In Phase. Out of Phase
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1 Superposition
2 Interference Waves ADD: Constructive Interference. Waves SUBTRACT: Destructive Interference. In Phase Out of Phase
3 Superposition Traveling waves move through each other, interfere, and keep on moving!
4 Pulsed Interference
5 Superposition Waves ADD in space. Any complex wave can be built from simple sine waves. Simply add them point by point. Simple Sine Wave Simple Sine Wave Complex Wave
6 Fourier Synthesis of a Square Wave Any periodic function can be represented as a series of sine and cosine terms in a Fourier series: y( t) ( A sin2 ƒ t B cos2 ƒ t) n n n n n
7 Superposition of Sinusoidal Waves Case 1: Identical, same direction, with phase difference (Interference) Case 2: Identical, opposite direction (standing waves) Case 3: Slightly different frequencies (Beats)
8 Superposition of Sinusoidal Waves Assume two waves are traveling in the same direction, with the same frequency, wavelength and amplitude The waves differ in phase y 1 = A sin (kx - wt) y 2 = A sin (kx - wt + f) y = y 1 +y 2 = 2A cos (f/2) sin (kx - wt + f/2) Resultant Amplitude Depends on phase: Spatial Interference Term
9 Sinusoidal Waves with Constructive Interference y = y1+y2 = 2A cos (f/2) sin (kx - wt + f /2) When f = 0, then cos (f/2) = 1 The amplitude of the resultant wave is 2A The crests of one wave coincide with the crests of the other wave The waves are everywhere in phase The waves interfere constructively
10 Sinusoidal Waves with Destructive Interference y = y1+y2 = 2A cos (f/2) sin (kx - wt + f /2) When f =, then cos (f/2) = 0 Also any even multiple of The amplitude of the resultant wave is 0 Crests of one wave coincide with troughs of the other wave The waves interfere destructively
11 Sinusoidal Waves Interference y = y1+y2 = 2A cos (f/2) sin (kx - wt + f /2) When f is other than 0 or an even multiple of, the amplitude of the resultant is between 0 and 2A The wave functions still add
12 Superposition of Sinusoidal Waves y = y 1 +y 2 = 2A cos (f/2) sin (kx - wt + f/2) The resultant wave function, y, is also sinusoidal The resultant wave has the same frequency and wavelength as the original waves The amplitude of the resultant wave is 2A cos (f/2) The phase of the resultant wave is f/2 Constructive Destructive Interference
13 Wave Interference y = y 1 +y 2 = 2A cos (f/2) sin (kx - wt + f/2) f Resultant Amplitude: 2Acos 2 Constructive Interference: f 2 n, n 0,1,2,3... Destructive Interference: f (2n 1), n 0,1, 2, 3...
