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# Conceptual Physics - Chapter 19: Vibrations and Waves

This Prezi contains the contents of Hewitt's "Conceptual Physics" text (Ninth Edition), Chapter 19, Vibrations and Waves.
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## David Sharp

on 14 March 2013

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#### Transcript of Conceptual Physics - Chapter 19: Vibrations and Waves

Vibrations and Waves Chapter 19 In a general sense, anything that switches back and forth, to and fro, side to side, in and out, off and on, loud and soft, or up and down is vibrating. A vibration is a wiggle in time. A wiggle in both space and time is a wave. A wave extends from one place to another. Introduction If we suspend a stone at the end of a piece of string, we have a simple pendulum. Pendulums swing to and fro with such regularity that for a long time they were used to control the motion of most clocks. They can still be found in grandfather clocks and cuckoo clocks. Galileo discovered that the time a pendulum takes to swing to and fro through small distances depends only on the length of the pendulum. Vibration of a Pendulum The to-and-fro vibratory motion (often called oscillatory motion) of a swinging pendulum in a small arc is called simple harmonic motion. The pendulum bob filled with sand in Figure 19.2 exhibits simple harmonic motion above a conveyor belt. When the conveyor belt is not moving (left), the sand traces out a straight line. More interestingly, when the conveyor belt is moving at constant speed (right), the sand traces out a special curve known as a sine curve. Wave Description Most information about our surroundings comes to us in some form of waves. It is through wave motion that sounds come to our ears, light to our eyes, and electromagnetic signals to our radios and television sets. Through wave motion, energy can be transferred from a source to a receiver without the transfer of matter between the two points. Wave Motion Fasten one end of a rope to a wall and hold the free end in your hand. If you suddenly twitch the free end up and then down, a pulse will travel along the rope and back (Figure 19.8). In this case the motion of the rope (up and down arrows) is at right angles to the direction of wave speed. The right-angled, or sideways, motion is called transverse motion. Now shake the rope with a regular, continuing up-and-down motion, and the series of pulses will produce a wave. Since the motion of the medium (the rope in this case) is transverse to the direction the wave travels, this type of wave is called a transverse wave. Transverse Waves Whereas a material object like a rock will not share its space with another rock, more than one vibration or wave can exist at the same time in the same space. If we drop two rocks in water, the waves produced by each can overlap and form an interference pattern. Within the pattern, wave effects may be increased, decreased, or neutralized. Interference 29. What is the minimum speed for orbiting the Earth in close orbit? The maximum speed? What happens above this speed?

Answer: The minimum speed for close Earth orbit is 8 km/s; the maximum speed is 1.2 km/s. Above that speed the object would escape from the Earth’s gravity. Answer: The sum of kinetic and potential energies is constant for both circular and elliptical orbits. 28. Is the sum of kinetic and potential energies a constant for satellites in circular orbits, elliptical orbits, or both? 27. With respect to the apogee and perigee of an elliptical orbit, where is the gravitational potential greatest? Least?

Answer: The gravitational potential energy is the greatest when it is the farthest away (the apogee); it is the least when it is closest (the perigee). 26. Why is kinetic energy a variable for a satellite in an elliptical orbit?

Answer: The kinetic energy changes because there is a component of force in the direction of motion, causing the velocity to change. 25. Why is kinetic energy a constant for a satellite in circular orbit?

Answer: Kinetic energy is constant because the force is perpendicular to the motion and therefore the velocity is constant. 24. After discovering his first two laws, how much more time did it take Kepler to discover his third law?

Answer: 10 years 23. In Kepler's thinking, what was the direction of force on a planet? In Newton's thinking, what was the direction of force?

Answer: Kepler thought the force acted in the direction of motion of the satellite; Newton realized that the force acted between the satellite and the Sun. 22. Did Kepler view planets as projectiles moving under the influence of the sun?

Answer: No. 21. What did Kepler discover about the speed of planets and their distance from the sun?

Answer: He discovered that the speed of the planets decreases as they get farther away from the sun. 20. Who gathered the data that showed that the planets travel in elliptical paths around the sun? Who discovered this fact? Who explained this fact?

Answer: Tycho Brahe gathered the data; Johannes Kepler analyzed the data and discovered that the planets travelled in elliptical paths. Isaac Newton explained why. 19. At what part of an elliptical orbit does a satellite have the greatest speed? The least speed?

Answer: A satellite has its greatest speed when it is closest to the Earth; it has its slowest speed when it is the farthest away. 18. Why does the force of gravity change the speed of a satellite in an elliptical orbit?

Answer: In the case of a satellite in an elliptical orbit, there is a component of gravity in the direction of motion, so as it is going away it slows down, and while it is coming toward the Earth, it will speed up. 17. For orbits of greater altitude, is the period greater or less?

