putting all of these conditions together meansthat only those frequencies that result in some multiple of half-wavelengths to be established inthe string will cause standing waves in the string. This situation is illustrated in the diagram onthe following page, where the first few allowed standing wave patterns for a string of length Lclamped at both ends are shown.

When a given frequency sets up a standing wave in an object, that frequency is said to bea resonant frequency of the object. The process of establishing a standing wave at a givenfrequency is called resonance. When resonance occurs in a given object, the amplitude of thestanding wave reaches a maximum, just as it did in your string in Exploration 2. If this situationoccurs in something like a bell, a drumhead or an organ pipe, this is a desirable effect.However, if an object is not built to withstand those vibrations, failure and break-up of the objectcan occur. For example, in 1850 a bridge over a river in France collapsed when soldiersmarching in step set up a standing wave in the bridge. Each soldier’s step created a wave, andsince the soldiers were marching in step, all the waves were in phase with each other. Theresulting standing wave had such a large amplitude that the bridge structure collapsed, killingover 200 soldiers. Another very famous example of this occurred in 1940 at the TacomaNarrows Bridge in Washington State. Perhaps your instructor will show you a tape of thisfamous example of resonance. Other, more everyday examples of resonance include therattling of your windows when a low-flying plane passes overhead, or the vibration of your chestwhen you are at a loud concert.

A consideration of resonance is also important in heavy machinery. When rotatingmachinery malfunctions, for example, if the shaft of an electric generator is out of balance,vibrations are produced which travel throughout the body of the machine. These vibrationscould potentially result in a resonance condition within the machine, causing vibrations of anamplitude large enough to cause the machine to break apart.

Resonance can occur in sound waves as well. This type of resonance is a very importantfactor in making wind instruments and organ pipes sound the way they do.

Standing Waves in Strings

1. Explain in your own words what a standing wave is and how it can be formed.

2. Shown below is a string clamped at both ends.

a) Draw the first four standing wave patterns that could be established in this string.

b) What are the wavelengths of the waves corresponding to the patterns sketched in parta?

c) If the speed of a wave in the string shown is 10 m/s, find the frequencies of each of thewaves in part a. Hint: Remember your wave relationship!

d) Is there any relationship between the frequencies you found in part c? If so, what isthat relationship?

3. Suppose that the string in Question 2 was half as long.

a) Would the first four standing wave patterns in this string look any different than thosein the original string? If there is any difference, what is it?

b) Would the first four standing wave frequencies for this string be higher or lower thanthe first four standing wave frequencies in the original string?

4. Suppose that two waves are interfering with each other. Will the amplitude of the resultingwave ever be larger than the amplitude of either of the interfering waves? Explain.

5. Shown below are graphs of two different waves traveling together in the same medium.Draw a quantitatively correct graph showing the overall displacement of the medium asthese two waves pass through.

Wave 1

-6

-4

-2

0

2

4

6

time0.15

0.35

0.55

0.75

0.95

1.15

1.35

1.55

1.75

1.95

time (s)

Dis

pla

cem

ent

(m)

Wave 2

-4-3-2-101234

time0.15

0.35

0.55

0.75

0.95

1.15

1.35

1.55

1.75

1.95

time (s)

dis

pla

cem

ent

(m)

-6

-4

-2

0

2

4

6

0 0.5 1 1.5 2

time (s)

Dis

pla

cem

en

t (m

)

The Mathematical Description of a Wave

You may have noticed that the diagrams used to illustrate the transverse wave look verymuch like a graph you may have seen in a math class: the sine function. This is not accidental.Wave motion is mathematically described by sinusoidal functions, but we have to be careful:there are two ways of looking at the wave.

Suppose you imagine setting up a wave in a rope, then taking a picture of the wave. Youwould get a picture similar to that shown below, where your picture represents the shape of thewave at the point in time when you took the picture. The picture shows a sinusoidal shape,which could be mathematically described with a sinusoidal function. The appropriate descriptionwould be

πλ=2sin,xyA

where A is the amplitude of the wave and λ isthe wavelength. This mathematical descriptionwould enable you to draw a picture of the waveas a function of position x along the direction oftravel of the wave. Again, it is important toremember that this description represents thewave at a certain point in time.

