Chapter 17 Waves in Two and Three Dimensions

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1 Chapter 17 Waves in Two and Three Dimensions Slide 17-1

2 Chapter 17: Waves in Two and Three Dimensions Concepts Slide 17-2

3 Section 17.1: Wavefronts The figure shows cutaway views of a periodic surface wave at two instants that are half a period apart. Slide 17-3

4 Section 17.1: Wavefronts When the source of the wavefront can be localized to a single point, the source is said to be a point source. The figure shows a periodic surface wave spreading out from a point source. The curves (or surfaces) in the medium on which all points have the same phase is called a wavefront. Slide 17-4

5 Section 17.1: Wavefronts Consider the figure. If we assume that there is no energy dissipation, then there is no loss of energy as the wave moves outward. As the wavefront spreads, the circumference increases, and hence the energy per unit length decreases. Slide 17-5

6 Checkpoint Let t 2 = 2t 1 in Figure (a) How does R 1 compare with R 2? (b) If the energy in the wave is E and there is no dissipation of energy, what is the energy per unit length along the circumference at R 1? At R 2? (c) How does the energy per unit length along a wavefront vary with radial distance r? Slide 17-6

7 Checkpoint (a) The wave speed c is constant, so in twice the time it covers twice the distance, R 2 = 2R 1 (b) Energy per unit length? At 1: E/2πR 1. At 2: If the radius doubles, so does the circumference. Now at point 2, same energy but double the circumference, so E/2πR 2 = E/4πR 1 (c) Energy per unit length goes as 1/r since circumference increases with r Slide 17-7

8 Section 17.1: Wavefronts The expansion of the circular wavefronts causes the energy per unit length along the wavefront to decrease as 1/r. In Chapter 16 we saw E λ = ½(µλ)ω 2 A 2 (Eq ), Therefore, it follows that for waves in two dimensions e.g., water waves A 1/ r. Slide 17-8

9 Section 17.1: Wavefronts The waves that spread out in 3 dimensions are called spherical waves. The energy carried by a spherical wavefront is spread out over a spherical area of A = 4πr 2. So, for waves in three dimensions, E ~ 1/r 2, and therefore A ~ 1/r. e.g. sound waves Slide 17-9

10 Section 17.1: Wavefronts Example 17.1 Ripple amplitude The amplitude of a surface wave for which λ = m is 5.0 mm at a distance of 1.0 m from a point source. What is the amplitude of the wave (a) 10 m from the source and (b) 100 m from the source Slide 17-10

11 Section 17.1: Wavefronts Example 17.1 Ripple amplitude (cont.) ❶ GETTING STARTED I am given that the amplitude A = 5.0 mm at r = 1.0 m. As the wave spreads out, its amplitude diminishes, and I need to calculate the amplitude at r = 10 m and r = 100 m. In addition I need to determine by how much the wave attenuates as it propagates over a 100-period time interval past these two positions. Slide 17-11

12 Section 17.1: Wavefronts Example 17.1 Ripple amplitude (cont.) ❷ DEVISE PLAN Because the wave is a 2D surface wave, the amplitude is proportional to 1 r. I know the amplitude A 1.0 m at r = 1.0 m, so I can use this dependence to determine the amplitude at other distances from the source. For parts a and b, I need to determine A 10 m and A 100 m at r = 10 m and r = 100 m. Slide 17-12

13 Section 17.1: Wavefronts Example 17.1 Ripple amplitude (cont.) ❸ EXECUTE PLAN (a) The ratio of the amplitudes must go as the square root of the ratio of the distances. At 1.0 m and 10 m we have (1.0 m) (10 m = (1.0 / 10) = 0.32, and so the amplitude at 10 m is 0.32 (5.0 mm) = 1.6 mm. (b) At 100 m, (1.0 m) (100 m = 0.10, and so the amplitude is 0.10 (5.0 mm) = 0.50 mm. Slide 17-13

14 Section 17.1: Wavefronts Example 17.1 Ripple amplitude (cont.) ❹ EVALUATE RESULT The amplitudes at 10 m and 100 m are both smaller than the amplitude at 1.0 m, which is what I expect. From 1 m to 10 m, factor 3 decrease. From 10 m to 100 m also a factor of three, even though distance is 10 times larger. Amplitude decays more slowly than linear Slide 17-14

15 Section 17.1: Wavefronts Far from a point source, the spherical wavefronts essentially become a two-dimensional flat wavefront called a planar wavefront. Slide 17-15

