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Wave Motion

Wave Motion. Chapter 15. t. t + D t. ConcepTest 15.4 Out to Sea. 1) 1 second 2) 2 seconds 3) 4 seconds 4) 8 seconds 5) 16 seconds.

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Wave Motion

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  1. Wave Motion Chapter 15

  2. t t + Dt ConcepTest 15.4Out to Sea 1)1 second 2) 2 seconds 3) 4 seconds 4) 8 seconds 5) 16 seconds A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 m and the speed of the waves is 5 m/s, how long does it take the boat to go from the top of a crest to the bottom of a trough ?

  3. t t + Dt ConcepTest 15.4Out to Sea 1)1 second 2) 2 seconds 3) 4 seconds 4) 8 seconds 5) 16 seconds A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 m and the speed of the waves is 5 m/s, how long does it take the boat to go from the top of a crest to the bottom of a trough ? We know that v = f l = l / T, hence T = l / v. If l = 20 mand v = 5 m/s, then T = 4 secs. The time to go from a crest to a trough is only T/2 (half aperiod), so it takes2 secs!!

  4. ConcepTest 15.6bWave Speed II A wave pulse is sent down a rope of a certain thickness and a certain tension. A second rope made of the same material is twice as thick, but is held at the same tension. How will the wave speed in the second rope compare to that of the first? 1) speed increases 2) speed does not change 3) speed decreases

  5. ConcepTest 15.6bWave Speed II A wave pulse is sent down a rope of a certain thickness and a certain tension. A second rope made of the same material is twice as thick, but is held at the same tension. How will the wave speed in the second rope compare to that of the first? 1) speed increases 2) speed does not change 3) speed decreases The wave speed goes inversely as the square root of the mass per unit length, which is a measure of the inertia of the rope. So in a thicker (more massive) rope at the same tension, the wave speed will decrease.

  6. 15-4 Mathematical Representation of a Traveling Wave Suppose the shape of a wave is given by:

  7. 15-4 Mathematical Representation of a Traveling Wave After a time t, the wave crest has traveled a distance vt, so we write:

  8. 15-4 Mathematical Representation of a Traveling Wave Example 15-5: A traveling wave. The left-hand end of a long horizontal stretched cord oscillates transversely in SHM with frequency f = 250 Hz and amplitude 2.6 cm. The cord is under a tension of 140 N and has a linear density μ = 0.12 kg/m. At t = 0, the end of the cord has an upward displacement of 1.6 cm and is falling. Determine (a) the wavelength of waves produced and (b) the equation for the traveling wave.

  9. 15-5 The Wave Equation Look at a segment of string under tension:

  10. 15-5 The Wave Equation This is the one-dimensional wave equation; it is a linear second-order partial differential equation in x and t. Its solutions are all sinusoidal waves satisfying v=/T.

  11. 15-6 The Principle of Superposition Superposition: The displacement at any point is the vector sum of the displacements of all waves passing through that point at that instant. Fourier’s theorem: Any complex periodic wave can be written as the sum of sinusoidal waves of different amplitudes, frequencies, and phases.

  12. 15-6 The Principle of Superposition Conceptual Example 15-7: Making a square wave. At t= 0, three waves are given by D1=Acoskx, D2= -1/3Acos3kx, and D3=1/5Acos5kx, where A= 1.0 m and k= 10 m-1. Plot the sum of the three waves from x= -0.4 m to +0.4 m. (These three waves are the first three Fourier components of a “square wave.”)

  13. 15-7 Reflection and Transmission A wave reaching the end of its medium, but where the medium is still free to move, will be reflected (b), and its reflection will be upright. A wave hitting an obstacle will be reflected (a), and its reflection will be inverted.

  14. 15-7 Reflection and Transmission A wave encountering a denser medium will be partly reflected and partly transmitted; if the wave speed is less in the denser medium, the wavelength will be shorter.

  15. 15-7 Reflection and Transmission Two- or three-dimensional waves can be represented by wave fronts, which are curves of surfaces where all the waves have the same phase. Lines perpendicular to the wave fronts are called rays; they point in the direction of propagation of the wave.

  16. 15-7 Reflection and Transmission The law of reflection: the angle of incidence equals the angle of reflection.