What if we were interested in what was happening at a certain point along the wave as afunction of time? In other words, what is the medium that carries the wave doing at a certainposition? For example, in Demonstration 1 you placed a piece of tape at a fixed position on therope, then watched the tape’s motion as the wave passed by. You saw that the tape,representing a position in the medium, moved up and down in an oscillatory motion. Like mostoscillatory motion, the motion of that fixed point in the medium can be described by a sinusoidalfunction as well. If you were to draw a graph representing the position of the fixed point in themedium as a function of time, you would get a graph similar to the one shown below. Thismotion can also be described with a sinusoidal function. The appropriate description would be

( ),2sin ftAy π=

where A is the amplitude of the wave and f is itsfrequency. This mathematical description wouldenable you to draw a representation of theposition of a certain point in the medium carryingthe wave as a function of time. Again, it isimportant to remember that this descriptionrepresents a fixed point in the medium as thewave travels through that medium.

In reality, of course, both of these things are happening together. The wave travelsthrough the medium, displacing many points in the medium as it travels through. Any completemathematical description would have to take all points in the medium into account at all times asthe wave travels. For problem solving purposes, however, we can look at these the twodescriptions separately.

The same descriptions can be used to describe longitudinal waves. The wavelength andfrequency of a longitudinal wave are described the same way as they are for transverse waves.The only change may be in what the amplitude term in the mathematical descriptionsrepresents. While the amplitude for a longitudinal wave can indeed still represent the maximumdisplacement of a particle in the medium from its normal, undisturbed position, it can be definedto represent other changes as well. For example, for a sound wave the amplitude term mayrepresent a maximum pressure change in the air as the wave passes. This idea will be lookedat later in the module.

Example 1

A given wave traveling through a medium is described by the function .25

sin10 cmx

y

=

π Does

this represent the wave as a snapshot at a fixed time or at a point in the medium? What is theamplitude of the wave? What is the wavelength of the wave?

Solution

These questions are answered by comparing the given function to the standard functions

explained in this Dialog. We see that the given function is of the form

=

λπ x

Ay2

sin , so the

function must represent the wave at a fixed point in time.By comparing terms between the standard function and the given function, we see that the

amplitude of the wave is 10 cm. Also by comparison, we see that 25

2 πλπ= , which means that

the wavelength of the wave must be 50 cm.

Example 2

The motion of a given point in the medium carrying the wave in Example 1 can be described by( )ty π30sin10= . What is the frequency of the wave? What is the speed of the wave?

Solution

Again, the answer is found by comparing to the standard description of the motion of a point inthe medium as a function of time, ( )ftAy π2sin= . By comparison, we see that ππ 302 =f , whichmeans that the frequency f = 15 Hz. Since for all waves the frequency and wavelength of thewave are related to the speed of the wave by fv λ= , we get ,/750)15)(50( scmHzcmv == wherewe have taken the wavelength from Example 1.

A note on the units: remember that 1 Hz is the same as 1 cycle/second. When we multiplied cmby cycle/sec, the “cycle” dropped out, giving our final answer units of cm/s.

The Sinusoidal Descriptions of Waves

1. A wave has a wavelength of 20 cm and a frequency of 100 Hz.

a) What is the speed of the wave?

b) Write a mathematical function to describe the shape of the wave at a fixed point intime.

c) Write a mathematical description of the motion of a point in the medium carrying thewave as a function of time.

2. A wave can be described by ( )cmxy π70sin50= and ( )cmty π50sin50= . Find thewavelength, the frequency and the speed of the wave.

3. For the wave in Question 2, sketch the two functions in the space below, using appropriateunits on the axes of each graph. Choose a scale to make the graphs fill your page andinclude at least two cycles of the wave in both position and time.

time (s) time (s)

4. A graphical depiction of a wave is seen in the diagram below. Write an appropriatemathematical function to describe the wave, based on the information given in the graph.

5. A graphical depiction of a wave is seen in the diagram below. Write an appropriatemathematical function to describe the wave, based on the information given in the graph.

6. The graphs in Questions 4 and 5 represent the same wave in different ways. What is thespeed of that wave?

How are sound waves created?

How are sound waves produced? What basic property do all sources of sound have incommon? Your instructor will carry out a demonstration to help you to answer these questions.

7. What do you think is necessary for a sound wave to be produced? List any requirementsbelow, indicating for each one why you think each is necessary.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

8. Your instructor will strike a tuning fork. Listen to the sound produced. How do you thinkthe tuning fork makes sound? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

9. Watch as your instructor inserts the ends of the tuning fork into a beaker of water.Describe the effect you see. What does the tuning fork seem to be doing as it producesthe sound? ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

10. The strobe light is a device that uses bright flashes of light to help us see motion that is toofast for our unaided eye to interpret. Your instructor will shine a strobe light on the tuningfork. Describe what you see._________________________________________________________________________________________________________________________________________________________ ____________________________________________________________

11. Listen as your instructor strikes another tuning fork. Why do you think this new fork makesa different sound than the previous one? ____________________________________________________________________________________________________________________________________________________________________________________