16 Checkpoint Notice that in the views of the surface wave in Figure 17.1 the amplitude does not decrease with increasing radial distance r. How could such waves be generated? Slide 17-16

17 Checkpoint Would work to decrease the source amplitude as a function of time. First wave out is diminished when the second one is created, so make the second one smaller to compensate. By the time the third one comes out, both the first and second are smaller (but still equal), so make the third one even smaller Makes it uniform over space, but not in time uniformly decreases over entire wave pattern. Slide 17-17

18 Section 17.1 Question 1 Which of the following factors plays a role in how much a wave s amplitude decreases as the wave travels away from its source? Answer all that apply. 1. Dissipation of the wave s energy 2. Dimensionality of the wave 3. Destructive interference by waves created by other sources Slide 17-18

19 Section 17.1 Question 1 Which of the following factors plays a role in how much a wave s amplitude decreases as the wave travels away from its source? Answer all that apply. 1. Dissipation of the wave s energy 2. Dimensionality of the wave 3. Destructive interference by waves created by other sources (don t lose any energy/amplitude this way!) Slide 17-19

20 Section 17.2: Sound Section Goals You will learn to Define the physical characteristics of sound. Represent sound graphically. Slide 17-20

21 Section 17.2: Sound Longitudinal waves propagating through any kind of material is what we call sound. The human ear can detect longitudinal waves at frequencies from 20 Hz to 20 khz. Sound waves consist of an alternating series of compressions and rarefactions. For dry air at 20C, the speed of sound is ~343 m/s. Slide 17-21

22 Section 17.2: Sound Exercise 17.2 Wavelength of audible sound Given that the speed of sound waves in dry air is 343 m/s, determine the wavelengths at the lower and upper ends of the audible frequency range (20 Hz 20 khz). Slide 17-22

23 Section 17.2: Sound Exercise 17.2 Wavelength of audible sound (cont.) SOLUTION The wavelength is equal to the distance traveled in one period. At 20 Hz, the period is 1/(20 Hz) = 1/(20 s 1 ) = s, so the wavelength is (343 m/s)(0.050 s) = 17 m. The period of a wave of 20 khz is 1/(20,000 Hz) = s, so the wavelength is (343 m/s)( s) = 17 mm. Conveniently, the size of everyday objects Slide 17-23

24 Section 17.2: Sound The figure illustrates a mechanical model for a longitudinal waves. Slide 17-24

25 Checkpoint Does the wave speed along the chain shown in Figure 17.9 increase or decrease when (a) the spring constant of the springs is increased and (b) the mass of the beads is increased? Slide 17-25

26 Checkpoint (a) Increase the greater spring constant, the faster any disturbance is passed along. Just like increasing tension in a string (same mechanical model)! (b) Decrease greater mass slows down the transmission of the wave just like with beads on a string. Slide 17-26

27 Checkpoint (a) Plot the velocity of the beads along the chain in Figure 17.9b as a function of their equilibrium position x. (b) Plot the linear density (number of beads per unit length) as a function of x. Slide 17-27

28 Checkpoint 17.4 v " = dd " dt Slide 17-28

29 Section 17.2: Sound Longitudinal waves can also be represented by plotting the linear density of the medium as a function of position. The compressions and rarefactions in longitudinal waves occur at the locations where the medium displacement is zero. Slide 17-29

30 Section 17.2: Sound The figure shows a sound wave generated by an oscillating tuning fork. At any fixed position: oscillates in time At any given time: spatial oscillation Slide 17-30

31 Section 17.3: Interference Section Goals You will learn to Visualize the superposition of two or more two- or threedimensional waves traveling through the same region of a medium at the same time. Define and represent visually the nodal and antinodal lines for interference in two dimensions. Slide 17-31

32 Section 17.3: Interference Let us now consider the superposition of overlapping waves in two and three dimensions. The figure shows the interference of two identical circular wave pulses as they spread out on the surface of a liquid. Slide 17-32

33 Section 17.3: Interference Sources that emit waves having a constant phase difference are called coherent sources. The pattern produces by overlapping circular wavefronts is called a Moiré pattern. Along nodal lines the two waves cancel each other and the vector sum of the displacement is always zero. Slide 17-33

34 Section 17.3: Interference The figure shows a magnified view of the interference pattern seen on the previous slide. Along antinodal lines the displacement is a maximum. Slide 17-34

35 Section 17.3: Interference One consequence of nodal regions is illustrated in the figure. When the waves from two coherent sources interfere, the amplitude of the sum of these waves in certain directions is less than that of a single wave. Slide 17-35

