16 - Wave Motion

1611 is one solution of the linear wave equation eq

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Unformatted text preview: 16.26). Although we do not prove it here, the linear Summary wave equation is satisfied by any wave function having the form y f (x vt ). Furthermore, we have seen that the linear wave equation is a direct consequence of Newton’s second law applied to any segment of the string. SUMMARY A transverse wave is one in which the particles of the medium move in a direction perpendicular to the direction of the wave velocity. An example is a wave on a taut string. A longitudinal wave is one in which the particles of the medium move in a direction parallel to the direction of the wave velocity. Sound waves in fluids are longitudinal. You should be able to identify examples of both types of waves. Any one-dimensional wave traveling with a speed v in the x direction can be represented by a wave function of the form y f (x (16.1, 16.2) vt ) where the positive sign applies to a wave traveling in the negative x direction and the negative sign applies to a wave traveling in the positive x direction. The shape of the wave at any instant in time (a snapshot of the wave) is obtained by holding t constant. The superposition principle specifies that when two or more waves move through a medium, the resultant wave function equals the algebraic sum of the individual wave functions. When two waves combine in space, they interfere to produce a resultant wave. The interference may be constructive (when the individual displacements are in the same direction) or destructive (when the displacements are in opposite directions). The speed of a wave traveling on a taut string of mass per unit length and tension T is T (16.4) v √ A wave is totally or partially reflected when it reaches the end of the medium in which it propagates or when it reaches a boundary where its speed changes discontinuously. If a wave pulse traveling on a string meets a fixed end, the pulse is reflected and inverted. If the pulse reaches a free end, it is reflected but not inverted. The wave function for a one-dimensional sinusoidal wave traveling to the right can be expressed as y A sin 2 (x vt ) A sin(kx t) (16.6, 16.11) where A is the amplitude, is the wavelength, k is the angular wave number, and is the angular frequency. If T is the period and f the frequency, v, k and can be written v k T f 2 2 T (16.7, 16.14) (16.9) 2f (16.10, 16.12) You should know how to find the equation describing the motion of particles in a wave from a given set of physical parameters. The power transmitted by a sinusoidal wave on a stretched string is 1 2 2A2v (16.21) 511 512 CHAPTER 16 Wave Motion QUESTIONS 1. Why is a wave pulse traveling on a string considered a transverse wave? 2. How would you set up a longitudinal wave in a stretched spring? Would it be possible to set up a transverse wave in a spring? 3. By what factor would you have to increase the tension in a taut string to double the wave speed? 4. When traveling on a taut string, does a wave pulse always invert upon reflection? Explain. 5. Can two pulses traveling in opposite directions on the same string reflect from each other? Explain. 6. Does the vertical speed of a segment of a horizontal, taut string, through which a wave is traveling, depend on the wave speed? 7. If you were to shake one end of a taut rope periodically three times each second, what would be the period of the sinusoidal waves set up in the rope? 8. A vibrating source generates a sinusoidal wave on a string under constant tension. If the power delivered to the string is doubled, by what factor does the amplitude change? Does the wave speed change under these circumstances? 9. Consider a wave traveling on a taut rope. What is the difference, if any, between the speed of the wave and the speed of a small segment of the rope? 10. If a long rope is hung from a ceiling and waves are sent up the rope from its lower end, they do not ascend with constant speed. Explain. 11. What happens to the wavelength of a wave on a string when the frequency is doubled? Assume that the tension in the string remains the same. 12. What happens to the speed of a wave on a taut string when the frequency is doubled? Assume that the tension in the string remains the same. 13. How do transverse waves differ from longitudinal waves? 14. When all the strings on a guitar are stretched to the same tension, will the speed of a wave along the more massive bass strings be faster or slower than the speed of a wave on the lighter strings? 15. If you stretch a rubber hose and pluck it, you can observe a pulse traveling up and down the hose. What happens to the speed of the pulse if you stretch the hose more tightly? What happens to the speed if you fill the hose with water? 16. In a longitudinal wave in a spring, the coils move back and forth in the direction of wave motion. Does the speed of the wave depend on the maximum speed of each coil? 17. When two waves interfere, can the amplitude of the resultant wave be greater than either of the two original waves? Under what conditions? 18. A so...
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