12. Tuning forks are usually labeled with a frequency value. How do you think the frequencylabel on the tuning fork relates to the vibrations of the fork and/or the sound the forkproduces? Be specific. __________________________________________________________________________________________________________________________________________________________________________________________________

13. Listen and observe while your instructor plucks a tight wire and shows its motion with thestrobe light. What does the motion of the wire as it produces sound have in common withthe motion of the tuning fork as it produces sound? ___________________________________________________________________________________________________________________________________________________________________

14. What do you think would happen to the sound if the wire was pulled to a greater tension?______________________________________________ What if it was pulled to lesstension? ______________________________ Your instructor will demonstrate thesechanges. Record your observations.______________________________________________________________________________________________________________________________________________

15. The wire will be placed under tension and clamped at both ends. Listen to the soundproduced by the wire when it is plucked near its center. What effect would you expect tohear in the sound produced by the wire if the tension was not changed, but the wire wasclamped at a point near its center, so that the length was effectively cut in half?____________________________________________________________ Listen to thesound produced when your instructor makes this change. How does this sound compareto the original sound? ____________________________________________________What would happen to the sound if the length was halved again? ______________________________________________________________ Listen to and describe this change.______________________________________________________________________________________________________________________________________________

16. Summarize your observations and comparisons of the sounds made by a string of differentlengths. In your summary, include an explanation of why you think the sounds changedthe way they did. ________________________________________________________________________________________________________________________________________________________________________________________________Your instructor will connect a frequency generator to a speaker. Listen to the soundproduced by the speaker as the input frequency changes. How does the sound producedby the speaker relate to the input frequency form the generator?_____________________________________________________________________________________________________________________________________________________________________________________________________________________

17. Does the speaker vibrate as it makes sound? _______________________ Check yourprediction by observing the speaker with the strobe or by lightly laying your fingertips onthe speaker cone while it is producing sound. Describe what you felt.______________________________________________________________________________________________________________________________________________Summarize your observations from this Demonstration. What seems to be necessary for anobject to produce sound? _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Does sound need a medium to travel from one point toanother?

How does sound get from the source to your ear? What requirements are there for thepropagation of sound? This Demonstration shouldhelp you answer those questions.

The basic apparatus used is in this Demonstration isa Bell jar that can be evacuated (this means thatthe air can be removed from inside the jar with apump). An electric buzzer is the source of sound.This apparatus is shown in the photo at right. To dothe Demonstration, your instructor will turn on thebuzzer, and then slowly remove the air from insidethe jar.

18. In the space below, predict whether there willbe any changes in how you see the lightand/or hear the bell as the air is removed.Explain your reasoning.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

19. Watch and listen carefully as the bell is turned on and the air is removed from the jar.Take particular note of the way the bell sounds. Record your observations.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

20. Why do you think the sound behaved the way it did? _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

The Speed of Sound

In this activity you will measure the speed of sound waves in air. You can do this by timing howlong it takes for sound waves to travel from one point to another. You will send sound wavesdown a hollow tube which is closed at one end. As you have observed, when waves on a Slinkyreach the clamped end, they are reflected back. Similarly, the sound waves are reflected backwhen they reach the closed end of the tube. You can measure the time for the waves to godown to the end of the tube and come back again.

Equipment• Sound Analysis Center software• Microphone• Computer• long hollow tube closed at one end (Use 4” Cardboard or Styrene Tubes)• two-meter meterstick

1. Open the Sound Analysis Center software again.2. Set up the tube and microphone as shown below.

Length

This end closedMicrophone

3. Click on Start and make a popping sound with your lips, clapping your hands or a clickingsound by hitting two objects together. The idea is to make as brief a sound as possible.Look at the wave pattern on the screen. If the microphone picked up the original wave andthe wave reflected back from the closed end, you should see several almost identical wavepatterns on the screen separated by a time interval. (If this isn't what you see, try again.)Keep trying until you get a wave pattern which clearly shows the initial and reflected waves.You may want to change the position and angle of the microphone, or try a different sound.

When you get a good wave pattern click on Stop, and sketch it below.

4. Measure the time interval between the beginning of the original wave and the beginning ofthe reflected wave in seconds. (This is the time for the wave to go from your mouth downthe tube to the closed end, and back to the microphone.) Measure the period using thecursors on the graph. Notice that as you move the yellow or red cursor, the values displayedin the amplitude and time graph change. Also measure the time interval for a round trip forany other reflected waves, and the length of the tube.