Answer: For orbits of higher altitude, the orbital speed is less, the distance is greater, and therefore the period is longer. 16. How much time is taken for a complete revolution of a satellite in close orbit about the Earth?

Answer: 90 minutes. 15. Why doesn't the force of gravity change the speed of a satellite in circular orbit?

Answer: Because the force of gravity is always perpendicular to the velocity of the satellite (there is no component of the force in the direction of motion, so it has no effect on the velocity.) 14. Are the planets of the solar system simply projectiles falling around and around the sun?

Answer: Yes. 13. Why is it important that the projectile in the last question be above the Earth's atmosphere?

Answer: If the projectile is travelling through the atmosphere, it will slow down and not follow the curvature of the Earth and will eventually hit the ground. 12. Why will a projectile that moves horizontally at 8 km/s follow a curve that matches the curvature of the Earth?

Answer: The projectile will fall 5 m for every 8 km it travels sideways, and that matches the curvature of the Earth. 11. How can a projectile “fall around the Earth”?

Answer: If the projectile is launched fast enough so that it falls parallel to the curvature of the Earth. 10. A projectile is launched vertically at 100 m/s. If air resistance can be neglected, at what speed will it return to its initial level?

Answer: It will return at the same speed, 100 m/s. 9. A projectile is launched upward at an angle of 75° from the horizontal and strikes the ground a certain distance down range. For what other angle of launch at the same speed would this projectile land just as far away?

Answer: The complement of the angle, or 15 degrees. 8. Does your answer to the last question depend on the angle at which the projectile is launched?

Answer: No. The distances fallen will be the same regardless of the initial launch angle. 7. A projectile falls beneath the straight-line path it would follow if there were no gravity. How many meters does it fall below this line if it has been traveling for 1 s? For 2 s?

Answer: After 1 second, it will fall 5 m below the straight-line path (5t2 with t=1); after 2 seconds, it will have fallen 20 m below that line (5t2 with t=2). 6. A rock is thrown upward at an angle. What happens to the vertical component of its velocity as it rises? As it falls?

Answer: The vertical component decreases as it rises (gravity is slowing it down) and increases as it falls (gravity causes it to speed up). 5. A rock is thrown upward at an angle. What happens to the horizontal component of its velocity as it rises? As it falls?

Answer: The remains constant throughout its motion, both as it rises and as it falls. 4. True or false: When air resistance does not affect the motion of a projectile, its horizontal and vertical components of velocity remain constant.

Answer: False. The horizontal component of the velocity remains constant, but the vertical velocity component changes because of gravity. 3. Why does the vertical component of velocity for a projectile change with time, whereas the horizontal component of velocity doesn't?

Answer: Because the force of gravity acts vertically to change the vertical component of the velocity, but there is no force acting horizontally to change the horizontal component. 2. What exactly is a projectile?

Answer: Any object that is projected by some means and continues in motion by its own inertia. Why does a horizontally moving projectile have to have a large speed to become an earth satellite?

Answer: It must be travelling fast enough so that it follows the curvature of the Earth and doesn’t hit the ground. Review Questions Ellipse The oval path followed by a satellite. The sum of the distances from any point on the path to two points called foci is a constant. When the foci are together at one point, the ellipse is a circle. As the foci get farther apart, the ellipse gets more “eccentric.”

Escape Speed The speed that a projectile, space probe, or similar object must reach to escape the gravitational influence of the Earth or celestial body to which it is attracted. Summary of Terms, cont. Projectile Any object that moves through the air or through space under the influence of gravity.

Parabola The curved path followed by a projectile near the Earth under the influence of gravity only.

