16 - Wave Motion

14 the tension at the free end is maintained because

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Unformatted text preview: string that is free to move vertically, as shown in Figure 16.14. The tension at the free end is maintained because the string is tied to a ring of negligible mass that is free to slide vertically on a smooth post. Again, the pulse is reflected, but this time it is not inverted. When it reaches the post, the pulse exerts a force on the free end of the string, causing the ring to accelerate upward. The ring overshoots the height of the incoming pulse, and then the downward component of the tension force pulls the ring back down. This movement of the ring produces a reflected pulse that is not inverted and that has the same amplitude as the incoming pulse. Finally, we may have a situation in which the boundary is intermediate between these two extremes. In this case, part of the incident pulse is reflected and part undergoes transmission — that is, some of the pulse passes through the boundary. For instance, suppose a light string is attached to a heavier string, as shown in Figure 16.15. When a pulse traveling on the light string reaches the boundary between the two, part of the pulse is reflected and inverted and part is transmitted to the heavier string. The reflected pulse is inverted for the same reasons described earlier in the case of the string rigidly attached to a support. Note that the reflected pulse has a smaller amplitude than the incident pulse. In Section 16.8, we shall learn that the energy carried by a wave is related to its amplitude. Thus, according to the principle of the conservation of energy, when the pulse breaks up into a reflected pulse and a transmitted pulse at the boundary, the sum of the energies of these two pulses must equal the energy of the incident pulse. Because the reflected pulse contains only part of the energy of the incident pulse, its amplitude must be smaller. (b) Incident pulse (c) (a) Reflected pulse Transmitted pulse (d) Figure 16.14 The reflection of a traveling wave pulse at the free end of a stretched string. The reflected pulse is not inverted. Reflected pulse (b) Figure 16.15 (a) A pulse traveling to the right on a light string attached to a heavier string. (b) Part of the incident pulse is reflected (and inverted), and part is transmitted to the heavier string. 503 16.7 Sinusoidal Waves Incident pulse (a) Figure 16.16 Reflected pulse (a) A pulse traveling to the right on a heavy string attached to a lighter string. (b) The incident pulse is partially reflected and partially transmitted, and the reflected pulse is not inverted. Transmitted pulse (b) When a pulse traveling on a heavy string strikes the boundary between the heavy string and a lighter one, as shown in Figure 16.16, again part is reflected and part is transmitted. In this case, the reflected pulse is not inverted. In either case, the relative heights of the reflected and transmitted pulses depend on the relative densities of the two strings. If the strings are identical, there is no discontinuity at the boundary and no reflection takes place. According to Equation 16.4, the speed of a wave on a string increases as the mass per unit length of the string decreases. In other words, a pulse travels more slowly on a heavy string than on a light string if both are under the same tension. The following general rules apply to reflected waves: When a wave pulse travels from medium A to medium B and vA Q vB (that is, when B is denser than A), the pulse is inverted upon reflection. When a wave pulse travels from medium A to medium B and vA P vB (that is, when A is denser than B), the pulse is not inverted upon reflection. 16.7 SINUSOIDAL WAVES In this section, we introduce an important wave function whose shape is shown in Figure 16.17. The wave represented by this curve is called a sinusoidal wave because the curve is the same as that of the function sin plotted against . The sinusoidal wave is the simplest example of a periodic continuous wave and can be used to build more complex waves, as we shall see in Section 18.8. The red curve represents a snapshot of a traveling sinusoidal wave at t 0, and the blue curve represents a snapshot of the wave at some later time t. At t 0, the function describing the positions of the particles of the medium through which the sinusoidal wave is traveling can be written y A sin 2 A sin 2 (x vt v (16.5) x where the constant A represents the wave amplitude and the constant is the wavelength. Thus, we see that the position of a particle of the medium is the same whenever x is increased by an integral multiple of . If the wave moves to the right with a speed v, then the wave function at some later time t is y y vt) (16.6) That is, the traveling sinusoidal wave moves to the right a distance vt in the time t, as shown in Figure 16.17. Note that the wave function has the form f (x vt ) and x t=0 Figure 16.17 t A one-dimensional sinusoidal wave traveling to the right with a speed v. The red curve represents a snapshot of the wave at t 0 , and the blue curve represents a sna...
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This note was uploaded on 03/24/2010 for the course PHYSICS 2202 taught by Professor Mihalisin during the Spring '09 term at Temple.

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