  17. 15-8 Interference The superposition principle says that when two waves pass through the same point, the displacement is the arithmetic sum of the individual displacements. In the figure below, (a) exhibits destructive interference and (b) exhibits constructive interference.

  18. 15-8 Interference These graphs show the sum of two waves. In (a) they add constructively; in (b) they add destructively; and in (c) they add partially destructively.

  19. 15-9 Standing Waves; Resonance Standing waves occur when both ends of a string are fixed. In that case, only waves which are motionless at the ends of the string can persist. There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value.

  20. 15-9 Standing Waves; Resonance The frequencies of the standing waves on a particular string are called resonant frequencies. They are also referred to as the fundamental and harmonics.

  21. 15-9 Standing Waves; Resonance The wavelengths and frequencies of standing waves are: and

  22. 15-9 Standing Waves; Resonance Example 15-8: Piano string. A piano string is 1.10 m long and has a mass of 9.00 g. (a) How much tension must the string be under if it is to vibrate at a fundamental frequency of 131 Hz? (b) What are the frequencies of the first four harmonics?

  23. a c b ConcepTest 15.7bStanding Waves II 1)is zero everywhere 2) is positive everywhere 3) is negative everywhere 4) depends on the position along the string A string is clamped at both ends and plucked so it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position c, the instantaneous velocity of points on the string:

  24. a c b ConcepTest 15.7bStanding Waves II 1)is zero everywhere 2) is positive everywhere 3) is negative everywhere 4) depends on the position along the string A string is clamped at both ends and plucked so it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position c, the instantaneous velocity of points on the string: When the string is flat, all points are moving through the equilibrium position and are therefore at their maximum velocity. However, the direction depends on the location of the point. Some points are moving upward rapidly, and some points are moving downward rapidly.

  25. 15-9 Standing Waves; Resonance Example 15-9: Wave forms. Two waves traveling in opposite directions on a string fixed at x = 0 are described by the functions D1 = (0.20 m)sin(2.0x – 4.0t) and D2 = (0.20m)sin(2.0x + 4.0t) (where x is in m, t is in s), and they produce a standing wave pattern. Determine (a) the function for the standing wave, (b) the maximum amplitude at x = 0.45 m, (c) where the other end is fixed (x > 0), (d) the maximum amplitude, and where it occurs.

  26. 15-10 Refraction If the wave enters a medium where the wave speed is different, it will be refracted—its wave fronts and rays will change direction. We can calculate the angle of refraction, which depends on both wave speeds:

  27. 15-10 Refraction The law of refraction works both ways—a wave going from a slower medium to a faster one would follow the red line in the other direction.

  28. 15-10 Refraction Example 15-10: Refraction of an earthquake wave. An earthquake P wave passes across a boundary in rock where its velocity increases from 6.5 km/s to 8.0 km/s. If it strikes this boundary at 30°, what is the angle of refraction?

  29. 15-11 Diffraction When waves encounter an obstacle, they bend around it, leaving a “shadow region.” This is called diffraction.

  30. 15-11 Diffraction The amount of diffraction depends on the size of the obstacle compared to the wavelength. If the obstacle is much smaller than the wavelength, the wave is barely affected (a). If the object is comparable to, or larger than, the wavelength, diffraction is much more significant (b, c, d).

  31. Summary of Chapter 15 • Vibrating objects are sources of waves, which may be either pulses or continuous. • Wavelength: distance between successive crests • Frequency: number of crests that pass a given point per unit time • Amplitude: maximum height of crest • Wave velocity:

  32. Summary of Chapter 15 • Transverse wave: oscillations perpendicular to direction of wave motion • Longitudinal wave: oscillations parallel to direction of wave motion • Intensity: energy per unit time crossing unit area (W/m2): • Angle of reflection is equal to angle of incidence

  33. Summary of Chapter 15 • When two waves pass through the same region of space, they interfere. Interference may be either constructive or destructive. • Standing waves can be produced on a string with both ends fixed. The waves that persist are at the resonant frequencies. • Nodes occur where there is no motion; antinodes where the amplitude is maximum. • Waves refract when entering a medium of different wave speed, and diffract around obstacles.