You can test this yourself: throw a heavy boulder horizontally, then vertically—you'll find the horizontal throw to be considerably faster than the vertical throw. So maximum range for heavy projectiles thrown by humans is attained for angles less than 45 degrees—and not because of air drag. Upwardly Launched Projectiles, cont. Without air drag, maximum range for a baseball would occur when it is batted 45 degrees above the horizontal. Because of air drag, best range occurs when the ball leaves the bat at about 43 degrees. Air drag is more significant for golf balls, where angles of about 38 degrees result in maximum range. For heavy projectiles like javelins and the shot, air drag has less effect on range. A javelin, being heavy and presenting a very small cross section to the air, follows an almost perfect parabola when thrown. So does a shot. For such a projectile, maximum range for equal launch speeds would occur for a launch angle of about 45 degrees (slightly less because the launching height is above ground level). Upwardly Launched Projectiles, cont. Ranges of a projectile shot at the same speed at different projection angles. Figure 10.9 Figure 10.9 shows the paths of several projectiles, all with the same initial speed but different launching angles. The figure neglects the effects of air drag, so the trajectories are all parabolas. Notice that these projectiles reach different altitudes, or heights above the ground. They also have different horizontal ranges, or distances traveled horizontally. The remarkable thing to note from Figure 10.9 is that the same range is obtained from two different launching angles when the angles add up to 90 degrees! An object thrown into the air at an angle of 60 degrees, for example, will have the same range as if it were thrown at the same speed at an angle of 30 degrees. For the smaller angle, of course, the object remains in the air for a shorter time. The greatest range occurs when the launching angle is 45 degrees—and when air drag is negligible. Upwardly Launched Projectiles, cont. Figure 10.8 shows the trajectory traced by a projectile launched with the same speed at a steeper angle. Notice the initial velocity vector has a greater vertical component than when the angle of launch is smaller. This greater component results in a trajectory that reaches a greater height. But the horizontal component is less, and the range is less. Upwardly Launched Projectiles, cont. The velocity of a projectile at various points along its trajectory. Note that the vertical component changes and the horizontal component is the same everywhere. Figure 10.7 In Figure 10.7 we see vectors representing both horizontal and vertical components of velocity for a projectile following a parabolic trajectory. Notice that the horizontal component is everywhere the same, and only the vertical component changes. Note also that the actual velocity is represented by the vector that forms the diagonal of the rectangle formed by the vector components. At the top of the trajectory the vertical component is zero, so the actual velocity there is only the horizontal component of velocity. Everywhere else the magnitude of velocity is greater (just as the diagonal of a rectangle is greater than either of its sides). Upwardly Launched Projectiles, cont. 2. With no air drag the cannonball will travel a horizontal distance of 100 m [d = = (20 m/s)(5 s) = 100 m]. Note that since gravity acts only vertically and there is no acceleration in the horizontal direction, the cannonball travels equal horizontal distances in equal times. This distance is simply its horizontal component of velocity multiplied by the time (and not 5t2, which applies only to vertical motion under the acceleration of gravity). Check Your Answer 2. If the horizontal component of the cannonball's velocity were 20 m/s, how far downrange would the cannonball be at the end of 5 s? Check Yourself 1. Vertical distance beneath the dashed line at the end of 5 s is 125 m [d = 5t2 = 5(5)2 = 5(25) = 125 m]. Interestingly enough, this distance doesn't depend on the angle of the cannon. If air drag is neglected, any projectile will fall 5t2 meters below where it would have reached if there were no gravity. Check Your Answer Note another thing from Figure 10.6. The cannonball moves equal horizontal distances in equal time intervals. That's because no acceleration takes place horizontally. The only acceleration is vertical, in the direction of Earth's gravity. The vertical distance it falls below the imaginary straight-line path during equal time intervals continuously increases with time.

Drop two balls of different mass and they accelerate at g. Let them slide without friction down the same incline and they slide together at the same fraction of g. Tie them to strings of the same length so they are pendulums, and they swing to and fro in unison. In all cases, the motions are independent of mass. A long pendulum has a longer period than a short pendulum; that is, it swings to and fro less frequently than a short pendulum. A grandfather's clock pendulum with a length of about 1 m, for example, swings with a leisurely period of 2 s, while the much shorter pendulum of a cuckoo clock swings with a period that is less than a second. In addition to length, the period of a pendulum depends on the acceleration of gravity. Oil and mineral prospectors use very sensitive pendulums to detect slight differences in this acceleration. The acceleration due to gravity varies due to the variety of underlying formations. Figure 19.2
Frank Oppenheimer at the San Francisco Exploratorium demonstrates (a) a straight line traced by a swinging pendulum bob that leaks sand on the stationary conveyor belt. (b) When the conveyor belt is uniformly moving, a sine curve is traced. A sine curve can also be traced by a bob attached to a spring undergoing vertical simple harmonic motion (Figure 19.3). A sine curve is a pictorial representation of a wave. Just as with a water wave, the high points of a sine wave are called crests, and the low points are called troughs. The straight dashed line in the figure represents the “home” position, or midpoint of the vibration. The term amplitude refers to the distance from the midpoint to the crest (or trough) of the wave. So the amplitude equals the maximum displacement from equilibrium. Figure 19.3
Electrons in the transmitting antenna vibrate 940,000 times each second and produce 940-kHz radio waves. If an object's frequency is known, its period can be calculated, and vice versa. Suppose, for example, that a pendulum makes two vibrations in one second. Its frequency is 2 Hz. The time needed to complete one vibration--that is, the period of vibration--is ½ second. Or if the vibration frequency is 3 Hz, then the period is 1/3 second. The frequency and period are the inverse of each other:

