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Chapter 16 Wave Motion

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Chapter 16 Wave Motion Chapter 16 Wave Motion CHAPTER OUTLINE 16.1 Propagation of a Disturbance 16.2 Sinusoidal Waves 16.3 The Speed of Waves on Strings 16.4 Reflection and Transmission 16.5 Rate of Energy Transfer by Sinusoidal Waves on Strings 16.6 The Linear Wave Equation L The rich sound of a piano is due to waves on strings that are under tension. Many such strings can be seen in this photograph. Waves also travel on the soundboard, which is visible below the strings. In this chapter, we study the fundamental principles of wave phenomena. (Kathy Ferguson Johnson/PhotoEdit/PictureQuest) 486 Most of us experienced waves as children when we dropped a pebble into a pond. At the point where the pebble hits the water’s surface, waves are created. These waves move outward from the creation point in expanding circles until they reach the shore. If you were to examine carefully the motion of a beach ball floating on the disturbed water, you would see that the ball moves vertically and horizontally about its original position but does not undergo any net displacement away from or toward the point where the pebble hit the water. The small elements of water in contact with the beach ball, as well as all the other water elements on the pond’s surface, behave in the same way. That is, the water wave moves from the point of origin to the shore, but the water is not carried with it. The world is full of waves, the two main types being mechanical waves and electromag- netic waves. In the case of mechanical waves, some physical medium is being dis- turbed—in our pebble and beach ball example, elements of water are disturbed. Elec- tromagnetic waves do not require a medium to propagate; some examples of electromagnetic waves are visible light, radio waves, television signals, and x-rays. Here, in this part of the book, we study only mechanical waves. The wave concept is abstract. When we observe what we call a water wave, what we see is a rearrangement of the water’s surface. Without the water, there would be no wave. A wave traveling on a string would not exist without the string. Sound waves could not travel from one point to another if there were no air molecules between the two points. With mechanical waves, what we interpret as a wave corresponds to the propagation of a disturbance through a medium. Considering further the beach ball floating on the water, note that we have caused the ball to move at one point in the water by dropping a pebble at another location. The ball has gained kinetic energy from our action, so energy must have transferred from the point at which we drop the pebble to the position of the ball. This is a central feature of wave motion—energy is transferred over a distance, but matter is not. All waves carry energy, but the amount of energy transmitted through a medium and the mechanism responsible for that transport of energy differ from case to case. For instance, the power of ocean waves during a storm is much greater than the power of sound waves generated by a single human voice. 16.1 Propagation of a Disturbance In the introduction, we alluded to the essence of wave motion—the transfer of energy through space without the accompanying transfer of matter. In the list of energy transfer mechanisms in Chapter 7, two mechanisms depend on waves—mechanical waves and electromagnetic radiation. By contrast, in another mechanism—matter transfer—the energy transfer is accompanied by a movement of matter through space. All mechanical waves require (1) some source of disturbance, (2) a medium that can be disturbed, and (3) some physical mechanism through which elements of the medium can influence each other. One way to demonstrate wave 487 488 CHAPTER 16 • Wave Motion motion is to flick one end of a long rope that is under tension and has its opposite end fixed, as shown in Figure 16.1. In this manner, a single bump (called a pulse) is formed and travels along the rope with a definite speed. Figure 16.1 represents four consecu- tive “snapshots” of the creation and propagation of the traveling pulse. The rope is the medium through which the pulse travels. The pulse has a definite height and a definite speed of propagation along the medium (the rope). As we shall see later, the proper- ties of this particular medium that determine the speed of the disturbance are the ten- sion in the rope and its mass per unit length. The shape of the pulse changes very little as it travels along the rope.1 We shall first focus our attention on a pulse traveling through a medium. Once we have explored the behavior of a pulse, we will then turn our attention to a wave, which is a periodic disturbance traveling through a medium. We created a pulse on our rope by flicking the end of the rope once, as in Figure 16.1. If we were to move the end of the rope up and down repeatedly, we would create a traveling wave, which has characteris- tics that a pulse does not have. We shall explore these characteristics in Section 16.2. As the pulse in Figure 16.1 travels, each disturbed element of the rope moves in a Figure 16.1 A pulse traveling direction perpendicular to the direction of propagation. Figure 16.2 illustrates this point down a stretched rope. The shape for one particular element, labeled P. Note that no part of the rope ever moves in the of the pulse is approximately un- direction of the propagation. changed as it travels along the rope. A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave. P Compare this with another type of pulse—one moving down a long, stretched spring, as shown in Figure 16.3. The left end of the spring is pushed briefly to the right and then pulled briefly to the left. This movement creates a sudden compression of a P region of the coils. The compressed region travels along the spring (to the right in Fig- ure 16.3). The compressed region is followed by a region where the coils are extended. Notice that the direction of the displacement of the coils is parallel to the direction of propagation of the compressed region. P A traveling wave or pulse that causes the elements of the medium to move parallel to the direction of propagation is called a longitudinal wave. P Sound waves, which we shall discuss in Chapter 17, are another example of longitu- dinal waves. The disturbance in a sound wave is a series of high-pressure and low- pressure regions that travel through air. Figure 16.2 A transverse pulse Some waves in nature exhibit a combination of transverse and longitudinal dis- traveling on a stretched rope. The direction of motion of any element placements. Surface water waves are a good example. When a water wave travels on the P of the rope (blue arrows) is per- surface of deep water, elements of water at the surface move in nearly circular paths, as pendicular to the direction of shown in Figure 16.4. Note that the disturbance has both transverse and longitudinal propagation (red arrows). Compressed Compressed Stretched Stretched Figure 16.3 A longitudinal pulse along a stretched spring. The displacement of the coils is parallel to the direction of the propagation. 1 In reality, the pulse changes shape and gradually spreads out during the motion. This effect is called dispersion and is common to many mechanical waves as well as to electromagnetic waves. We do not consider dispersion in this chapter. SECTION 16.1 • Propagation of a Disturbance 489 Velocity of propagation Active Figure 16.4 The motion of water elements on the surface of deep water in which a wave is propagating is a combination of transverse and longitudinal displace- ments, with the result that elements at the surface move in nearly circular paths. Each element is displaced both horizontally and vertically from its equilibrium position. At the Active Figures link at http://www.pse6.com, you can observe the displacement of water elements at the surface of the moving waves. components. The transverse displacements seen in Figure 16.4 represent the variations in vertical position of the water elements. The longitudinal displacement can be ex- plained as follows: as the wave passes over the water’s surface, water elements at the highest points move in the direction of propagation of the wave, whereas elements at the lowest points move in the direction opposite the propagation. The three-dimensional waves that travel out from points under the Earth’s surface along a fault at which an earthquake occurs are of both types—transverse and longitu- dinal. The longitudinal waves are the faster of the two, traveling at speeds in the range of 7 to 8 km/s near the surface. These are called P waves (with “P” standing for pri- mary) because they travel faster than the transverse waves and arrive at a seismograph (a device used to detect waves due to earthquakes) first. The slower transverse waves, called S waves (with “S” standing for secondary), travel through the Earth at 4 to 5 km/s near the surface. By recording the time interval between the arrivals of these two types of waves at a seismograph, the distance from the seismograph to the point of ori- gin of the waves can be determined.
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