Introduction to Waves and Sound

Transcription

1 Introduction to Waves and Sound Principal Authors: Martin Mason, Mt. San Antonio College and Christine Carmichael, Woodbury University Based on the work of John 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 21 st Century Project supported by Joliet Junior College (IL), Lee College (TX), and a grant (# ) 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 # in the Advanced Technological Education Program of the National Science Foundation. The National Science Foundation 4201 Wilson Boulevard Arlington, Virginia 22230, USA

2 The Properties of Waves You 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 feel but cannot see, such as the winds during a storm or the sensation of riding in a car on a bumpy road. You may have been told that light is a wave, and while you can see light you can t see the wave nature of the light. You also may know that sound is a wave. In this section you will learn what all of these types of waves have in common and what distinguishes them from each other. You will also learn the terminology that applies to all waves, so that a discussion of the specific properties of sound can be related to waves in general. Finally, you will learn about the sources of waves and how the waves travel from one place to another.

3 What is a wave? You have experienced waves many times in many different situations. In this introductory Exploration you will be asked to think about what you know about waves, so that you have an information base on which to build additional knowledge about waves. 1. The term wave has more than on meaning in everyday language. List as many different meanings 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 least five different examples. 4. Look at your answers to the previous question. What, if any, things do all of your examples have in common? 5. Now consider the specific example of a sound wave. List all the ways you can think of to make a sound wave. Again, try to give at least five different ways to make the sound wave. 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 sound travels? 7. What types of materials do you think sound travels through? What evidence do you have to support your answer? When a thunderstorm is overhead what have you noticed about the time it takes for you to see the lightning and to hear the thunder? What does this suggest about the differences between light and sound?

4 Making Waves In this Demonstration you will create two different types of waves and compare their properties. Equipment Slinky Rope or Long Spring Masking tape 8. Your instructor will ask a student volunteer to help stretch the rope or spring across the room or hall. The student will generate a wave in the rope by moving his or her hand in an up-and-down motion. Observe the wave created in the rope by this motion. Describe the wave as completely as possible, including information such as direction of motion of the wave, direction of motion of the rope, and anything 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 student volunteer will create a wave pulse in the rope by giving one end of the rope a quick flip while your instructor holds the other end stationary. Observe the motion of the pulse down the 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 and describe 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 at the same speed as in Step 1, but with a greater up-and-down motion. What changes do you see in the wave? 12. Your instructor will ask the student to generate the wave by moving his or her hand in a faster up-and-down motion than in Step 3. What changes in the wave do you notice? What if the student moves the rope up and down slower than in Step 3? What property or

5 properties of the wave seem to change when the speed of the motion of the hand generating 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 about 1.5 meters long. The student will move one end of the Slinky back and forth, parallel to the tabletop. Observe the wave created in the Slinky by this motion. Describe the wave as completely 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 sketch of 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 the Slinky. The student will generate a wave pulse in the Slinky by giving the end of the Slinky a 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 piece of tape as the pulse passes the tape. How do these motions compare? Describe your observations. 15. Now the student will generate a continuous wave in the Slinky by moving his or her hand back and forth, as in Step 6. Observe and describe the motion of the piece of tape as the wave travels in the Slinky. 16. Your instructor will now ask the volunteer to move his/her hand back and forth faster than in Step 8. What changes in the wave do you notice? What if the hand moves back and forth slower than in Step 8? What property or properties of the wave seem to change when the speed hand generating the waves changes? Record your observations for these changes. 17. Summarize what you have observed about the similarities and differences between the type of wave generated in the rope and the type of wave generated in the Slinky.

6 Sound: Waves in Air Previously you looked at waves in a string. Sound waves behave the same way, except that since they travel in air, we can t see them. To study and identify the basic properties of sound waves 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 of particles in the medium. In a gas, like air, sound waves may also be thought of as a pressure disturbance (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 ( Microphone Computer various sources of sound waves. In this activity you will see how to collect and various sound waves on the screen. display 1. Plug the microphone into the input of your soundcard(this is often identified with a small picture of a microphone). Open the Sound Analysis Center software. You should see Record and 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 the loudness of the sound constant. The graph you see represents the sound wave (in volts). If you 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 the screen and compare them (one being music and the other noise). Try to get graphs on the screen with sound waves large enough to be seen clearly but not touching the top and bottom of the display box. Get good, clear sound waves captured in the box so that you can study them carefully.

7 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 you crumple a piece of paper. Crumple the paper close enough to the microphone to get a fairly large 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 first display 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 only similar? 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. What characteristic of the waves seems to represent the difference between loud and soft in all of these pairs? What seems to stay the same?

8 3. Compare high and low sounds. Capture pairs of high-pitched and low-pitched sounds. For example, sing "ah" as a high note once and then as a low note. Repeat for many kinds of sounds. Try to make the pairs of sounds about just as loud. What characteristic of the waves 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 a sound wave? Can you find any similarities in sound waves? In this activity, you will be making several sounds. Try to decide whether each sound wave shows a repeating pattern. 1. Clap your hands close to the microphone to trigger a sound wave. Try different distances and loudness as you clap your hands until you get a sound wave that almost fills the box but does not touch the top or bottom lines. When you get a good representation of a clap, click on Stop. 2. Draw a sketch to show the approximate shape of the sound wave you captured. 3. Record other sounds.

9 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 repeating patterns. The time the wave takes to repeat is called the period of the wave. How do repeating and non-repeating waves sound differently? Amplitude is the name given to the height of a wave. The amplitude of the wave below is indicated by the double arrow and labeled A. In the next activity you will compare loud sounds and soft sounds. You will examine the relationship 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 the microphone close to your mouth. Hum, and adjust the position of the microphone until the wave almost fills the height of the box. Wait several seconds in each position of the microphone so that the graph you see definitely represents that position. Move the microphone away from your mouth while you continue to hum just as loud. Again wait several 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 your mouth?