36 Section 17.3: Interference The effect that the separation between the two point sources have on the appearance of nodal lines is shown in the figure. If two coherent sources located a distance d apart emit identical waves of wavelength λ, then the number of nodal lines on either side of a straight line running through the centers of the sources is the greatest integer smaller than or equal to 2(d/λ). Slide 17-36

37 Section 17.3: Interference With more than two coherent sources? Do one pair first, then add a third source to the resultant of that pair. Repeat. Find path lengths from either source, divide by λ Difference is ½ integer: destructive Difference is integer: constructive Slide 17-37

38 Section 17.3: Interference The figure shows what happens when 100 coherent sources are placed close to each other: When many coherent point sources are placed close together along a straight line, the waves nearly cancel out in all directions except the direction perpendicular to the axis of the sources. Slide 17-38

39 Checkpoint How does the wave amplitude along the beam of wavefronts in Figure change with distance from the row of sources? It doesn t very much! Neighboring sources shore each other up Slide 17-39

40 Section 17.4: Diffraction Section Goals You will learn to Define the physical causes of diffraction. Represent diffraction graphically. Slide 17-40

41 Section 17.4: Diffraction Huygens principle states that any wavefront can be regarded as a collection of closely spaced, coherent point sources. All these point sources emit wavelets, and these forward-moving wavelets combine to form the next wavefront. Slide 17-41

42 Section 17.4: Diffraction The figure shows planar wavefronts incident on gaps of varying size. Obstacles or apertures whose width is smaller than the wavelength of an incident wave give rise to considerable spreading of that wave. The spreading is called diffraction. Slide 17-42

43 Checkpoint Suppose the barriers in Figure were held at an angle to the incident wavefronts. Sketch the transmitted wavefronts for the case where the width of the gap is much smaller than the wavelength of the incident waves. Slide 17-43

44 Checkpoint Doesn t make a difference: the gap causes the same diffraction regardless. Only relies on incident waves causing the gap to become a point source. Slide 17-44

45 Examples Slide 17-45

46 Examples Slide 17-46

47 Examples Slide 17-47

48 This happens with sound too! Slide 17-48

49 Slide 17-49

50 Chapter 17: Self-Quiz #5 Because sound waves diffract around an open doorway, you can hear sounds coming from outside the doorway. You cannot, however, see objects outside the doorway unless you are directly in line with them. What does this observation imply about the wavelength of light? Slide 17-50

51 Chapter 17: Self-Quiz #5 Answer Because light does not diffract as it travels through the doorway, this observation implies that the wavelength of the light must be smaller than the width of the doorway. Given that visible light has wavelengths between m and m and most doorways are about 1 m wide and 2 m tall, this is indeed the case. Slide 17-51

52 Chapter 17: Waves in Two and Three Dimensions Quantitative Tools Slide 17-52

53 Section 17.5: Intensity Section Goals You will learn to Define the intensity of a wave. Calculate the intensity of a wave using the decibel scale. Slide 17-53

54 Section 17.5: Intensity For waves in three dimensions, intensity I is defined as I P A P is the power delivered by the wave over an area A. SI units: W/m 2 If the power delivered by a point source is P s, the intensity at a distance r from the source is I = P s = P s (uniformly radiating point source) A sphere 4πr 2 For two-dimensional surface waves, the intensity is SI units: W/m I surf P L Slide 17-54

55 Section 17.5: Intensity The human ear can handle an extremely wide range of intensities, from the threshold of hearing I th 1 x W/m 2 to the threshold of pain at 1.0 W/m 2. To deal with this vast range of intensities, it s convenient to use a logarithmic scale and it s logical to place the zero of the scale at the threshold of hearing. To do so, we define the intensity level β, expressed in decibels (db), as I β (10 db)log I th where I th = 1 x W/m 2. Slide 17-55

56 Section 17.5: Intensity Average auditory response of the human ear. Most sensitive at 3kHz. (lower magnitude means more sensitive. 3kHz is very annoying.) Slide 17-56

57 Section 17.5: Intensity Slide 17-57

58 Section 17.5: Intensity Exercise 17.5 Doubling the intensity A clarinet can produce about 70 db of sound. By how much does the intensity level increase if a second clarinet is played at the same time? Slide 17-58

59 Section 17.5: Intensity Exercise 17.5 Doubling the intensity (cont.) SOLUTION If the intensity of the sound produced by one clarinet is I c, the intensity level of one clarinet is β 1 = (10 db)log I c = 70 db. I th Slide 17-59