14 Ch 18 HO Problem #1 y = y 1 +y 2 = 2A cos (f/2) sin (kx - wt + f/2)
15 Reflected PULSE: Free End Bound End
16 Reflected PULSE:
17 Standing Waves Created by Boundary Conditions
18 Standing Waves on Strings
19 Transverse Standing Wave Produced by the superposition of two identical waves moving in opposite directions.
20 Standing Waves Superposition of two identical waves moving in opposite directions. y A sin ( kx - wt) y A sin ( kx wt) 1 2 y (2Asin kx)cos wt There is no kx wt term, and therefore it is not a traveling wave! Every element in the medium oscillates in simple harmonic motion with the same frequency, w: coswt The amplitude of the simple harmonic motion depends on the location of the element within the medium: (2Asinkx)
21 Note on Amplitudes y (2Asin kx)cos wt There are three types of amplitudes used in describing waves The amplitude of the individual waves, A The amplitude of the simple harmonic motion of the elements in the medium,2a sin kx The amplitude of the standing wave, 2A A given element in a standing wave vibrates within the constraints of the envelope function 2Asin kx, where x is the position of the element in the medium
22 Node & Antinodes A node occurs at a point of zero amplitude n x n 0,1, 2 An antinode occurs at a point of maximum displacement, 2A x n 4 n 1,3,
23 Antinode #5 Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 2 sin ( x) cos (3 t) where x is in m and t is in s. What is the distance (in m) between the first two antinodes? a. 8 b. 2 c. 4 y (2Asin kx)cos wt d. 1 e. 0.5
24 Standing Waves on a String Harmonics
25 Standing Waves on a String Harmonics
26 Standing Waves on a String 2L 1 L 2 3 2L 3
27 Standing Waves on a String n 2L n f n v/ n v f n n 2 L
28 Standing Wave on a String v f n n 2 L v T v f
29 v fn n 2 L v T v f
30 Strings & Atoms are Quantized The possible frequency and energy states of an electron in an atomic orbit or of a wave on a string are quantized. f v n 2 l En nhf, n= 0,1,2,3, h 6.626x10 Js
31 Superposition Sound Waves
32 Longitudinal Standing Wave
33 Standing Waves in a Tube Open at both ends. v f n n 2 L 2L Resonant Frequencies: fn nf 1 L Same as a string fixed at both ends.
34 Standing Waves in a Tube Open at one end. n 4L n odd f n n odd v 4L
35
36 Multiple Harmonics can be present at the same time.
37 Which harmonics (modes) are present on the string? The Fundamental and third harmonic.
38 What is the difference between Noise and Music? Regular Repeating Patterns
39 The amount that each harmonic is present determines the quality or timbre of the sound for each instrument.
40 Any complex wave can be built from simple sine waves.
41
42
43 Standing Waves in Membranes Two-dimensional oscillations may be set up in a flexible membrane stretched over a circular hoop The resulting sound is not harmonic because the standing waves have frequencies that are not related by integer multiples The fundamental frequency contains one nodal curve
44 Standing Waves Standing waves form in certain MODES based on the length of the string or tube or the shape of drum or wire. Not all frequencies are permitted!
45
46 Standing Waves: Membranes
47 Standing Waves: Membranes
48 Interference
49 Interference: Beats beats frequency = difference in frequencies
50 Interference: Beats f f f f B ave 2 1 f f 2 1 2
51 Interference: Beats
52 Beat Frequency ƒ ƒ Aresultant 2Acos2 t The number of amplitude maxima one hears per second is the beat frequency: ƒ beat = ƒ 1 ƒ 2 The human ear can detect a beat frequency up to about 20 beats/sec
53 You hear three beats per second when two sound tones are generated. The frequency of one tone is known to be 610 Hz. The frequency of the other is A. 604 Hz. B. 607 Hz. C. 613 Hz. D. 616 Hz. E. Either b or c.
54 You hear three beats per second when two sound tones are generated. The frequency of one tone is known to be 610 Hz. The frequency of the other is A. 604 Hz. B. 607 Hz. C. 613 Hz. D. 616 Hz. E. Either b or c.
55 Beat Frequency #11 In certain ranges of a piano keyboard, more than one string is tuned to the same note to provide extra loudness. For example, the note at 110 Hz has two strings at this frequency. If one string slips from its normal tension of 600 N to 540 N, what beat frequency is heard when the hammer strikes the two strings simultaneously?