T =

L =

5. Calculate the speed of sound waves in air in m/sec. Show your calculation.

6. Why is it important for the sound you use to be as brief as possible?

7. The speed of sound in air at room temperature T (°C) can be calculated fromv = 331+ 0.60 T (m/s). Calculate the speed of sound using this formula. Do these valuesagree?

8. Calculate the % error between your measured value (5.) and the actual value (7.).

9. What reasons can you think of which might explain any inaccuracies in your measurementof the speed of sound?

Producing and Transmitting Sound

Previously you saw that vibrating objects generated sound. Then you saw that a medium wasrequired for the propagation of sound. These two observations are in agreement with Dialog 1,where it was stated that both a vibrating source and a medium are necessary to generate amechanical wave. In other words, sound must be a mechanical wave.

How does sound travel through a medium such as air? You saw previously that a mechanicalwave travels by interactions between molecules, each molecule transmitting the disturbancethrough the medium by interaction with its nearest neighbors. In air the molecules are so widelyseparated that there are virtually no forces between them. In a gas, such as air, the moleculesare in random thermal motion. There are no constantly acting elastic forces between molecules,as there would be in a liquid or a solid. This means that the molecules only interact with eachother during collisions.

Sound is a longitudinal wave. The longitudinal sound waves traveling through air areresponsible for sound traveling from a source and reaching our ears. As the sound wave travelsthrough the air, a portion of the air is suddenly compressed, crowding the air moleculestogether. These crowded molecules tend to move to adjoining portions of the air, where thedensity of the molecules is not so great. This results in the density of molecules increasing inthis new area, causing the molecules that are there to move to a new area. This processcontinues, thereby causing the original disturbance (compression) that started the sound waveto be propagated through the air. The fact that the sound waves are compressing moleculestogether into a greater density as they pass is the reason that sound waves are sometimescalled compressional waves. These compression waves correspond to density changes in thegas, as illustrated in the diagram below.

For sound waves movingin air that have frequenciesin the range of humanhearing (roughly 20 –20,000 Hz), the oscillationsin the gas density take place very rapidly. The speed of sound in air, or any gas, is somewhatdependent upon the temperature of the gas. This is because the molecules of the gas are, onaverage, moving faster at higher temperatures. Higher average speeds for the moleculestranslate to a higher speed of the sound wave through the gas. An empirical formula for thespeed of sound in air at temperature T (oC) is v = (331.5 + 0.6T) m/s. Knowing the airtemperature you can determine the speed of sound in that air.

What about sound traveling in other materials? You may have noticed when swimming that youcould hear sounds under water, so sound must travel through liquids as well as gases.Likewise, placing a metal pipe near your ear and tapping at the other end shows that soundtravels through solids as well. What is happening on a microscopic level when soundpropagates through a solid or a liquid? In actuality, since there are bonds acting between themolecules of a liquid or a solid, the transmission of sound is easier to understand in thesecases.

When a deforming force acting on a material causes a change in the volume or shape, but thematerial resumes its original size and shape after the force is removed, the material is said to beelastic. The elasticity of a given material depends upon the strength of the bonds that connectthe molecules of the material together. When sound travels through a solid or a liquid, itstretches, or deforms, these bonds. This stretching of the bonds between the molecules isresponsible for the propagation of the sound through the material. As the sound wave passes,the bonds are compressed, and then stretched, thereby transmitting the disturbance through thematerial. The speed of sound in a material is governed by the elasticity of the material and,sometimes, by the temperature of the material. In general, the stronger the bonds between themolecules of a material, the faster sound will travel in that material. As a result, the speed ofsound will be higher in liquids than in gases, and higher in solids than in either liquids or gases.

Sound Waves

1. A student connects a frequency generator to a speaker in a warm room. Describe whathappens to the sound wave produced by the speaker if the student carries it outside whereit is 32o F (0o C).

2. A speaker sitting on the deck of a pool produces sound waves. What happens to thesound waves as they enter the water in the pool?

3. Ultrasound pulses are used to image tissue in the human body. Typical frequencies usedrange from 1 MHz to 10 MHz. If the speed of sound in human tissue is 1,550 m/s, find thewavelengths corresponding to this range of frequencies.

4. The speed of sound in steel is about 5,130 m/s. If a charge of dynamite is set off near arailroad track, how long will it take before a person with their ear to the steel rail to hear thesound if they are 8 km from the point of the explosion?

5. The diagram at right represents a wave.The time covered by the picture is 10seconds. Using this information and theinformation on the diagram find a) theperiod of the wave; b) the frequency of thewave; c) the wavelength; d) the amplitude;and e) the speed of the wave.

