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Chapter 15

Chapter 15. Mechanical Waves. Introduction. At right, you’ll see the piles of rubble from a highway that absorbed just a little of the energy from a wave propagating through the earth in California. In this chapter, we’ll focus on ripples of disturbance moving through various media.

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Chapter 15

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  1. Chapter 15 Mechanical Waves

  2. Introduction • At right, you’ll see the piles of rubble from a highway that absorbed just a little of the energy from a wave propagating through the earth in California. In this chapter, we’ll focus on ripples of disturbance moving through various media.

  3. Waves are for SURFING… yeah! Monterey, CA Cayucos, CA

  4. Get outta there Ghost tree

  5. Types of mechanical waves • Waves that have compressions and rarefactions parallel to the direction of wave propagation are longitudinal. • Waves that have compressions and rarefactions perpendicular to the direction of propagation.

  6. Periodic waves • A detailed look at periodic transverse waves will allow us to extract parameters.

  7. Q15.1 If you double the wavelength l of a wave on a string, what happens to the wave speed v and the wave frequency f? A. v is doubled and f is doubled. B. v is doubled and f is unchanged. C. v is unchanged and f is halved. D. v is unchanged and f is doubled. E. v is halved and f is unchanged.

  8. A15.1 If you double the wavelength l of a wave on a string, what happens to the wave speed v and the wave frequency f? A. v is doubled and f is doubled. B. v is doubled and f is unchanged. C. v is unchanged and f is halved. D. v is unchanged and f is doubled. E. v is halved and f is unchanged.

  9. Mathematical description of a wave • When the description of the wave needs to be more complete, we can generate a wave function with y(x,t).

  10. Q15.2 Which of the following wave functions describe a wave that moves in the –x-direction? A. y(x,t) = A sin (–kx – wt) B. y(x,t) = A sin (kx + wt) C. y(x,t) = A cos (kx + wt) D. both B. and C. E. all of A., B., and C.

  11. A15.2 Which of the following wave functions describe a wave that moves in the –x-direction? A. y(x,t) = A sin (–kx – wt) B. y(x,t) = A sin (kx + wt) C. y(x,t) = A cos (kx + wt) D. both B. and C. E. all of A., B., and C.

  12. A new swell A swell arrived last night and the buoys indicated an 18 second period with 3m height. Sweet!! The wave velocity where the buoys are located is 28 m/s. What is the amplitude, angular frequency, wavelength and wave number of the wave? Write a wave function describing the wave. What is the velocity of the buoy 5 s after the peak of the wave passes under it? Rincon

  13. The speed of a transverse wave • In the first method we will consider a pulse on a string. • Figure 15.11 will show one approach.

  14. Q15.3 y A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. At this time, what is the velocity of a particle of the string at x = a? 0 x a A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide

  15. A15.3 y A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. At this time, what is the velocity of a particle of the string at x = a? 0 x a A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide

  16. Q15.4 y A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. At this time, what is the acceleration of a particle of the string at x = a? 0 x a A. The acceleration is upward. B. The acceleration is downward. C. The acceleration is zero. D. not enough information given to decide

  17. A15.4 y A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. At this time, what is the acceleration of a particle of the string at x = a? 0 x a A. The acceleration is upward. B. The acceleration is downward. C. The acceleration is zero. D. not enough information given to decide

  18. Q15.5 y A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. At this time, what is the velocity of a particle of the string at x = b? 0 x b A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide

  19. A15.5 y A wave on a string is moving to the right. This graph of y(x, t) versus coordinate x for a specific time t shows the shape of part of the string at that time. At this time, what is the velocity of a particle of the string at x = b? 0 x b A. The velocity is upward. B. The velocity is downward. C. The velocity is zero. D. not enough information given to decide

  20. The speed of a transverse wave II • Nylon rope is tied to a stationary support at the top of a vertical mine shaft 80 m deep. The rope is stretched taut by a 20 kg box of mineral samples at the bottom. The mass of the rope is 2 kg. The geologist at the bottom signals by jerking the rope sideways. • What is the speed of the transverse wave on the rope? • If the rope is given SHM with frequency 2 Hz, how many cycles are there in the rope’s length?

  21. Q15.8 The four strings of a musical instrument are all made of the same material and are under the same tension, but have different thicknesses. Waves travel A. fastest on the thickest string. B. fastest on the thinnest string. C. at the same speed on all strings. D. not enough information given to decide

  22. A15.8 The four strings of a musical instrument are all made of the same material and are under the same tension, but have different thicknesses. Waves travel A. fastest on the thickest string. B. fastest on the thinnest string. C. at the same speed on all strings. D. not enough information given to decide

  23. Wave intensity • Go beyond the wave on a string and visualize, say … a sound wave spreading from a speaker. That wave has intensity dropping as 1/r2.

  24. Thumping bass • You are 3m from your friend’s car, who is pumping obnoxiously loud, bass-heavy music through his subwoofer-equipped car, causing the entire car to rattle. He yells to you that he’s got a 800W subwoofer in his car and its cranked to full volume. Unimpressed, you measure a peak sound intensity of 4.4 W/m2 on your handy Radioshack soundmeter. How many watts is your friend’s subwoofer putting out? • What is the sound intensity for your friend in the car, at 1 m away from the subwoofer?

  25. Wave interference, boundaries, and superposition • Waves in motion from one boundary (the source) to another boundary (the endpoint) will travel and reflect.

  26. Vertical applications of SHM • As wave pulses travel, reflect, travel back, and repeat the whole cycle again, waves in phase will add and waves out of phase will cancel.

  27. Standing waves on a string • Fixed at both ends, the resonator was have waveforms that match. In this case, the standing waveform must have nodes at both ends. Differences arise only from increased energy in the waveform.

  28. The formation of a standing wave • The process seems complicated at first, but it is nothing more than waveforms adding constructively when they’re in phase and destructively when they’re not. • Refer to Problem-Solving Strategy 15.2 and Example 15.6.

  29. Complex standing waves • As the shape and composition of the resonator change, the standing wave changes also. Regard Figure 15.27, a multidimensional standing wave. Figure 15.29 provides many such multidimensional shapes.

  30. Monster bass • You want to build a huge bass guitar with a 5m long bass string and tune it to give a 20 Hz fundamental frequency (the lowest humans can hear). • Calculate the tension of the string • Calculate the frequency and wavelength of the second harmonic.

  31. Q15.9 While a guitar string is vibrating, you gently touch the midpoint of the string to ensure that the string does not vibrate at that point. The lowest-frequency standing wave that could be present on the string A. vibrates at the fundamental frequency. B. vibrates at twice the fundamental frequency. C. vibrates at three times the fundamental frequency. D. vibrates at four times the fundamental frequency. E. not enough information given to decide

  32. A15.9 While a guitar string is vibrating, you gently touch the midpoint of the string to ensure that the string does not vibrate at that point. The lowest-frequency standing wave that could be present on the string A. vibrates at the fundamental frequency. B. vibrates at twice the fundamental frequency. C. vibrates at three times the fundamental frequency. D. vibrates at four times the fundamental frequency. E. not enough information given to decide.

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