56 1-D Sound Wave Interference
57 2-D Wave Interference? P
58 2-D Phase Difference Different than 1-D You have to consider the Path Difference! v 2 2 f w t 2 f t 2 t ( v t) r 2-D Phase Difference at P: f is different from the phase difference f between the two source waves! 2 Phase Difference at P: f r Path Difference at P: r f 2 f f2 f1 P
59 Spherically Symmetric Waves
60
61
62
63 Intensity
64 Quiet Loud Quiet Loud Min Max Min Max
65 Contour Map of Interference Pattern of Two Sources
66 Interference: Two Spherical Sources
67 Constructive or Destructive? (Identical in phase sources) 2 Phase Difference at P: f r f0 P 2 f (1 ) 2 Constructive! f Resultant Amplitude: 2Acos 2 Constructive Interference: r n, f 2 n, n 0,1, 2, 3... Destructive Interference: r (2n 1), f (2n 1), n 0,1, 2,3... 2
68 Constructive or Destructive? (Source out of Phase by 180 degrees) 2 Phase Difference at P: f r f0 P 2 f (1 ) 3 Destructive! f Resultant Amplitude: 2Acos 2 Constructive Interference: r n, f 2 n, n 0,1, 2, 3... Destructive Interference: r (2n 1), f (2n 1), n 0,1, 2,3... 2
69 2-D Phase Difference Different than 1-D You have to consider the Path Difference! v 2 2 f w t 2 f t 2 t ( v t) r 2-D Phase Difference at P: f is different from the phase difference f between the two source waves! 2 Phase Difference at P: f r Path Difference at P: r f 2 f f2 f1 P
70 In Phase or Out of Phase? B A
71 Constructive or Destructive? A B
72 The interference at point C in the figure at the right is A. maximum constructive. B. destructive, but not perfect. C. constructive, but less than maximum. D. perfect destructive. E. there is no interference at point C.
73 The interference at point C in the figure at the right is A. maximum constructive. B. destructive, but not perfect. C. constructive, but less than maximum. D. perfect destructive. E. there is no interference at point C.
74 Ch 18 HO Problem #2 2 Phase Difference at P: f r Path Difference at P: r f 2
75 Ch 18 HO Problem #3 You Try
76 Intensity
77 Quiet Loud Quiet Loud Min Max Min Max
78 Same Idea with Superposition of Light Waves
79 Next time: Interference of 2 Light Sources
80 These two loudspeakers are in phase. They emit equal-amplitude sound waves with a wavelength of 1.0 m. At the point indicated, is the interference maximum constructive, perfect destructive or something in between? A. perfect destructive B. maximum constructive C. something in between
81 These two loudspeakers are in phase. They emit equal-amplitude sound waves with a wavelength of 1.0 m. At the point indicated, is the interference maximum constructive, perfect destructive or something in between? A. perfect destructive B. maximum constructive C. something in between
82 Two loudspeakers emit waves with = 2.0 m. Speaker 2 is 1.0 m in front of speaker 1. What, if anything, must be done to cause constructive interference between the two waves? A. Move speaker 1 forward (to the right) 0.5 m. B. Move speaker 1 backward (to the left) 1.0 m. C. Move speaker 1 forward (to the right) 1.0 m. D. Move speaker 1 backward (to the left) 0.5 m. E. Nothing. The situation shown already causes constructive interference.
83 Two loudspeakers emit waves with = 2.0 m. Speaker 2 is 1.0 m in front of speaker 1. What, if anything, must be done to cause constructive interference between the two waves? A. Move speaker 1 forward (to the right) 0.5 m. B. Move speaker 1 backward (to the left) 1.0 m. C. Move speaker 1 forward (to the right) 1.0 m. D. Move speaker 1 backward (to the left) 0.5 m. E. Nothing. The situation shown already causes constructive interference.
84 An open-open tube of air supports standing waves at frequencies of 300 Hz and 400 Hz, and at no frequencies between these two. The second harmonic of this tube has frequency A. 800 Hz. B. 200 Hz. C. 600 Hz. D. 400 Hz. E. 100 Hz.
85 An open-open tube of air supports standing waves at frequencies of 300 Hz and 400 Hz, and at no frequencies between these two. The second harmonic of this tube has frequency A. 800 Hz. B. 200 Hz. C. 600 Hz. D. 400 Hz. E. 100 Hz.
Waves ADD: Constructive Interference. Waves SUBTRACT: Destructive Interference. In Phase. Out of Phase
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