10 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 a shill higher pitched tone. Position the microphone close to the flute. Save the wave and sketch the screen below. 3. Capture a softer sound wave. Now play the same note until volume sounds about half as loud 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 same frequency? How is the amplitude of a wave related to the quality of a sound which you hear called loudness? 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) of the wave. Be sure that you understand how to determine the period, as indicated in the following diagram.

11 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 many cycles 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. Another name 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 sound before you click on Stop to capture the picture. Get a large wave, but do not let it touch the top 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 as your 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 complete cycles are displayed in the box for your high-pitched sound. Read the total time for all of these cycles in milliseconds using the cursors on the graph. Notice that as you move the yellow 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 your calculations. High-Pitched Sound Low-Pitched Sound

12 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 called pitch? What types of sounds have high frequencies? Low frequencies? Is pitch a very precisely defined physical concept? Explain.

13 Transverse and Longitudinal Waves Previously waves were created in a rope and a Slinky. The volunteer held the end of the rope or Slinky and moved his or her hand in a regular manner, either up and down or back and forth. This caused a vibratory motion, or disturbance, in the rope or the Slinky. The motion of a disturbance in a medium is called a wave. (By the word medium we mean some sort of matter or substance that must be present to carry the wave.) In the demonstration the medium was the rope in one case and the Slinky coils in the other. In the sound experiment, the medium was the air. Waves in the ocean are produced by a disturbance in the water, so water is the medium for ocean waves. These examples all represent mechanical waves. A mechanical wave is a wave that travels in a material medium such as a rope, a spring, air or water. An additional property of mechanical waves is that Newton s laws govern their motion and that of the material carrying the waves. When a wave moves through a medium, it is said to propagate through the medium. Both a vibrating source and a medium are necessary to generate a mechanical wave. The wave is created by a disturbance of the medium. After the disturbance creates the wave, the wave propagates through the medium. These two events are independent of each other: a change in the way the wave is created does not change the way in which it propagates. The propagation of the wave is determined by the medium, not by the source. In particular, the speed at which the 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 the disturbance? What was the medium that carried each of the waves 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, the molecules that make up the material are densely packed and interact with each other through intermolecular forces that cause them to act like small masses connected by tiny springs. A disturbance in the material disturbs molecules, which stretch against their bonds. These bonds act like springs, stretching or compressing, and affecting the surrounding molecules. In liquids the bonds between molecules are not as strong as they are for solids. Because of this, surface tension and other factors play an important role in the propagation of waves through liquids. In gases such as air intermolecular bonds are virtually non-existent, yet waves can and do travel through gases. In this case, nearest neighbor molecules play a role in the propagation of the wave; as a wave passes 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 air molecules. This rearrangement of the air molecules causes the wave to propagate through the gas. You can probably think of many examples of waves traveling, or propagating, through a medium: sound waves in air or in solids, waves in a string, vibrations in a metal rod clamped at one end, earthquake waves, water waves, waves in the head of a drum, and vibrations in machinery. Even the wave performed by fans at sporting events has many of the same

14 properties scientists attribute to waves. In Demonstration 1 you looked at examples of transverse and longitudinal waves. A transverse wave is a wave that is produced when the particles of a medium vibrate in a direction perpendicular to the direction of propagation of the wave itself. A longitudinal wave is a wave that is produced when the vibration of the particles 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 the height of the wave change. If you imagine a line that represents the undisturbed position of the rope, you can see that moving your hand higher or lower creates a wave that is higher or lower with respect to the undisturbed position of the rope. The displacement, y, is the distance from the undisturbed position. The maximum distance that the rope moves, either above or below its undisturbed position, is called the amplitude(a) of the wave. This is illustrated in the diagram below. For a wave in a rope or water, displacement and amplitude are measured in distance units such as meters or centimeters. The amplitude thus represents the maximum displacement of a particle in the medium carrying the wave as the wave passes by. The maximum upward displacement point on a transverse wave is called a crest, while the maximum downward displacement point is called a trough. For sound waves the amplitude 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 distance between one complete cycle of wave disturbances; that is, the distance from one point of the disturbance to another similar point. In other words, the wavelength is the distance along the wave before the wave pattern starts to repeat itself. To measure the wavelength on a graph of

15 the wave, the x-axis must show position along the wave. This is like taking a photograph of the wave at a point in time; that is, the time is held constant while you measure wavelength. For a transverse wave this distance can be measured from one crest to the next, from one trough to the next, or from one undisturbed position to the second one following. For a longitudinal wave, the easiest way to measure the wavelength is from the center of one compression to the center of the next compression. This is illustrated in the diagram below. Wavelength is measured in distance units, such as meters or centimeters, and is symbolized by the Greek letter λ (lambda). When you move your hand up and down to create waves in a rope, you will notice that the faster you move your hand the faster the waves you generate move through the rope. By moving your hand fast you will also create a larger number of waves in a given amount of time than if you moved your hand slowly. The number of cycles, 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. In Demonstration 1, the movement of the student s hand was the source of the wave, and the rope or Slinky was the medium through which the wave propagated. The frequency at which the hand 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 for a complete wave to be formed. This quantity is called the period of the wave, symbolized by T, and is measured in time units such as seconds. The period of a wave is the time between one crest of the wave and the next. To measure this on a graph of a wave, the x-axis must show time. That is, the position of the wave is held constant while measuring the period. This can be thought of as standing in one position with a 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 the period is how many seconds in a wave. This suggests a relationship between frequency and period. The period of a wave is just the inverse 1Tf= of the frequency: 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 the water waves is 3 Hz. The period is the inverse of the frequency: T 1 1 = = s = 0. s f 3 33.