6. The temperature of the air in a certain room is found to be 27o C. What is the speed ofsound in the room? What would the wavelength of a sound of frequency 256 Hz be?

Standing Waves and the Speed of Sound

An important piece of information to know about a wave is the speed that it travels. As a wavetravels through matter, it moves at a speed that is different from the speed at which the particlesof matter carrying the wave are moving. If you could look closely enough, you would find that asa sound wave travels through air, the molecules in the air move back and forth. However, themotion of the molecules is not the same as the overall motion of the wave itself. The moleculesoscillate back and forth at a given location while the wave travels from one place to another. Aswe discussed in Part One, a sound wave is a longitudinal wave in air.

Usually the speed of the particles of matter that carry the wave is of little interest, while thespeed of the wave is very important. For instance, if an earthquake creates a tidal wave at seathat approaches the shore, the inhabitants waiting for its arrival probably do not care how fastthe surface of the water bobs up and down as the wave passes by. On the other hand, theyhave a major interest in how long it will be until the tidal wave reaches them, a time that can bedetermined by the speed of the wave itself.

Previously you used standing waves to helpmeasure the speed of sound in air. You learnedin that a node is a position on a standing wavewhere the displacement of the medium is zero,while an antinode is a point where the mediumoscillates between maximum positive andmaximum negative displacement. If we set astring clamped at both ends into vibration, theclamped ends of the string are nodes, since theyare unable to move. For the first frequency thatwill create a standing wave in the string, the pointmidway between the two ends is then anantinode. These same situations exist in astanding sound wave as well. If a sound waveexists in a tube that is closed at both ends, thenboth ends of the tube will be nodes. Thesenodes would be locations where the moleculesare not able to vibrate. These ends must benodes in order for the sound wave to reflect fromone end of the tube and travel back to the other.(Remember that a standing wave is the result ofa wave interfering constructively anddestructively with its reflection.) With both endsof the tube being nodes, the first frequency thatwill create a standing wave in the closed tube willbe that frequency which will cause an antinode atthe center of the tube.

The frequencies that cause standing waves to be created in a medium are called resonantfrequencies. For a string clamped at both ends or a tube closed at both ends, these resonantfrequencies will all have nodes at both ends. The term frequently applied to describe theseresonant frequencies is harmonics. The wave with the lowest frequency and longest

wavelength that causes a standing wave is called the first harmonic. Subsequent harmonics arenamed according to their order above the first. Illustrated below are the first few harmonics for astring clamped at both ends and a tube closed at both ends.

The situation is slightly different when the tube is open at both ends. In this situation, the airmolecules at the ends of the tube are free tomove. This means that the ends of the tubewill be antinodes rather than nodes. But sincethere must be a node in between antinodes,the center of the tube must be a node for thefirst harmonic. Comparing the first fewharmonics for a tube that is open at both endsto those for a tube that is closed at both ends,we can see that the allowed frequencies areexactly the same; the difference is in thelocations of the nodes and antinodes.

But what if the tube is closed at one end andopen at the other? In this case, the closed endwill be a node while the open end will be anantinode. These requirements will change thefrequency pattern from what we have seen sofar. In the case of the first harmonic, there isalready a node and an antinode, so the firstharmonic will be a wave that has a wavelength offour times the length of the tube instead of twotimes the length as was seen in the previous twocases. Following the allowed patterns of nodesand antinodes, we see that only the oddharmonics exist. The second resonant frequency is the third harmonic; the third resonantfrequency is the fifth harmonic, and so on. The even harmonics do not exist for this kind of tube.This situation is illustrated in the figure below.

One refinement to the model we have discussed so far takes into account that the antinode atthe end of an open pipe does not occur exactly at the end of the pipe, but rather a short distanceaway. This distance depends on the diameter of the pipe, and causes the standing waves in thepipe to be established as if the tube was actually slightly longer. This slightly longer length isknown as the effective length of the tube. To a good approximation the antinode actually occursat a point one-fourth the diameter of the pipe beyond the end of the pipe. This effective lengthfor a tube open only at one end would be L + d/4, where L is the true length and d the diameter.For a tube open at both ends this correction shows up at both ends, making the effective lengthof the tube L + (d/4) + (d/4), or L + d/2.

In summary, we can state that the conditions for resonance require that a clamped string end orclosed pipe end act as a node for the standing wave. Similarly, a loose string end or an openpipe end must act as an antinode for the standing wave. The relationship between wavelength,frequency and wave speed (v = λf) can be used to relate the resonant frequency of a standingwave to the wavelength of the standing wave and the speed of the wave in the material. For astanding wave in a pipe, this speed would be the speed of sound in air. Knowing the nature ofthe waves in a string or a pipe allows us to predict the resonant frequencies of the standingwaves.