or vice versa: 1. What is the frequency in vibrations per second of a 60-Hz wave? What is its period? Check Yourself A 60-Hz wave vibrates 60 times per second and has a period of 1/60 second. 2. Gusts of wind make the Sears Building in Chicago sway back and forth at a vibration frequency of about 0.1 Hz. What is its period of vibration? The period is 1/frequency = 1/(0.1 Hz) = 1/(0.1 vibration/s) = 10 s. Each vibration therefore takes 10 seconds. Wave motion can be most easily understood by first considering the simple case of a horizontally stretched rope. If one end of such a rope is shaken up and down, a rhythmic disturbance travels along the rope. Each particle of the rope moves up and down, while at the same time the disturbance moves along the length of the rope. The medium, rope or whatever, returns to its initial condition after the disturbance has passed. What is propagated is the disturbance, not the medium itself. Perhaps a more familiar example of wave motion is provided by a water wave. If a stone is dropped into a quiet pond, waves will travel outward in expanding circles, the centers of which are at the source of the disturbance. In this case we might think that water is being transported with the waves, since water is splashed onto previously dry ground when the waves meet the shore. We should realize, however, that barring obstacles the water will run back into the pond, and things will be much as they were in the beginning: The surface of the water will have been disturbed, but the water itself will have gone nowhere. A leaf on the surface will bob up and down as the waves pass, but will end up where it started. Again, the medium returns to its initial condition after the disturbance has passed. Let us consider another example of a wave to illustrate that what is transported from one place to another is a disturbance in a medium, not the medium itself. If you view a field of tall grass from an elevated position on a gusty day, you will see waves travel across the grass. The individual stems of grass do not leave their places; instead, they swing to and fro. Furthermore, if you stand in a narrow footpath, the grass that blows over the edge of the path, brushing against your legs, is very much like the water that doused the shore in our earlier example. While wave motion continues, the tall grass swings back and forth, vibrating between definite limits but going nowhere. When the wave motion stops, the grass returns to its initial position. Wave Speed The speed of periodic wave motion is related to the frequency and wavelength of the waves. We can understand this by considering the simple case of water waves (Figures 19.5 and 19.6). Figure 19.5 - Water waves Figure 19.6 - A top view of water waves. Imagine that we fix our eyes on a stationary point on the surface of water and observe the waves passing by this point. We can measure how much time passes between the arrival of one crest and the arrival of the next one (the period), and also observe the distance between crests (the wavelength). We know that speed is defined as distance divided by time. In this case, the distance is one wavelength and the time is one period, so wave speed 5 wavelength/period. For example, if the wavelength is 10 meters and the time between crests at a point on the surface is 0.5 second, the wave moves 10 meters in 0.5 seconds and its speed is 10 meters divided by 0.5 seconds, or 20 meters per second.

Since period is equal to the inverse of frequency, the formula wave speed = wavelength/period can also be written