16 Properties of Waves Previously you observed examples of transverse and longitudinal waves and saw that the frequency of the wave was determined by the source of the wave, not the medium through which the wave traveled. In this Exploration you will use transverse waves to investigate how the frequency of a wave is related to another property the wavelength. Equipment Section I Wave generator String Pulley Masses and mass hanger Meter stick 1. Open the Sound Analysis Center Wave Generator software. Be sure that the cable from the Wave Generator box is plugged into the speaker output of your computer and the Wave Generator box is receiving power. 2. Set up the wave generator as seen in the photo. Connect a string to the paperclip vibrator on the wave generator and run it to a pulley on a support pole a distance away as in the photo. The string should be horizontal and at a convenient height for viewing. After passing the string over the pulley, connect a mass hanger to the end of the string and add a 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 the value 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 through which the wave traveled. For the wave to be created by the wave generator, what is the medium through which the wave will travel? For a wave in a string, the speed of the wave is determined by the tension in the string; the greater the tension the faster µ=sv the wave will travel. The speed of the wave set up in the string can be calculated from, 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 value here. How will you determine the tension in the string? When you do this Exploration, take your time and be sure to measure a complete wave. 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 a clean wave is observed in the string. If you are not sure if your wave is clean enough,

17 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 in the table below. (DO NOT USE MORE THEN 200 GRAMS ON THE STRING!) Number of Loops Total Mass on String (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 by adding or removing mass from the hanger until another clean wave goes through the string. Be sure that the number of loops is different from the number of loops you had in Step 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 in the 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 space below Wavespeed (m/s) Wavelength (m) 10. What are the units of your slope? What physical quantity do you think is represented by the slope of your graph?

18 Section II 11. Place an amount of mass on the mass hanger that is about the middle value of the range of 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 in the string. Measure the wavelength and the frequency and record the values in the table below. Frequency (Hz) Wavelength, λ (m) 13. Repeat Step 9 for four other values of frequency, leaving the tension and thus the velocity constant. 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 Wavelength (m) 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 your equation. 16. Summarize what you have discovered about the relationship between the speed of a wave, its frequency and its wavelength.

19 The Speed of a Wave Previously you saw that as the speed of the wave increased at a fixed frequency, the wavelength 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 relationship that 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 traveled divided 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 of the wave divided by the time required for one complete cycle of the wave. Since a wave travels at a constant speed in a given medium, the average speed of the wave is the only speed in that medium. 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 wave is given by v = Since the period of the wave is the inverse of the frequency, this can also be written as d t λ = T λ λ v = = = λ f. 1 T 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 graph should 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? How does the value of your slope compare to the true value? The second graph from the previous activity should have shown an inverse relationship. An A inverse relationship is one that has the mathematical form y =, where A is a constant. Based x on your results and the discussion in this dialog, what physical quantity is represented by the constant term in your graph? How does the value of your constant compare to the true value? Do your graphs correspond to the mathematical relationship between wave speed, wavelength and frequency? f

20 Example A sound wave travels in air with a speed of 345 m/s. If the wavelength of the sound wave is 5 m, 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= vfl=, which we will need to rearrange to solve for f:. 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 the string move from one place to another as the wave was carried in it? Or did the string only vibrate in place? A wave was previously defined as being a disturbance traveling through a medium. This definition tells us the answer to the questions above. The disturbance travels through the medium; the medium that the wave travels in is not moved from one place to another. The particles that make up the medium oscillate in place, but once the wave passes they move back to their original positions. Think about Exploration 2: once you finished the experiment and turned off the generator, the string was in the same position it was originally. Likewise, if you have ever seen wind blowing across a field of grain, you can see the waves traveling across the field, but the grain remains in place. Even the wave, often done by sports fans by moving back and forth in their seats, demonstrates this property of waves. The fans move back and forth in place, 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 place in the medium. It takes energy to create the original disturbance that generates the wave, and as the disturbance moves through the medium, the energy of the disturbance moves with it. In a way, a wave can be considered as an energy transport system.

21 Working with Wave Relationships 1. A student connects a wave generator to a string that has a fixed amount of tension applied to it. Describe what happens to the wave in the string as the frequency of the generator is increased. 2. A student connects a wave generator that has a fixed frequency to a string. What happens to 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 a material?. 4. A despondent young man sits at the end of a pier, contemplating his lost love. As he watches the waves come across the end of the pier, he notes that the wave crests are 3 m apart, 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 buildings are designed to have a certain amount of sway in windy conditions. If the oscillation frequency of the building is 0.1 Hz, what is the period of the motion?

22 6. A woman competing in a fishing tournament sits in a 6 meter long fishing boat that is 0.70 m high. The fish aren t biting, and in her boredom the lady notices that the vertical distance between a crest and a trough of a water wave passing the boat is one-quarter the height of the boat. At any time there are 10 crests from the front of the boat to the back of the boat. Find the amplitude and the wavelength of the waves. 7. A wave travels through the air at 3 x 10 8 m/s. If the wavelength of the wave is 5 cm, find the 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 the wave.