Where do we see examples of standing waves? One good example would be in the large pipesthat are used in a pipe organ. Resonant frequencies of sound in the pipes create the musicalsounds that we hear. All musical instruments, whether stringed or wind, rely on the properties ofstanding waves to create musical notes. You may have even seen barrier ribbons aroundconstruction sites blowing in the wind, creating standing waves you can both see and hear.

An example of a more complicated standing wave is seen in a drumhead, such as might beseen on a tympani drum. The fundamental frequency of the drumhead results in a wave patternthat is circularly symmetric, with the center of the drumhead oscillating with maximum amplitudewhile the outside edge of the drumhead, along the rim, does not move at all. We say that thecenter of the drumhead is an antinode while the outside edge is a node.

In this more complicated system the frequencies of the higher harmonics are found not to bemultiples of the fundamental frequency as was the case in the more basic systems we looked atin earlier Explorations. The frequencies that create standing wave patterns on the drumheadare called characteristic frequencies, and can be computed with good accuracy. However, themathematics required for these calculations is quite complex and beyond the scope of thiscourse.

These characteristic frequencies can be experimentally determined. This is accomplished byusing a speaker and a frequency generator to apply vibrations of a known frequency to thedrumhead. The loudspeaker is placed directly above the horizontal drumhead and the output ofthe frequency generator is used to drive the speaker. When the frequency of the generatormatches a characteristic frequency of the drumhead, the drumhead will begin to vibrate withlarge amplitude oscillations. If the drumhead is covered with chalk dust, the dust will collectalong the nodes of the vibrations. By noting the pattern of the nodes and the frequency at whichthe pattern occurs, the resonant frequencies of the drumhead can be determined.

Standing waves occur in other systems as well. Sometimes these standing waves can bebeneficial, but in other systems the effects of a standing wave can be detrimental or evencatastrophic. For example, a beam secured at one end and free on the other is called acantilever beam. This beam also has characteristic modes of vibration that are morecomplicated than the systems observed in this module. But an understanding of the modes ofvibration is very important in the design of a number of different devices, including airplanes. Anairplane wing is essentially a cantilever beam, and an understanding of the modes of vibration ofthe wing is critical in the design of safe and efficient airliners. Another example of theimportance of understanding standing waves and resonance comes from the history of aircraft.One of the first models of commercial jet aircraft had a design defect that caused undampedvibrations in the engines. These vibrations became so large that the engine could break looseand smash into the main cabin area of the aircraft. More than one aircraft crashed, killing allaboard. A simple modification of the engine design eliminated the resonant vibrations and keptthe engine in place.

You have seen how standing waves are established in simple systems such as strings and airtubes. You have also seen that standing waves can occur in many other applications as well,such as a drumhead or in machinery. The standing waves that are established in morecomplicated systems require correspondingly more complicated mathematics to describe thewaves. However, the three main types of vibrational systems we have discussed – strings,organ pipes, and drumheads – are the three types of standing wave systems found in nature.All standing wave systems, no matter how complex, can be categorized as variations of one ofthese three main types.

Applications of Standing Waves

7. A pipe organ has pipes of many different lengths. Which pipes (long or short) do you thinkwould produce low frequency notes? Which would produce high frequency notes?

8. Describe the type of pipe that would have the standing waves described in each situationbelow.

a) The wave has antinodes at both ends of the tube.

b) The wave has an antinode at one end of the tube and a node at the other end of thetube.

c) The wave has nodes at both ends of the tube.

9. A tube that is closed at both ends has a length of 2 m. The speed of sound in the area ofthe tube is 400 m/s. Resonant sound waves with wavelengths of 40 cm (or 80 mm) areobserved in the tube. Find the frequency and the number of the harmonic for each ofthese waves.

10. While driving to work, Jill notices that workers have strung plastic ribbon around theirworksite to prevent people from walking into the hole. Jill notes that between two supportpoles the ribbon is 1.5 m long and is vibrating in a standing wave pattern with 5 loops andthat the vibrating ribbon is creating a sound that corresponds to the note A (f = 420 Hz).What is the wavelength of the standing wave? What is the speed of sound in the ribbon?

11. A spaceship lands on a new planet. The crew decides to measure the speed of sound inthe planets atmosphere. Taking a tube with a length of 50 cm, they find that one resonantfrequency occurs at 2,520 Hz, and the next resonant frequency at 2,940 Hz. What is thespeed of the sound? What is the number of each harmonic? What type of pipe is the crewusing (open-open, open-closed, closed-closed)?