This relationship holds true for all kinds of waves, whether they are water waves, sound waves, or light waves. Check Yourself 1. If a train of freight cars, each 10 m long, rolls by you at the rate of three cars each second, what is the speed of the train? 30 m/s. We can see this in two ways. (a) According to the speed definition from Chapter 2, v = d/t = (3 × 10 m)/1 s = 30 m/s, since 30 m of train passes you in 1 s. (b) If we compare our train to wave motion, where wavelength corresponds to 10 m and frequency is 3 Hz, then Speed = wavelength × frequency = 10 m × 3 Hz = 10 m × 3/s = 30 m/s. 2. If a water wave oscillates up and down three times each second and the distance between wave crests is 2 m, what is its frequency? Its wavelength? Its wave speed? The frequency of the wave is 3 Hz, its wavelength is 2 m, and its wave speed = wavelength × frequency = 2 m × 3/s = 6 m/s. It is customary to express this as the equation v = f where v is wave speed, (the Greek letter lambda) is wavelength, and f is wave frequency. Figure 19.8 - A transverse wave. Waves in the stretched strings of musical instruments and upon the surfaces of liquids are transverse. We will see later that electromagnetic waves, which make up radio waves and light, are also transverse. Longitudinal Waves Not all waves are transverse. Sometimes parts that make up a medium move to and fro in the same direction in which the wave travels. Motion is along the direction of the wave rather than at right angles to it. This produces a longitudinal wave. Both a transverse and a longitudinal wave can be demonstrated with a spring or a Slinky stretched out on the floor, as shown in Figure 19.9. A transverse wave is demonstrated by shaking the end of a Slinky from side to side. A longitudinal wave is demonstrated by rapidly pulling and pushing the end of the Slinky toward and away from you. In this case we see that the medium vibrates parallel to the direction of energy transfer. Part of the Slinky is compressed, and a wave of compression travels along the spring. In between successive compressions is a stretched region, called a rarefaction. Both compressions and rarefactions travel in the same direction along the Slinky. Sound waves are longitudinal waves. Figure 19.9
Both waves transfer energy from left to right. When the end of the Slinky is shaken from side to side, a transverse wave is produced. When it's pushed and pulled rapidly along its length, a longitudinal wave is produced. Figure 19.10
Waves generated by an earthquake. P waves are longitudinal and travel through both molten and solid materials. S waves are transverse and travel only through solid materials. Reflections and refractions of the waves provide information about the Earth's interior. Waves that travel in the ground generated by earthquakes are of two main types: longitudinal P waves, and transverse S waves. (Geology students often remember P waves as “push-pull” waves, and S waves as “side-to-side” waves.) S waves cannot travel through liquid matter, while P waves can travel through both molten and solid parts of the Earth's interior. Study of these waves reveals much about the Earth's interior. The wavelength of a longitudinal wave is the distance between successive compressions or equivalently, the distance between successive rarefactions. The most common example of longitudinal waves is sound in air. Elements of air vibrate to and fro about some equilibrium position as the waves move by. We will treat sound waves in detail in the next chapter. When more than one wave occupies the same space at the same time, the displacements add at every point. This is the superposition principle. So when the crest of one wave overlaps the crest of another, their individual effects add together to produce a wave of increased amplitude. This is called constructive interference (Figure 19.11). When the crest of one wave overlaps the trough of another, their individual effects are reduced. The high part of one wave simply fills in the low part of another. This is called destructive interference. Figure 19.11
Constructive and destructive interference in a transverse wave. Wave interference is easiest to see in water. In Figure 19.12 we see the interference pattern made when two vibrating objects touch the surface of water. We can see the regions where a crest of one wave overlaps the trough of another to produce regions of zero amplitude. At points along these regions, the waves arrive out of step. We say they are out of phase with each other. Figure 19.12
Two sets of overlapping water waves produce an interference pattern. The left diagram is an idealized drawing of the expanding waves from the two sources. The right diagram is a photograph of an actual interference pattern. Interference is characteristic of all wave motion, whether the waves are water waves, sound waves, or light waves. We will treat the interference of sound in the next chapter and the interference of light in Chapter 28. Standing Waves If we tie a rope to a wall and shake the free end up and down, we produce a train of waves in the rope. The wall is too rigid to shake, so the waves are reflected back along the rope. By shaking the rope just right, we can cause the incident and reflected waves to form a standing wave, where parts of the rope, called the nodes, are stationary. Nodes are the regions of minimal or zero displacement, with minimal or zero energy. Antinodes (not labeled in Figure 19.13), on the other hand, are the regions of maximum displacement and maximum energy. You can hold your fingers just over and under the nodes and the rope doesn't touch them. Other parts of the rope, especially the antinodes, would make contact with your fingers. Antinodes occur halfway between nodes. Figure 19.13
The incident and reflected waves interfere to produce a standing wave. Standing waves are the result of interference (and as we will see in the next chapter, resonance). When two sets of waves of equal amplitude and wavelength pass through each other in opposite directions, the waves are steadily in and out of phase with each other. This occurs for a wave that reflects upon itself. Stable regions of constructive and destructive interference are produced It is easy to make standing waves yourself. Tie a rope, or better, a rubber tube between two firm supports. Shake the tube from side to side with your hand near one of the supports. If you shake the tube with the right frequency, you will set up a standing wave as shown in Figure 19.14a. Shake the tube with twice the frequency, and a standing wave of half the previous wavelength, having two loops, will result. (The distance between successive nodes is a half wavelength; two loops make up a full wavelength.) Triple the frequency, and a standing wave with one-third the original wavelength, having three loops, results, and so forth. Figure 19.14
(a) Shake the rope until you set up a standing wave of one segment (1/2 wavelength). (b) Shake with twice the frequency and produce a wave with two segments (1 wavelength). (c) Shake with three times the frequency and produce three segments (1½ wavelengths). Standing waves are set up in the strings of musical instruments when plucked, bowed, or struck. They are set up in the air in an organ pipe, a trumpet, or a clarinet, and the air of a soda-pop bottle when air is blown over the top. Standing waves can be set up in a tub of water or a cup of coffee by sloshing it back and forth with the right frequency. Standing waves can be produced with either transverse or longitudinal vibrations. 1. Is it possible for one wave to cancel another wave so that no amplitude remains? Check Yourself Yes. This is called destructive interference. In a standing wave in a rope, for example, parts of the rope have no amplitude--the nodes. 2. Suppose you set up a standing wave of three segments, as shown below. If you shake with twice as much frequency, how many wave segments will occur in your new standing wave? How many wavelengths? If you impart twice the frequency to the rope, you'll produce a standing wave with twice as many segments. You'll have six segments. Since a full wavelength has two segments, you'll have three complete wavelengths in your standing wave. Doppler Effect A pattern of water waves produced by a bug jiggling its legs and bobbing up and down in the middle of a quiet puddle is shown in Figure 19.15. The bug is not going anywhere but is merely treading water in a fixed position. The waves it makes are concentric circles, because wave speed is the same in all directions. If the bug bobs in the water at a constant frequency, the distance between wave crests (the wavelength) is the same in all directions. Waves encounter point A as frequently as they encounter point B. This means that the frequency of wave motion is the same at points A and B, or anywhere in the vicinity of the bug. This wave frequency is the same as the bobbing frequency of the bug. Figure 19.15