23 Wave Interference Previously you may have noticed that the waves you were working with formed fixed patterns in the string. That is, the wave pattern in the string did not seem to change. There were 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 of pattern you saw is seen in the diagram below. This type of a wave pattern is called a standing wave. In this Exploration you will discover some of the conditions that cause a standing wave to occur. 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 a wavelength 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 same colored pencil as you used to draw the wave, indicate the direction of motion of the wave by 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 1 and in phase with wave 1. This second wave will be wave 2, and it will be considered to be moving to the left. Using the same colored pencil you used to draw wave 2, indicate the direction of motion of wave 2 by an arrow on the graph. How far would this wave travel in 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 to disturb 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 second wave would cause a displacement of 2 cm at the same point, the total displacement of the medium at that point will be 7 cm. If the two waves you have drawn are traveling together in the same medium, what would be the maximum displacement of the medium at any point? How do you know? 4. Using the third colored pencil, draw the wave that represents the overall displacement of the medium when the two original waves are traveling in the same medium at the same time. What is the amplitude of this resultant wave? How far would the resultant 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 would appear a time equal to one-quarter of a period after the first graph. Use the same color for

24 each wave as you used previously. Remember that wave 1 is moving to the right and wave 2 is moving to the left. On the same graph, use the third colored pencil to draw the resultant 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 a period later than in the first graph. What is the amplitude of the resultant wave in this case? 7. Repeat Step 5 one more time, this time showing the original two waves at a time equal to three-quarters of a period later than in the first graph. What is the amplitude of the resultant wave in this case? 8. Which of the situations would you categorize as having a constructive effect taking place? 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 the medium 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 called nodes. Where are the nodes of the resultant wave located? 11. Again observe the resultant waves on your graphs. Are there any locations where the medium carrying the wave would oscillate in a vertical direction between maximum positive displacement and maximum negative displacement? Locations where the medium carrying a wave oscillates between maximum positive and negative displacement are called antinodes. Where are the antinodes of the resultant wave located? 12. Summarize what you have learned in this Exploration by making a general statement about the effect on the medium when two waves are in the same location at the same time.

25 Wave Worksheet y (cm) 2 t = 0 s x (cm) -1-2 y (cm) 2 t = (1/4)T s x (cm) -1-2 y (cm) 2 t = (1/2)T s x (cm) -1-2 y (cm) 2 t = (3/4)T s x (cm) -1-2

26 The Superposition of Waves Previously you looked at several situations in which two identical waves overlapped. If these two waves were traveling together in the same medium, the medium would react to both waves 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 and the same amplitude, oriented in such a way that the crests and troughs of the two waves are aligned with each other. Such waves are said to be in phase. The resultant wave that occurs when the two waves overlap and interfere with each other is a wave with twice the amplitude of the original waves. This is an example of constructive interference, and is illustrated in the diagram below. The overall displacement of any point in the medium is the algebraic sum of the displacements 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 wave align with troughs of the second wave. Such waves are said to be 180 o, or π radians, out of phase. This corresponds to a shift equivalent in time to one-half the period of the waves. The two waves now in effect cancel each other out, so that the resultant wave has an amplitude of zero. 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 180 o out of phase, but rather something in between? In situations such as these the waves still interfere with each other, but the result is not perfectly constructive or perfectly destructive. The overall displacement of the medium is still the algebraic sum of the individual displacements provided 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, the resulting amplitude can become quite large. This type of situation can be beneficial, or it can be disastrous. For example, for a musical instrument to create the sounds we generally regard as pleasant, 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 catastrophic failure of that machine as the vibrations created by the interfering waves become larger than the

27 ability of the structure of the machine to support them. This rather dramatic situation has occurred many times in the past, resulting in the collapse of bridges and the crashing of airliners. In your first experiment you saw a wave pattern that seemed to be stationary; certain places on the string appeared to never move, while others were in constant motion. 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 no displacement of the medium appears to take place are called nodes, while those places that oscillate between maximum positive and negative displacement are called antinodes. The standing wave is the result of the interference of the original wave traveling in one direction and its reflection traveling in the opposite direction. In the case of the string, the wave generator created a wave in the string, which traveled away from the generator towards the opposite end. The wave reflected off of the pulley at the other end of the string and traveled back towards the wave generator. Because the generator was continually generating a wave, there was continual reflection, and this reflected wave interfered with the wave from the generator. To understand the process by which the standing wave occurs, we can break the process down into small steps, looking at what is happening at various times during the period of the wave and its reflection. In the first diagram at right, the wave is traveling to the left, while the reflection travels to the right. At this point in the motion, the wave and its reflection are perfectly aligned, resulting in constructive interference. The wave and its reflection continue to travel in their respective directions. At a time equivalent to one-quarter of a period later, the two waves are now 180 o out of phase, resulting in complete destructive interference. After a time equal to one-half a period, the waves are again aligned, this time in the opposite orientation as before. Again, the overall result is constructive interference.

28 After three-quarters of a period have passed, the wave and its reflection are once again 180 o out of phase and destructive interference occurs. The cycle is complete after a time equal to 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 are certain points in the resultant wave pattern that always have a zero displacement. These are the 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 onehalf a period apart. The second constructive interference pattern is inverted with respect to the next one, and that after a time equal to one period of the wave, the constructive interference pattern is exactly the same as the first one. In other words, the period of the standing wave is the 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 or material 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 always zero, and that an antinode is a point on the standing wave where the displacement oscillates with maximum amplitude. Nodes on a standing wave in a string occur at distances one-half a wavelength apart, and antinodes always occur midway between nodes. If we consider the vibrations of a guitar string, or any taut string clamped at two points, the points where the vibrating string is clamped are nodes. (Why?) Putting all of these conditions together means that only those frequencies that result in some multiple of half-wavelengths to be established in the string will cause standing waves in the string. This situation is illustrated in the diagram on the following page, where the first few allowed standing wave patterns for a string of length L clamped at both ends are shown.