Echolocation

A common application of sound waves involves the detection of reflected sound waves todetermine the distance to an object. The simplest possible example of this is an echo. Whenyou yell at a distant cliff, the time it takes for you to hear the echo depends on both the speed ofsound in the air and how far it is to the cliff. This same concept applies to such diverseapplications as depth finding and the police radar gun, although this last application issomewhat more complicated since the object reflecting the sound is moving. In this Explorationyou will look at the basic principles of echolocation, where the position of the object isdetermined by the time it takes for a sound wave to “echo” from the object.

Equipment• Sound Analysis Center software• Microphone• White board• Thermometer

12. Suppose you are hiking in the mountains.Seeing a distant cliff and hoping to hear an echo,you yell and hear the echo of your shout. Howfar did the sound wave from your shout have totravel?_________________________________________________________________________________________________________________________

13. Suppose the speed of sound in the area is 325 m/s. If you hear your echo 2 seconds afteryou shout, how far away is the cliff that the sound wave reflectsfrom?___________________________________________________________________

14. Point the microphone toward the whiteboard and connect it to your computer. Open theSound Analysis Center Software and Your instructor will inform you of the correct settingsfor your system. Hold the whiteboard upright a distance of 1 to 2 m from the microphone.Start the data collection sequence, then place your hands next to the microphone and capthem together loudly. Observe the signal on the computer screen. How many signals doyou see? ______ What is the source of each of the signals that you observe?______________________________________________________________________________________________________________________________________________

15. Record the temperature of the air in the room from the thermometer. Using therelationship determined previously, determine the speed of sound in the room._______________________________________________________________________

16. Using the speed determined in Step 4 and the time between the signals observed on thecomputer screen, calculate the distance that the sound wave traveled.__________________ Use this result to determine how far away the whiteboard is fromthe microphone. _____________________ How does this result compare to the actual

distance? __________________________________________________________________________________________________________________________________

17. Repeat these measurements for a different distance between the plywood sheet and themicrophone. How does your measured distance compare to the actual distance in thiscase? _______________________________________________________________________________________________________________________________________

18. How could the results of this Exploration be used to determine the depth of a lake? Whatadditional information would you need to make this measurement?___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

19. Suppose pulses are sent to the bottom of the ocean, and the reflected signal is heard 2.5seconds later. How deep is the ocean? The speed of sound in the seawater is 1530 m/s.Show your work below.

Applications of Echolocation

Echolocation is used in a number of practical applications. In an application known as sonar,pulses of sound are transmitted outward from a source through water, and sensitive sensorsdetect the echoes of the sound waves. The longer the delay between the initial pulse and theecho, the further away the object is that causes the echo. Since the speed of sound in water isknown, the distance to the object can be determined. Such systems can be used to map out thefloor of a lake or the ocean, or to locate shipwrecks far beneath the surface. This same basicprinciple is used in fish locators, which locate fish by looking for the echoes of sound wavesbouncing of the bodies of the fish.

Similar systems are used to explore for subsurface oil or mineral deposits. Explosions are setoff underground, and sensors called geophones record sound signals from the explosions. Arecord is made of the time it takes after the explosion for the sound wave energy to reach thegeophones. By using the known speeds of sound in various materials, geologists are able tomap out the structure of the ground below the surface and determine the most likely locations tofind oil or other valuable materials.

Radar is also based on the same principle, but uses high frequency radio waves instead ofsound waves. Again, the wave speed is known, so the delay of the echoes can be used tolocate the reflecting object, such as a plane or a speeding car.

Nature uses the principles of echolocation as well. Bats use a sonar system to hunt insects forfood. They have very poor vision, and hunt mainly at night. The bats emit a high frequencysound and listen for the echoes from insects to locate their food. They also use the echoes fromlarger objects to avoid flying into obstacles.

A final example of the application of these principles is seen in ultrasound. Pulses ofultrasound, which are sound waves beyond the limit of human hearing, are used to image softtissue in the human body. Typical frequencies of ultrasound used in such imaging are from 1 to10 Megahertz. Special computer systems measure the time required for the ultrasound signalsto be reflected from tissues within the body. By mapping out the pattern of signals, an image ofthe internal organs and structures can be produced.

Alarm

Reflections of sound and standing waves occur naturally all around us. For example, the roomswe live in are full of reflecting surfaces. The following Exploration will show you how this factcan be used in a practical device – an intrusion alarm.