Top view of water waves made by a stationary bug jiggling in still water. Suppose the jiggling bug moves across the water at a speed less than the wave speed. In effect, the bug chases part of the waves it has produced. The wave pattern is distorted and is no longer made of concentric circles (Figure 19.16). The center of the outer wave was made when the bug was at the center of that circle. The center of the next smaller wave was made when the bug was at the center of that circle, and so forth. The centers of the circular waves move in the direction of the swimming bug. Although the bug maintains the same bobbing frequency as before, an observer at B would see the waves coming more often. The observer would measure a higher frequency. This is because each successive wave has a shorter distance to travel and therefore arrives at B more frequently than if the bug weren't moving toward B. An observer at A, on the other hand, measures a lower frequency because of the longer time between wave-crest arrivals. This is because to reach A, each crest has to travel farther than the one ahead of it due to the bug's motion. This change in frequency due to the motion of the source (or receiver) is called the Doppler effect (after the Austrian scientist Christian Doppler, 1803-1853). Figure 19.16
Water waves made by a bug swimming in still water toward point B. Water waves spread over the flat surface of the water. Sound and light waves, on the other hand, travel in three-dimensional space in all directions like an expanding balloon. Just as circular waves are closer together in front of the swimming bug, spherical sound or light waves ahead of a moving source are closer together and reach a receiver more frequently. The Doppler effect is evident when you hear the changing pitch of a car horn as the car passes you. When the car approaches, the pitch is higher than normal (higher like a higher note on a musical scale). This is because the crests of the sound waves are hitting your ear more frequently. And when the car passes and moves away, you hear a drop in pitch because the crests of the waves are hitting your ear less frequently. Figure 19.17
The pitch (frequency) of sound increases when a source moves toward you and decreases when the source moves away. The Doppler effect also occurs for light. When a light source approaches, there is an increase in its measured frequency; and when it recedes, there is a decrease in its frequency. An increase in frequency is called a blue shift, because the increase is toward the high frequency, or blue end of the color spectrum. A decrease in frequency is called a red shift, referring to a shift toward the lower-frequency, or red, end of the color spectrum. Distant galaxies, for example, show a red shift in the light they emit. A measurement of this shift permits a calculation of their speeds of recession. A rapidly spinning star shows a red shift on the side turning away from us and a blue shift on the side turning toward us. This enables a calculation of the star's spin rate. Check Yourself When a sound source moves toward you, do you measure an increase or decrease in wave speed? Neither! It is the frequency of a wave that undergoes a change where there is motion of the source, not the wave speed. Be clear about the distinction between frequency and speed. How frequently a wave vibrates is altogether different from how fast the disturbance moves from one place to another. Bow Waves When the speed of a source is as great as the speed of the waves it produces, something interesting happens. The waves pile up in front of the source. Consider the bug in our previous example when it swims as fast as the wave speed. Can you see that the bug will keep up with the waves it produces? Instead of the waves moving ahead of the bug, they superimpose and hump up on one another directly in front of the bug (Figure 19.18). The bug moves right along with the leading edge of the waves it is producing. Figure 19.18
Wave pattern made by a bug swimming at wave speed. A similar thing happens when an aircraft travels at the speed of sound. In the early days of jet aircraft, it was believed that this pile-up of sound waves in front of the airplane imposed a “sound barrier” and that to go faster than the speed of sound, the plane would have to “break the sound barrier.” What actually happens is that the overlapping wave crests disrupt the flow of air over the wings, making it more difficult to control the craft. But the barrier is not real. Just as a boat can easily travel faster than the waves it produces, with sufficient power an aircraft easily travels faster than the speed of sound. Then we say that it is supersonic. A supersonic airplane flies into smooth, undisturbed air because no sound wave can propagate out in front of it. Similarly, a bug swimming faster than the speed of water waves finds itself always entering into water with a smooth, unrippled surface. When the bug swims faster than wave speed, ideally it produces a wave pattern as shown in Figure 19.19. It outruns the waves it produces. The waves overlap at the edges, and the pattern made by these overlapping waves is a V shape, called a bow wave, which appears to be dragging behind the bug. The familiar bow wave generated by a speedboat knifing through the water is not a typical oscillatory wave. It is a disturbance produced by the overlapping of many circular waves. Figure 19.19
A bow wave, the pattern made by a bug swimming faster than wave speed. The points at which adjacent waves overlap (x) produce the V shape. Some wave patterns made by sources moving at various speeds are shown in Figure 19.20. Note that after the speed of the source exceeds wave speed, increased speed of the source produces a narrower V shape. Figure 19.20
Patterns made by a bug swimming at successively greater speeds. Overlapping at the edges occurs only when the bug swims faster than wave speed. Shock Waves A speedboat knifing through the water generates a two-dimensional bow wave. A supersonic aircraft similarly generates a three-dimensional shock wave. Just as a bow wave is produced by overlapping circles that form a V, a shock wave is produced by overlapping spheres that form a cone. And just as the bow wave of a speedboat spreads until it reaches the shore of a lake, the conical wake generated by a supersonic craft spreads until it reaches the ground. The bow wave of a speedboat that passes by can splash and douse you if you are at the water's edge. In a sense, you can say that you are hit by a “water boom.” In the same way, when the conical shell of compressed air that sweeps behind a supersonic aircraft reaches listeners on the ground below, the sharp crack they hear is described as a sonic boom. Figure 19.21
This aircraft has just cracked the wave barrier. The cloud is water vapor that has just condensed out of the rapidly expanding air in the rarefied region behind the wall of compressed air. We don't hear a sonic boom from slower-than-sound, or subsonic, aircraft because the sound waves reaching our ears are perceived as one continuous tone. Only when the craft moves faster than sound do the waves overlap to reach the listener in a single burst. The sudden increase in pressure is much the same in effect as the sudden expansion of air produced by an explosion. Both processes direct a burst of high pressure air to the listener. The ear is hard pressed to distinguish between the high pressure from an explosion and the high pressure from many overlapping waves. A water skier is familiar with the fact that next to the high hump of the V-shaped bow wave is a V-shaped depression. The same is true of a shock wave, which usually consists of two cones: a high-pressure cone generated at the bow of the supersonic aircraft and a low-pressure cone that follows at the tail of the craft. The edges of these cones are visible in the photograph of the supersonic bullet below. Figure 19.22
Shock wave of a bullet piercing a sheet of Plexiglas. Light deflecting as it passes through the compressed air makes the shock visible. Look carefully and see the second shock wave originating at the tail of the bullet. Between these two cones the air pressure rises sharply to above atmospheric pressure, then falls below atmospheric pressure before sharply returning to normal beyond the inner tail cone (Figure 19.24). This overpressure suddenly followed by underpressure intensifies the sonic boom. Figure 19.23 A shock wave. Figure 19.24
The shock wave is actually made up of two cones--a high-pressure cone with the apex at the bow of the aircraft and a low-pressure cone with the apex at the tail. A graph of the air pressure at ground level between the cones takes the shape of the letter N. A common misconception is that sonic booms are produced when an aircraft flies through the “sound barrier”--that is, just as the aircraft surpasses the speed of sound. This is the same as saying that a boat produces a bow wave when it first overtakes its own waves. This is not so. The fact is that a shock wave and its resulting sonic boom are swept continuously behind and below an aircraft traveling faster than sound, just as a bow wave is swept continuously behind a speedboat. In Figure 19.25, listener B is in the process of hearing a sonic boom. Listener C has already heard it, and listener A will hear it shortly. The aircraft that generated this shock wave may have broken through the sound barrier hours ago! Figure 19.25
The shock wave has not yet reached listener A, but is now reaching listener B and has already reached listener C. It is not necessary that the moving source be “noisy” to produce a shock wave. Once an object is moving faster than the speed of sound, it will make sound. A supersonic bullet passing overhead produces a crack, which is a small sonic boom. If the bullet were larger and disturbed more air in its path, the crack would be more boomlike.