29 When a given frequency sets up a standing wave in an object, that frequency is said to be a resonant frequency of the object. The process of establishing a standing wave at a given frequency is called resonance. When resonance occurs in a given object, the amplitude of the standing wave reaches a maximum, just as it did in your string in Exploration 2. If this situation occurs 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 object can occur. For example, in 1850 a bridge over a river in France collapsed when soldiers marching in step set up a standing wave in the bridge. Each soldier s step created a wave, and since the soldiers were marching in step, all the waves were in phase with each other. The resulting standing wave had such a large amplitude that the bridge structure collapsed, killing over 200 soldiers. Another very famous example of this occurred in 1940 at the Tacoma Narrows Bridge in Washington State. Perhaps your instructor will show you a tape of this famous example of resonance. Other, more everyday examples of resonance include the rattling of your windows when a low-flying plane passes overhead, or the vibration of your chest when you are at a loud concert. A consideration of resonance is also important in heavy machinery. When rotating machinery 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 vibrations could potentially result in a resonance condition within the machine, causing vibrations of an amplitude large enough to cause the machine to break apart. Resonance can occur in sound waves as well. This type of resonance is a very important factor in making wind instruments and organ pipes sound the way they do.

30 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 part a? c) If the speed of a wave in the string shown is 10 m/s, find the frequencies of each of the waves in part a. Hint: Remember your wave relationship! d) Is there any relationship between the frequencies you found in part c? If so, what is that relationship?

31 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 those in 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 than the first four standing wave frequencies in the original string? 4. Suppose that two waves are interfering with each other. Will the amplitude of the resulting wave 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 as these two waves pass through. Wave 1 Wave 2 Displacement (m) time time (s) displacement (m) time time (s) Displacement (m)

32 The Mathematical Description of a Wave You may have noticed that the diagrams used to illustrate the transverse wave look very much 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. You would get a picture similar to that shown below, where your picture represents the shape of the wave 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 description would be πλ = 2sin,xyA where A is the amplitude of the wave and λ is the wavelength. This mathematical description would enable you to draw a picture of the wave as a function of position x along the direction of travel of the wave. Again, it is important to remember that this description represents the wave at a certain point in time. What if we were interested in what was happening at a certain point along the wave as a function of time? In other words, what is the medium that carries the wave doing at a certain position? For example, in Demonstration 1 you placed a piece of tape at a fixed position on the rope, 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 most oscillatory motion, the motion of that fixed point in the medium can be described by a sinusoidal function as well. If you were to draw a graph representing the position of the fixed point in the medium as a function of time, you would get a graph similar to the one shown below. This motion can also be described with a sinusoidal function. The appropriate description would be y = Asin( 2πft), where A is the amplitude of the wave and f is its frequency. This mathematical description would enable you to draw a representation of the position of a certain point in the medium carrying the wave as a function of time. Again, it is important to remember that this description represents a fixed point in the medium as the wave travels through that medium.

33 In reality, of course, both of these things are happening together. The wave travels through the medium, displacing many points in the medium as it travels through. Any complete mathematical description would have to take all points in the medium into account at all times as the wave travels. For problem solving purposes, however, we can look at these the two descriptions separately. The same descriptions can be used to describe longitudinal waves. The wavelength and frequency 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 descriptions represents. While the amplitude for a longitudinal wave can indeed still represent the maximum displacement of a particle in the medium from its normal, undisturbed position, it can be defined to represent other changes as well. For example, for a sound wave the amplitude term may represent a maximum pressure change in the air as the wave passes. This idea will be looked at later in the module. Example 1 π x A given wave traveling through a medium is described by the function y = 10 sin cm. Does 25 this represent the wave as a snapshot at a fixed time or at a point in the medium? What is the amplitude of the wave? What is the wavelength of the wave? Solution These questions are answered by comparing the given function to the standard functions 2π x explained in this Dialog. We see that the given function is of the form y = Asin, so the λ 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 2π π amplitude of the wave is 10 cm. Also by comparison, we see that =, which means that λ 25 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 y = 10sin 30πt. 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 in the medium as a function of time, y = Asin( 2πft). By comparison, we see that 2 πf = 30π, which means that the frequency f = 15 Hz. Since for all waves the frequency and wavelength of the wave are related to the speed of the wave by v = λf, we get v = ( 50cm)(15 Hz) = 750cm / s, where we 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 cm by cycle/sec, the cycle dropped out, giving our final answer units of cm/s.

34 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 in time. c) Write a mathematical description of the motion of a point in the medium carrying the wave as a function of time. 2. A wave can be described by y = 50sin( 70πx)cm and y = 50sin( 50πt)cm. Find the wavelength, the frequency and the speed of the wave. 3. For the wave in Question 2, sketch the two functions in the space below, using appropriate units on the axes of each graph. Choose a scale to make the graphs fill your page and include at least two cycles of the wave in both position and time.

35 4. A graphical depiction of a wave is seen in the diagram below. Write an appropriate mathematical 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 appropriate mathematical 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 the speed of that wave?

36 How are sound waves created? How are sound waves produced? What basic property do all sources of sound have in common? 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 requirements below, 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 think the 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 produces the sound? 10. The strobe light is a device that uses bright flashes of light to help us see motion that is too fast for our unaided eye to interpret. Your instructor will shine a strobe light on the tuning fork. Describe what you see. 11. Listen as your instructor strikes another tuning fork. Why do you think this new fork makes a different sound than the previous one? 12. Tuning forks are usually labeled with a frequency value. How do you think the frequency label on the tuning fork relates to the vibrations of the fork and/or the sound the fork produces? Be specific. 13. Listen and observe while your instructor plucks a tight wire and shows its motion with the strobe light. What does the motion of the wire as it produces sound have in common with the motion of the tuning fork as it produces sound?