Equipment• Sound Analysis Center software• Microphone• Microphone stand or wood block• Three whiteboards• Duct tape• Large Speaker• Sound Analysis Center Wave Generator

20. Set up a model room using the threewhiteboards to form three walls. Secure the walls of the room together with duct tape.The walls of the model room should rest directly on the lab table. Place the speakeragainst one wall of the room, as seen in the photo. If the speaker produces sound, whatwill happen when those sound waves strike the walls of the room?________________________________________________________________________________________________________________________________________________________________________________________________

21. Turn on the Sound Analysis Center wave generator to 2000 Hz. Set the microphone on itsstand or a wood block near the center of the model room. Start the Sound Analysis Centersoftware and observe the signal from the microphone. Describe the signal._____________________________________________________________________________________________________________________________________________________________________________________________________________________

22. Insert your hand or other objects into the model room. What happens to the signalobserved on the computer screen?______________________________________________________________________________________________________________________________________________

23. Why do you think the signal behaved the way it did? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

24. Could a system that operates in a manner similar to that in model room be used as anintrusion alarm? Why or why not? What changes would have to be made in order for sucha system to be practical? ____________________________________________________________________________________________________________________________________________________________________________________________

______________________________________________________________________________________________________________________________________________

Palm Pipes

You learned previously that one of the most common applications of standing waves was inmusical instruments. In this Exploration you will use a simple musical instrument to illustrate theproperties of standing waves.

Equipment• PVC pipes cut to various lengths• Music sheets• Thermometer• Metric ruler

25. A palm pipe is simply a length of pipe that creates a sound when one of the open ends isstruck against the palm of the hand. Look at the set of palm pipes provided by yourinstructor. Which pipes (longer or shorter) do you think will create the lowest frequencysounds? ___________________ The highest frequency sounds? _________________Why do you think this is so? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

26. Your instructor will provide palm pipes to everyone in the class. Holding your pipe in onehand, strike the palm of the opposite hand with the open end of the pipe and listen to thesound it produces. Compare the sound from your pipe with that of another student’s pipe.Was your prediction in Step 1 correct?______________________________________________________________________

27. Determine the temperature of the room. Using the relationship for the speed of sound invarious temperatures of air, as discussed previously, calculate the speed of sound in theroom. ________________________________________________________________________________________________________________________________________

28. Using what you have learned about the allowed standing wave patterns for sound waves ina pipe, determine what frequencies of sound would establish standing waves in your pipe._________________________________________________________________________________________________________________________________________________________________________________________________________________The range of human hearing is generally considered to be from 20 Hz to 20,000 Hz. Whichof the frequencies you calculated would you be able to hear? ________________________________________________________________________________________

29. Your instructor will provide you with a music sheet and act as the director to lead the classin playing a song with the palm pipes.

Materials: for constructing a complete set of 2 octaves (15 pipes). Supplies available from HomeDepot or other Builder’s Supply stores.

1 standard 10 foot length of 1/2 inch CPVC pipe for 180°F water

1 plastic pipe and tube cutter (for cutting pipes)

1 set colored permanent markers (for labeling pipes)

Safety: Wash pipes with alcohol or a solution of 2 teaspoons of Chlorox per gallon of water ifstudents blow across them.

No. Note Length of pipe Frequency

#1 F 23.60 cm 349 Hertz

#2 G 21.00 cm 392 Hertz

#3 A 18.75 cm 440 Hertz

#4 Bflat 17.50 cm 446 Hertz

#5 C 15.80 cm 523 Hertz

#6 D 14.00 cm 587 Hertz

#7 E 12.50 cm 659 Hertz

#8 F 11.80 cm 698 Hertz

#9 G 10.50 cm 748 Hertz

#10 A 9.40 cm 880 Hertz

#11 Bflat 9.20 cm 892 Hertz

#12 C 7.90 cm 1049 Hertz

#13 D 7.00 cm 1174 Hertz

#14 E 6.25 cm 1318 Hertz

#15 F 5.90 cm 1397 Hertz

How does a Telephone Work?

How does a telephone route your call to the phone number you dial? Isthere a relationship between the frequencies of the sounds produced by eachbutton and the number on the button? Is a “code” involved? Are the soundspure tones or combinations of tones?

Using Sound Analysis Center, microphones, or any other equipment youfeel is appropriate, analyze the tone structure and sequence of the buttons ona telephone dial pad. You can use an actual telephone, or your instructor mayhave a tone dialer available for you to use. A tone dialer is a device thatduplicates the tones made by the buttons on a telephone. As you carry outyour investigation, be sure to answer the following questions:

1. Is each sound a pure tone or a combination of frequencies?

2. Is there a pattern to the frequencies used in the dialer? Is there apattern in the rows of buttons? The columns? In the number sequence?

3. How does the phone company route your call to the phone numberyou dialed?