When a lion tamer cracks a circus whip, the cracking sound is actually a sonic boom produced by the tip of the whip when it travels faster than the speed of sound. Both the bullet and the whip are not vibrating so they are not sound sources. But when traveling at supersonic speeds they produce their own sound as they generate shock waves. Summary of Terms Sine curve A wave form traced by simple harmonic motion, which can be made visible on a moving conveyor belt by a pendulum swinging at right angles above the moving belt.

Amplitude For a wave or vibration, the maximum displacement on either side of the equilibrium (midpoint) position. Wavelength The distance between successive crests, troughs, or identical parts of a wave.

Frequency For a vibrating body or medium, the number of vibrations per unit time. For a wave, the number of crests that pass a particular point per unit time. Hertz The SI unit of frequency. One hertz (symbol Hz) equals one vibration per second.

Period The time in which a vibration is completed. The period of a wave equals the period of the source, and is equal to 1/frequency. Wave speed The speed with which waves pass a particular point: Transverse wave A wave in which the medium vibrates in a direction perpendicular (transverse) to the direction in which the wave travels. Light waves and water waves are transverse. Longitudinal wave A wave in which the medium vibrates in a direction parallel (longitudinal) to the direction in which the wave travels. Sound waves are longitudinal. Interference pattern The pattern formed by superposition of different sets of waves that produces reinforcement in some places and cancellation in others.

Standing wave A stationary wave pattern formed in a medium when two sets of identical waves pass through the medium in opposite directions.