37 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 less tension? Your instructor will demonstrate these changes. Record your observations. 15. The wire will be placed under tension and clamped at both ends. Listen to the sound produced by the wire when it is plucked near its center. What effect would you expect to hear in the sound produced by the wire if the tension was not changed, but the wire was clamped at a point near its center, so that the length was effectively cut in half? Listen to the sound produced when your instructor makes this change. How does this sound compare to 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 different lengths. In your summary, include an explanation of why you think the sounds changed the way they did. Y our instructor will connect a frequency generator to a speaker. Listen to the sound produced by the speaker as the input frequency changes. How does the sound produced by the speaker relate to the input frequency form the generator? 17. Does the speaker vibrate as it makes sound? Check your prediction by observing the speaker with the strobe or by lightly laying your fingertips on the speaker cone while it is producing sound. Describe what you felt. S ummarize your observations from this Demonstration. What seems to be necessary for an object to produce sound?

38 Does sound need a medium to travel from one point to another? How does sound get from the source to your ear? What requirements are there for the propagation of sound? This Demonstration should help you answer those questions. The basic apparatus used is in this Demonstration is a Bell jar that can be evacuated (this means that the air can be removed from inside the jar with a pump). An electric buzzer is the source of sound. This apparatus is shown in the photo at right. To do the Demonstration, your instructor will turn on the buzzer, and then slowly remove the air from inside the jar. 18. In the space below, predict whether there will be any changes in how you see the light and/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?

39 The Speed of Sound In this activity you will measure the speed of sound waves in air. You can do this by timing how long it takes for sound waves to travel from one point to another. You will send sound waves down a hollow tube which is closed at one end. As you have observed, when waves on a Slinky reach the clamped end, they are reflected back. Similarly, the sound waves are reflected back when they reach the closed end of the tube. You can measure the time for the waves to go down 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. Microphone Length This end closed 3. Click on Start and make a popping sound with your lips, clapping your hands or a clicking sound 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 and the wave reflected back from the closed end, you should see several almost identical wave patterns 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.

40 4. Measure the time interval between the beginning of the original wave and the beginning of the reflected wave in seconds. (This is the time for the wave to go from your mouth down the tube to the closed end, and back to the microphone.) Measure the period using the cursors on the graph. Notice that as you move the yellow or red cursor, the values displayed in the amplitude and time graph change. Also measure the time interval for a round trip for any 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 from v = T (m/s). Calculate the speed of sound using this formula. Do these values agree? 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 measurement of the speed of sound?

41 Producing and Transmitting Sound Previously you saw that vibrating objects generated sound. Then you saw that a medium was required 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 a mechanical 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 mechanical wave travels by interactions between molecules, each molecule transmitting the disturbance through the medium by interaction with its nearest neighbors. In air the molecules are so widely separated that there are virtually no forces between them. In a gas, such as air, the molecules are 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 each other during collisions. Sound is a longitudinal wave. The longitudinal sound waves traveling through air are responsible for sound traveling from a source and reaching our ears. As the sound wave travels through the air, a portion of the air is suddenly compressed, crowding the air molecules together. These crowded molecules tend to move to adjoining portions of the air, where the density of the molecules is not so great. This results in the density of molecules increasing in this new area, causing the molecules that are there to move to a new area. This process continues, thereby causing the original disturbance (compression) that started the sound wave to be propagated through the air. The fact that the sound waves are compressing molecules together into a greater density as they pass is the reason that sound waves are sometimes called compressional waves. These compression waves correspond to density changes in the gas, as illustrated in the diagram below. For sound waves moving in air that have frequencies in the range of human hearing (roughly 20 20,000 Hz), the oscillations in the gas density take place very rapidly. The speed of sound in air, or any gas, is somewhat dependent upon the temperature of the gas. This is because the molecules of the gas are, on average, moving faster at higher temperatures. Higher average speeds for the molecules translate to a higher speed of the sound wave through the gas. An empirical formula for the speed of sound in air at temperature T ( o C) is v = ( T) m/s. Knowing the air temperature you can determine the speed of sound in that air. What about sound traveling in other materials? You may have noticed when swimming that you could 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 sound travels through solids as well. What is happening on a microscopic level when sound propagates through a solid or a liquid? In actuality, since there are bonds acting between the molecules of a liquid or a solid, the transmission of sound is easier to understand in these cases.

42 When a deforming force acting on a material causes a change in the volume or shape, but the material resumes its original size and shape after the force is removed, the material is said to be elastic. The elasticity of a given material depends upon the strength of the bonds that connect the molecules of the material together. When sound travels through a solid or a liquid, it stretches, or deforms, these bonds. This stretching of the bonds between the molecules is responsible 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 the material. 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 the molecules of a material, the faster sound will travel in that material. As a result, the speed of sound will be higher in liquids than in gases, and higher in solids than in either liquids or gases.

43 Sound Waves 1. A student connects a frequency generator to a speaker in a warm room. Describe what happens to the sound wave produced by the speaker if the student carries it outside where it is 32 o F (0 o C). 2. A speaker sitting on the deck of a pool produces sound waves. What happens to the sound waves as they enter the water in the pool? 3. Ultrasound pulses are used to image tissue in the human body. Typical frequencies used range from 1 MHz to 10 MHz. If the speed of sound in human tissue is 1,550 m/s, find the wavelengths 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 a railroad track, how long will it take before a person with their ear to the steel rail to hear the sound 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 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 the wave. 6. The temperature of the air in a certain room is found to be 27 o C. What is the speed of sound in the room? What would the wavelength of a sound of frequency 256 Hz be?