  34. Chapter 16Sound

  35. Units of Chapter 16 • Characteristics of Sound • Mathematical Representation of Longitudinal Waves • Intensity of Sound: Decibels • Sources of Sound: Vibrating Strings and Air Columns • Quality of Sound, and Noise; Superposition • Interference of Sound Waves; Beats

  36. Units of Chapter 16 • Doppler Effect • Shock Waves and the Sonic Boom • Applications: Sonar, Ultrasound, and Medical Imaging

  37. 16-1 Characteristics of Sound Sound can travel through any kind of matter, but not through a vacuum. The speed of sound is different in different materials; in general, it is slowest in gases, faster in liquids, and fastest in solids. The speed depends somewhat on temperature, especially for gases.

  38. 16-2 Mathematical Representation of Longitudinal Waves Longitudinal waves are often called pressure waves. The displacement is 90° out of phase with the pressure.

  39. 16-2 Mathematical Representation of Longitudinal Waves If the displacement is sinusoidal, we have where and

  40. ConcepTest 16.3Wishing Well You drop a rock into a well, and you hear the splash 1.5 s later. If the depth of the well were doubled, how long after you drop the rock would you hear the splash in this case? 1) more than 3 s later 2) 3 s later 3) between 1.5 s and 3 s later 4) 1.5 s later 5) less than 1.5 s later

  41. ConcepTest 16.3Wishing Well You drop a rock into a well, and you hear the splash 1.5 s later. If the depth of the well were doubled, how long after you drop the rock would you hear the splash in this case? 1) more than 3 s later 2) 3 s later 3) between 1.5 s and 3 s later 4) 1.5 s later 5) less than 1.5 s later Because the speed of sound is so much faster than the speed of the falling rock, we can essentially ignore the travel time of the sound. As for the falling rock, it is accelerating as it falls, so it covers the bottom half of the deeper well much quicker than the top half. The total time will not be exactly 3 s, but somewhat less. Follow-up: How long does the sound take to travel the depth of the well?

  42. 16-3 Intensity of Sound: Decibels The intensity of a wave is the energy transported per unit time across a unit area. The human ear can detect sounds with an intensity as low as 10-12 W/m2 and as high as 1 W/m2. Perceived loudness, however, is not proportional to the intensity.

  43. 16-3 Intensity of Sound The intensity can be written in terms of the maximum pressure variation. With some algebraic manipulation, we find:

  44. 16-3 Intensity of Sound • Example 16-7: How tiny the displacement is. • Calculate the displacement of air molecules for a sound having a frequency of 1000 Hz at the threshold of hearing. • (b) Determine the maximum pressure variation in such a sound wave.

  45. 16-3 Intensity of Sound: Decibels The loudness of a sound is much more closely related to the logarithm of the intensity. Sound level is measured in decibels (dB) and is defined as: I0 is taken to be the threshold of hearing:

  46. 16-3 Intensity of Sound: Decibels Example 16-3: Sound intensity on the street. At a busy street corner, the sound level is 75 dB. What is the intensity of sound there?

  47. 16-3 Intensity of Sound: Decibels Example 16-4: Loudspeaker response. A high-quality loudspeaker is advertised to reproduce, at full volume, frequencies from 30 Hz to 18,000 Hz with uniform sound level ± 3 dB. That is, over this frequency range, the sound level output does not vary by more than 3 dB for a given input level. By what factor does the intensity change for the maximum change of 3 dB in output sound level?

  48. 16-3 Intensity of Sound: Decibels Conceptual Example 16-5: Trumpet players. A trumpeter plays at a sound level of 75 dB. Three equally loud trumpet players join in. What is the new sound level?

  49. 16-3 Intensity of Sound: Decibels An increase in sound level of 3 dB, which is a doubling in intensity, is a very small change in loudness. In open areas, the intensity of sound diminishes with distance: However, in enclosed spaces this is complicated by reflections, and if sound travels through air, the higher frequencies get preferentially absorbed.

  50. 16-3 Intensity of Sound: Decibels Example 16-6: Airplane roar. The sound level measured 30 m from a jet plane is 140 dB. What is the sound level at 300 m? (Ignore reflections from the ground.)

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