60 Section 17.5: Intensity Exercise 17.5 Doubling the intensity (cont.) SOLUTION The second clarinet doubles the intensity, so the intensity level becomes β 2 = (10 db)log 2I c I th = (10 db) log2 + log I c I th = (10 db)log 2 + β 1, where I have used the logarithmic relationship log AB = log A + log B. Because log 2 0.3, the intensity level increases to β 2 (10 db)(0.3) + 70 db = 73 db. So, even though the intensity doubles, the intensity level increases by only 3 db. Slide 17-60

61 Checkpoint In Exercise 17.5, how many clarinets must play at the same time in order to increase the intensity level from 70 db to 80 db? 10 db means a factor of 10 increase in intensity (log scale!), so we need 10 clarinets playing at the same time. Slide 17-61

62 Section 17.6: Beats Section Goals You will learn to Establish the concept of beats, which arises from the overlap of equal amplitude waves with slightly different frequency. Derive the mathematical formula that relates the frequency of the beats to the frequencies of the overlapping waves. Slide 17-62

63 Section 17.6: Beats Part (a) shows the displacement curves for two waves of equal amplitude A, but slightly different frequencies. The superposition of the two waves result in a wave of oscillating amplitude as shown in part (b). This effect is called beating. Slide 17-63

64 Section 17.6: Beats The displacement caused by the two individual waves at some fixed point is given by D 1x = A sin(2πf 1 t) D 2x = A sin(2πf 2 t) The superposition of the two waves gives us D x = D 1x + D 2x = A(sin 2πf 1 t + sin 2πf 2 t) Using trigonometric identities, we can simplify the equation to D x = 2A cos 1 2π ( f 2 1 f 2 )t sin 1 2π ( f 2 1 f 2 )t Slide 17-64

65 Section 17.6: Beats Using Δf = f 1 f 2 and f av = ½ ( f 1 + f 2 ), we can write D x = 2A cos 2π ( 1 Δf )t 2 sin(2π f av t) We can see that the resulting wave has a frequency of f av. The frequency of the amplitude variation is ½Δf. However, since two beats occur in each cycle of this amplitude variation, the beat frequency is twice that f beat f 1 f This is how you know you re out of tune. Faster beating means farther apart. Slide 17-65

66 Section 17.6: Beats Exercise 17.7 Tuning a piano Your middle-c tuning fork oscillates at Hz. When you play the middle-c key on your piano together with the tuning fork, you hear 15 beats in 10 s. What are the possible frequencies emitted by this key? Slide 17-66

67 Section 17.6: Beats Exercise 17.7 Tuning a piano SOLUTION The beat frequency the number of beats per second is equal to the difference between the two frequencies (Eq. 17.8). I am given the frequency of the tuning fork, f t = Hz, and the beat frequency, f B = (15 beats)/(10 s) = 1.5 Hz. I do not know, however, whether the frequency f p of the struck middle-c piano key is higher or lower than that of the tuning fork. Slide 17-67

68 Section 17.6: Beats Exercise 17.7 Tuning a piano SOLUTION If it is higher, I have f B = f p f t. If it is lower, then f B = f t f p. So f p = f t ± f B = Hz ± 1.5 Hz and the possible frequencies emitted by the out-of-tune middle-c key are Hz and Hz. Slide 17-68

69 Section 17.6 Question 6 One way to tune a piano is to strike a tuning fork (which emits only one specific frequency), then immediately strike the piano key for the frequency being sounded by the fork, and listen for beats. In making an adjustment, a piano tuner working this way causes the beat frequency to increase slightly. Is she going in the right direction with that adjustment? 1. Yes 2. No Slide 17-69

70 Section 17.6 Question 6 One way to tune a piano is to strike a tuning fork (which emits only one specific frequency), then immediately strike the piano key for the frequency being sounded by the fork, and listen for beats. In making an adjustment, a piano tuner working this way causes the beat frequency to increase slightly. Is she going in the right direction with that adjustment? 1. Yes 2. No faster beating means larger difference in freq. Slide 17-70

71 Doppler Effect: moving relative to waves Slide 17-71

72 in one period T, you move closer to the source by vst the waves appear squashed together by v s T the apparent frequency (1/T) is still velocity / wavelength v vs Slide 17-72

73 Approaching the source: pitch (freq) seems higher Moving away from source: pitch (freq) seems lower Only has to do with RELATIVE motion! e.g., ambulance - driver hears no change similarly: doesn t matter who is moving happens for light too - receding galaxies have red shift (lower freq) Slide 17-73

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