44 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 wave travels through matter, it moves at a speed that is different from the speed at which the particles of matter carrying the wave are moving. If you could look closely enough, you would find that as a sound wave travels through air, the molecules in the air move back and forth. However, the motion of the molecules is not the same as the overall motion of the wave itself. The molecules oscillate back and forth at a given location while the wave travels from one place to another. As we 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 the speed of the wave is very important. For instance, if an earthquake creates a tidal wave at sea that approaches the shore, the inhabitants waiting for its arrival probably do not care how fast the surface of the water bobs up and down as the wave passes by. On the other hand, they have a major interest in how long it will be until the tidal wave reaches them, a time that can be determined by the speed of the wave itself. Previously you used standing waves to help measure the speed of sound in air. You learned in that a node is a position on a standing wave where the displacement of the medium is zero, while an antinode is a point where the medium oscillates between maximum positive and maximum negative displacement. If we set a string clamped at both ends into vibration, the clamped ends of the string are nodes, since they are unable to move. For the first frequency that will create a standing wave in the string, the point midway between the two ends is then an antinode. These same situations exist in a standing sound wave as well. If a sound wave exists in a tube that is closed at both ends, then both ends of the tube will be nodes. These nodes would be locations where the molecules are not able to vibrate. These ends must be nodes in order for the sound wave to reflect from one end of the tube and travel back to the other. (Remember that a standing wave is the result of a wave interfering constructively and destructively with its reflection.) With both ends of the tube being nodes, the first frequency that will create a standing wave in the closed tube will be that frequency which will cause an antinode at the center of the tube. The frequencies that cause standing waves to be created in a medium are called resonant frequencies. For a string clamped at both ends or a tube closed at both ends, these resonant frequencies will all have nodes at both ends. The term frequently applied to describe these resonant frequencies is harmonics. The wave with the lowest frequency and longest

45 wavelength that causes a standing wave is called the first harmonic. Subsequent harmonics are named according to their order above the first. Illustrated below are the first few harmonics for a string 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 air molecules at the ends of the tube are free to move. This means that the ends of the tube will be antinodes rather than nodes. But since there must be a node in between antinodes, the center of the tube must be a node for the first harmonic. Comparing the first few harmonics for a tube that is open at both ends to those for a tube that is closed at both ends, we can see that the allowed frequencies are exactly the same; the difference is in the locations of the nodes and antinodes. But what if the tube is closed at one end and open at the other? In this case, the closed end will be a node while the open end will be an antinode. These requirements will change the frequency pattern from what we have seen so far. In the case of the first harmonic, there is already a node and an antinode, so the first harmonic will be a wave that has a wavelength of four times the length of the tube instead of two times the length as was seen in the previous two cases. Following the allowed patterns of nodes and antinodes, we see that only the odd harmonics exist. The second resonant frequency is the third harmonic; the third resonant frequency 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 at the end of an open pipe does not occur exactly at the end of the pipe, but rather a short distance away. This distance depends on the diameter of the pipe, and causes the standing waves in the pipe to be established as if the tube was actually slightly longer. This slightly longer length is known as the effective length of the tube. To a good approximation the antinode actually occurs at a point one-fourth the diameter of the pipe beyond the end of the pipe. This effective length for 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 length of 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 or closed pipe end act as a node for the standing wave. Similarly, a loose string end or an open pipe 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 standing wave to the wavelength of the standing wave and the speed of the wave in the material. For a standing wave in a pipe, this speed would be the speed of sound in air. Knowing the nature of the waves in a string or a pipe allows us to predict the resonant frequencies of the standing waves.

46 Where do we see examples of standing waves? One good example would be in the large pipes that are used in a pipe organ. Resonant frequencies of sound in the pipes create the musical sounds that we hear. All musical instruments, whether stringed or wind, rely on the properties of standing waves to create musical notes. You may have even seen barrier ribbons around construction 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 be seen on a tympani drum. The fundamental frequency of the drumhead results in a wave pattern that is circularly symmetric, with the center of the drumhead oscillating with maximum amplitude while the outside edge of the drumhead, along the rim, does not move at all. We say that the center 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 be multiples of the fundamental frequency as was the case in the more basic systems we looked at in earlier Explorations. The frequencies that create standing wave patterns on the drumhead are called characteristic frequencies, and can be computed with good accuracy. However, the mathematics required for these calculations is quite complex and beyond the scope of this course. These characteristic frequencies can be experimentally determined. This is accomplished by using a speaker and a frequency generator to apply vibrations of a known frequency to the drumhead. The loudspeaker is placed directly above the horizontal drumhead and the output of the frequency generator is used to drive the speaker. When the frequency of the generator matches a characteristic frequency of the drumhead, the drumhead will begin to vibrate with large amplitude oscillations. If the drumhead is covered with chalk dust, the dust will collect along the nodes of the vibrations. By noting the pattern of the nodes and the frequency at which the pattern occurs, the resonant frequencies of the drumhead can be determined. Standing waves occur in other systems as well. Sometimes these standing waves can be beneficial, but in other systems the effects of a standing wave can be detrimental or even catastrophic. For example, a beam secured at one end and free on the other is called a cantilever beam. This beam also has characteristic modes of vibration that are more complicated than the systems observed in this module. But an understanding of the modes of vibration is very important in the design of a number of different devices, including airplanes. An airplane wing is essentially a cantilever beam, and an understanding of the modes of vibration of the wing is critical in the design of safe and efficient airliners. Another example of the importance 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 undamped vibrations in the engines. These vibrations became so large that the engine could break loose and smash into the main cabin area of the aircraft. More than one aircraft crashed, killing all aboard. A simple modification of the engine design eliminated the resonant vibrations and kept the engine in place. You have seen how standing waves are established in simple systems such as strings and air tubes. 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 more complicated systems require correspondingly more complicated mathematics to describe the waves. 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 of these three main types.

