Physics 102 Superposition Moza M. Al-Rabban Professor of Physics Lecture 7 Interference

2 Interference interference The combination, or superposition, of waves is often called interference. A standing wave is the interference pattern produced when two waves of equal frequency travel in opposite directions. same Now, we will look at the interference of two waves traveling in the same direction

3 Interference in One Dimension Overlapping light waves and overlapping sound waves both obey the principle of superposition, and so both show the effects of interference. We will assume that the waves are sinusoidal, have the same frequency and amplitude, and travel to the right along the x-axis. source

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5 Note > Perfect destructive interference occurs only if the two waves have equal amplitudes.

6 The Phase Difference Path-length differenceInherent phase difference

7 The Phase Difference The condition of being in phase, where crests are aligned with crests and troughs with troughs, is that  = 0, 2 , 4 , or any integer multiple of 2 . For identical sources,  0 = 0 rad, maximum constructive interference occurs when  x = m, Two identical sources produce maximum constructive interference when the path-length difference is an integer number of wavelengths.

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9 The Phase Difference The condition of being out of phase, where crests are aligned with troughs of other, that is,  = , 3 , 5  or any odd multiple of . For identical sources,  0 = 0 rad, maximum constructive interference occurs when  x = (m+ ½ ), Two identical sources produce perfect destructive interference when the path-length difference is half-integer number of wavelengths.

10 Constructive and Destructive Interference

11 Destructive Interference Three Ways

12 Example 8: Interference Between Two Sound Waves You are standing in front of two side-by-side loudspeakers playing sounds at the same frequency. Initially there is almost no sound at all. Then speaker 2 is moved slowly back, and the intensity increases until it is moved back 0.75 m. As speaker 2 continues to move back, the sound begins to decrease. What is the distance at which the sound intensity is again a minimum? SOLVE: A minimum sound intensity implies that the two sound waves are interfering destructively. Initially the loudspeakers are side-by-side, so

13 Moving one of the speakers does not change But it does change the path-length difference. Constructive interference condition is, The next point of destructive interference, with m=1, occurs when

14 Clicker Question 1 Two loudspeakers emit sound waves with =2.0 m with the phases indicated in the figure. Speaker 2 is 1.0 m in front of Speaker 1. What, if anything, should be done to arrange for constructive interference between the two waves? (a)Move Speaker 1 forward 1.0 m; (b)Move Speaker 1 forward 0.5 m; (c)Move Speaker 1 back 1.0 m; (d)Move Speaker 1 back 0.5 m; (e)Nothing. Constructive interference is already present.

15 The Mathematics of Interference

16 Example: More Interference of Sound Waves Two loudspeakers emit 500 Hz sound waves with amplitudes of a=0.10 mm. Speaker 2 is 1.00 m behind Speaker 1, and the phase difference between the two speakers is What is the amplitude of sound waves at a point 2.00 m in front of Speaker 1?

17 Interference in 2D and 3D The motion of the waves does not affect the points of constructive and destructive interference.

18 Example11: 2D Interference between Loudspeakers Two loudspeakers are 2.0 m apart and produce 700 Hz sound waves in phase, in a room where the speed of sound is 341 m/s. A listener stands 5.0 m from one loudspeaker and 2.0 m to one side of center. Is the interference there constructive, destructive, or something in between? How will this result differ if the speakers are out of phase? Destructive interference if  0 =0; Constructive interference if  0 = .

19 Visualizing Interference

20 Example: Intensity of Two Interfering Loudspeakers Two loudspeakers are 6.0 m apart and produce in-phase equal amplitude sound waves with a wavelength of 1.0 m. Each speaker alone produces an intensity I 0. An observer at point A is 10.0 m in front of the plane containing the speakers on their line of symmetry. A second observer at point B is 10.o m directly in front of one of the speakers. In terms of I 0, what is the intensity I A at point A and I B at point B ?

21 Clicker Question 3 Two loudspeakers emit equal-amplitude sound waves in phase with a wavelength of 1.0 m. At the point indicated, the interference is: (a)Constructive; (b)Perfect destructive; (c)Something in between; (d)Cannot tell without knowing the speaker separation.

22 Beats and Modulation If you listen to two sounds with very different frequencies, you hear two distinct tones. But if the frequency difference is very small, just one or two Hz, then you hear a single tone whose intensity is modulated once or twice every second. That is, the sound goes up and down in volume, loud, soft, loud, soft, ……, making a distinctive sound pattern called beats.

23 Beats and Modulation The periodically varying amplitude is called a modulation of the wave.

24 For 1.The two waves have the same amplitude a, 2.A detector is located at the origin (x=0), 3.The two sources are in phase, 4.The source phase happen to be The modulation frequency is: And the beat frequency is,

25 Example 13: Listening to Beats One flutist plays a note of 510 Hz while a second plays a note of 512 Hz. What frequency do you hear? What is the beat frequency?

26 Graphical Beats

27 Clicker Question 1 You hear three beats per second when two sound tones are generated. The frequency of one tone is 610 Hz. The frequency of the other tone is: (a) 604 Hz; (b) 607 Hz; (c) 613 Hz; (d) 616 Hz; (e) Either (a) or (d) (f) Either (b) or (c)

28 Chapter 21 - Summary (1)

29 Chapter 21 - Summary (2)

30 Chapter 21 - Summary (3)

End of Lecture 7