Doppler effect The shift in received frequency due to motion of a vibrating source toward or away from a receiver. Bow wave The V-shaped disturbance made by an object moving across a liquid surface at a speed greater than the wave speed.

Shock wave The cone-shaped disturbance made by an object moving at supersonic speed through a fluid.

Sonic boom The loud sound resulting from the incidence of a shock wave. Review Questions 1. What is a wiggle in time called? A wiggle in space and time? A wiggle in time is called a vibration; a wiggle in space and time is called a wave. 2. Distinguish between the propagation of sound waves and light waves. Sound is the propagation of vibrations through a material medium--a solid, liquid, or gas. Light is a vibration of electric and magnetic fields. 3. What is the source of all waves? The source of all waves is something that is vibrating. 4. What feature about a pendulum makes it useful in a grandfather clock? The period of a pendulum does not depend on the mass of the pendulum or on the size of the arc through which it swings. 5. What is meant by the period of a pendulum? The period of a pendulum is the time it takes to swing back and forth. 6. Which has the longer period, a short or a long pendulum? The longer the length of a pendulum, the longer its period. 7. How is a sine curve related to a wave? A sine curve is a pictorial representation of a wave. 8. Distinguish between these different parts of a wave: period, amplitude, wavelength, and frequency. Period is the time for one complete vibration; Amplitude is the distance from the midpoint to the crest (or trough) of the wave; the wavelength is the distance from the top of one crest to the top of the next one; the frequency is the number of complete vibrations in a given time (usually one second). 9. How many vibrations per second are represented in a radio wave of 101.7 MHz? The number of vibrations per second is equal to the frequency--so there are 101.7 million vibrations per second. 10. How do frequency and period relate to each other? Frequency and period are reciprocals of each other. 11. In one word, what is it that moves from source to receiver in wave motion? Energy No. A vibrating rope transfers energy along the rope while the rope moves back and forth; a water wave transfers energy from the ocean to the shore but the water moves up and down. 12. Does the medium in which a wave travels move with the wave? Give examples to support your answer. 13. What is the relationship among frequency, wavelength, and wave speed? Speed =
frequency x wavelength 14. In what direction are the vibrations relative to the direction of wave travel in a transverse wave? The vibrations are perpendicular to the direction of wave travel. 15. In what direction are the vibrations relative to the direction of wave travel in a longitudinal wave? In a longitudinal wave, the vibrations are in the same direction (parallel) to the travel direction of the wave. 16. The wavelength of a transverse wave is the distance between successive crests (or troughs). What is the wavelength of a longitudinal wave? The distance between the high pressure (compressions) or low pressure (rarefactions) regions in the wave. 17. What is meant by the superposition principle? When more than one wave occupies the same space at the same time, the displacements add at every point. 18. Distinguish between constructive interference and destructive interference. In constructive interference, the crest of one wave adds to the crest of another, and the amplitude increases. In destructive interference, the crest of one wave adds to the trough of another and the amplitude decreases. 19. What kinds of waves can show interference? Interference is characteristic of all wave motion. 20. What causes a standing wave? A standing waves results when an incident wave and a reflective wave interfere and cause parts of the rope to remain stationary. 21. What is a node? What is an antinode? A node is a region of minimal or zero displacement, with minimal or zero energy. An antinode is a region of maximum displacement and maximum energy. 22. In the Doppler effect, does frequency change? Does wavelength change? Does wave speed change? In the Doppler effect, the frequency of the wave changes and the wavelength changes, but the wave velocity does not change. 23. Can the Doppler effect be observed with longitudinal waves, transverse waves, or both? The Doppler effect can be observed with both types of waves. 24. What is meant by a blue shift and a red shift for light? A red shift is a decrease in frequency of a light wave (toward the red end of the spectrum) and a blue shift is an increase in frequency (toward the blue end of the spectrum). 25. How fast must a bug swim to keep up with the waves it produces? How fast must it move to produce a bow wave? It will keep up with the waves it produces if it swims at the same speed as the wave travels. If it swims at a higher speed, it will produce a bow wave. 26. How fast does a supersonic aircraft fly compared with the speed of sound? Faster than the speed of sound. 27. How does the V shape of a bow wave depend on the speed of the source? As the speed of the source increases past the wave speed, the source produces a narrower V-shape. 28. A bow wave on the surface of water is two-dimensional. How about a shock wave in air? A shock wave in air is three-dimensional, producing overlapping spheres that form a cone. 29. True or false: A sonic boom occurs only when an aircraft is breaking through the sound barrier. Defend your answer. False. A sonic boom is created continuously as long as an aircraft is flying faster than the speed of sound. It is only heard when the bow wave created by the plane reaches the ear of a listener. 30. True or false: In order for an object to produce a sonic boom, it must be “noisy.” Give two examples to support your answer. False. Once an object is moving faster than the speed of sound, it will make sound. A bullet and a whip are two examples.
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