47 Applications of Standing Waves 7. A pipe organ has pipes of many different lengths. Which pipes (long or short) do you think would 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 situation below. 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 the tube. 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 of the tube is 400 m/s. Resonant sound waves with wavelengths of 40 cm (or 80 mm) are observed in the tube. Find the frequency and the number of the harmonic for each of these waves. 10. While driving to work, Jill notices that workers have strung plastic ribbon around their worksite to prevent people from walking into the hole. Jill notes that between two support poles the ribbon is 1.5 m long and is vibrating in a standing wave pattern with 5 loops and that 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 in the planets atmosphere. Taking a tube with a length of 50 cm, they find that one resonant frequency occurs at 2,520 Hz, and the next resonant frequency at 2,940 Hz. What is the speed of the sound? What is the number of each harmonic? What type of pipe is the crew using (open-open, open-closed, closed-closed)?

48 Echolocation A common application of sound waves involves the detection of reflected sound waves to determine the distance to an object. The simplest possible example of this is an echo. When you yell at a distant cliff, the time it takes for you to hear the echo depends on both the speed of sound in the air and how far it is to the cliff. This same concept applies to such diverse applications as depth finding and the police radar gun, although this last application is somewhat more complicated since the object reflecting the sound is moving. In this Exploration you will look at the basic principles of echolocation, where the position of the object is determined 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. How far did the sound wave from your shout have to travel? 13. Suppose the speed of sound in the area is 325 m/s. If you hear your echo 2 seconds after you shout, how far away is the cliff that the sound wave reflects from? 14. Point the microphone toward the whiteboard and connect it to your computer. Open the Sound Analysis Center Software and Your instructor will inform you of the correct settings for 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 cap them together loudly. Observe the signal on the computer screen. How many signals do you 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 the relationship 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 the computer screen, calculate the distance that the sound wave traveled. Use this result to determine how far away the whiteboard is from the microphone. How does this result compare to the actual

49 distance? 17. Repeat these measurements for a different distance between the plywood sheet and the microphone. How does your measured distance compare to the actual distance in this case? 18. How could the results of this Exploration be used to determine the depth of a lake? What additional 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 2.5 seconds later. How deep is the ocean? The speed of sound in the seawater is 1530 m/s. Show your work below.

50 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 sensors detect the echoes of the sound waves. The longer the delay between the initial pulse and the echo, the further away the object is that causes the echo. Since the speed of sound in water is known, the distance to the object can be determined. Such systems can be used to map out the floor of a lake or the ocean, or to locate shipwrecks far beneath the surface. This same basic principle is used in fish locators, which locate fish by looking for the echoes of sound waves bouncing of the bodies of the fish. Similar systems are used to explore for subsurface oil or mineral deposits. Explosions are set off underground, and sensors called geophones record sound signals from the explosions. A record is made of the time it takes after the explosion for the sound wave energy to reach the geophones. By using the known speeds of sound in various materials, geologists are able to map out the structure of the ground below the surface and determine the most likely locations to find oil or other valuable materials. Radar is also based on the same principle, but uses high frequency radio waves instead of sound waves. Again, the wave speed is known, so the delay of the echoes can be used to locate 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 for food. They have very poor vision, and hunt mainly at night. The bats emit a high frequency sound and listen for the echoes from insects to locate their food. They also use the echoes from larger objects to avoid flying into obstacles. A final example of the application of these principles is seen in ultrasound. Pulses of ultrasound, which are sound waves beyond the limit of human hearing, are used to image soft tissue in the human body. Typical frequencies of ultrasound used in such imaging are from 1 to 10 Megahertz. Special computer systems measure the time required for the ultrasound signals to be reflected from tissues within the body. By mapping out the pattern of signals, an image of the internal organs and structures can be produced.

51 Alarm Reflections of sound and standing waves occur naturally all around us. For example, the rooms we live in are full of reflecting surfaces. The following Exploration will show you how this fact can 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 three whiteboards 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 speaker against one wall of the room, as seen in the photo. If the speaker produces sound, what will 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 its stand or a wood block near the center of the model room. Start the Sound Analysis Center software 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 signal observed 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 an intrusion alarm? Why or why not? What changes would have to be made in order for such a system to be practical?

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53 Palm Pipes You learned previously that one of the most common applications of standing waves was in musical instruments. In this Exploration you will use a simple musical instrument to illustrate the properties 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 is struck against the palm of the hand. Look at the set of palm pipes provided by your instructor. Which pipes (longer or shorter) do you think will create the lowest frequency sounds? 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 one hand, strike the palm of the opposite hand with the open end of the pipe and listen to the sound 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 in various temperatures of air, as discussed previously, calculate the speed of sound in the room. 28. Using what you have learned about the allowed standing wave patterns for sound waves in a pipe, determine what frequencies of sound would establish standing waves in your pipe. Th e range of human hearing is generally considered to be from 20 Hz to 20,000 Hz. Which of 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 class in playing a song with the palm pipes.

54 Materials: for constructing a complete set of 2 octaves (15 pipes). Supplies available from Home Depot 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 if students blow across them. No. Note Length of pipe Frequency #1 F cm 349 Hertz #2 G cm 392 Hertz #3 A cm 440 Hertz #4 Bflat cm 446 Hertz #5 C cm 523 Hertz #6 D cm 587 Hertz #7 E cm 659 Hertz #8 F cm 698 Hertz #9 G 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

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57 How does a Telephone Work? How does a telephone route your call to the phone number you dial? Is there a relationship between the frequencies of the sounds produced by each button and the number on the button? Is a code involved? Are the sounds pure tones or combinations of tones? Using Sound Analysis Center, microphones, or any other equipment you feel is appropriate, analyze the tone structure and sequence of the buttons on a telephone dial pad. You can use an actual telephone, or your instructor may have a tone dialer available for you to use. A tone dialer is a device that duplicates the tones made by the buttons on a telephone. As you carry out your 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 a pattern in the rows of buttons? The columns? In the number sequence? 3. How does the phone company route your call to the phone number you dialed?