Chapter4: Superposition and Interference

Transcription

1 Chapter4: Superposition and Interference Sections Superposition Principle Superposition of Sinusoidal Waves Interference of Sound Waves Standing Waves Beats: Interference in Time Nonsinusoidal Wave Patterns

2 Superposition Principle Superposition principle states that if two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves.

3 Superposition Principle (a d) Two pulses traveling on a stretched string in opposite directions pass through each other.

4 (a e) Two pulses traveling in opposite directions and having displacements that are inverted relative to each other

5 Interference Interference pattern is a result of the superpositions of waves. When two or more waves meet, they superpose or combine at a particular point. The waves are said to interfere. Interference is the superposition of two waves originating from two coherent sources. Sources which are coherent produce waves of the same frequency (f), amplitude (a) and in phase.

6 Interference If the waves are in phase, that is crests and troughs are aligned, the amplitude is increased. This is called constructive interference. If the waves are out off phase, that is crests and troughs are completely misaligned, the amplitude is decreased and can even be zero. This is called destructive interference.

7 Constructive Interference When two waves always meet condensation-to-condensation and rarefaction-to-rarefaction, they are said to be exactly in phase and to exhibit constructive interference.

8 Destructive Interference When two waves always meet condensation-to-rarefaction, they are said to be exactly out of phase and to exhibit destructive interference.

9 Superposition (Interference) of Sinusoidal Waves They move in the same direction and have the same frequency and velocity. To derive an equation for the combined wave, we use the equations for the two waves, the principle of superposition, and a trigonometric identity. Two traveling waves, y 1 and y 2, whose phases differ only by the constant φ, have amplitude A, angular wave number k and angular frequency ω.

10 Superposition (Interference) of Sinusoidal Waves To simplify the analysis, the first wave is assumed to have a phase constant of zero, so the phase difference φ is the phase constant of the second wave. The principle of superposition says we can add these equations to determine the equation for the combined wave, y c. We use the trigonometric identity sin a sin b 2cos a b 2 sin a b 2 to derive the equation for the combined wave.

11 Superposition (Interference) of Sinusoidal Waves y = transverse displacement A = amplitude k = wave number ω = angular frequency φ = phase constant

12 Superposition (Interference) of Sinusoidal Waves y=2acos (φ/2) sin(kx wt+φ/2) The superposition of two identical waves y 1 and y 2 (blue and green) to yield a resultant wave (red). (a) When y 1 and y 2 are in phase, the result is constructive interference. (b) When y 1 and y 2 are π rad out of phase, the result is destructive interference. (c) When the phase angle has a value other than 0 or π rad, the resultant wave y falls somewhere between the extremes shown in (a) and (b)

13 Superposition (Interference) of Sound Waves

14 Superposition (Interference) of Sound Waves Phase difference f can be related to path length difference DL, by noting that a phase difference of 2p rad corresponds to one wavelength. Therefore, Fully constructive interference occurs when f is zero, 2p, or any integer multiple of 2p. Fully destructive interference occurs when f is an odd multiple of p:

15 Example: What Does a Listener Hear? Two in-phase loudspeakers, A and B, are separated by 3.20 m. A listener is stationed at C, which is 2.40 m in front of speaker B. Both speakers are playing identical 214-Hz tones, and the speed of sound is 343 m/s. Does the listener hear a loud sound, or no sound? Calculate the path length difference m 2.40 m 2.40 m 1.60 m Calculate the wavelength. f v 343 m s 214 Hz 1.60 m Because the path length difference is equal to an integer (1) number of wavelengths, there is constructive interference, which means there is a loud sound.

16 Application: Noise Cancelling Headphones

17 Standing Waves Two waves with the same frequency, wavelength, and amplitude traveling in opposite directions will interfere and produce standing waves. Let two functions as Where; y 1 represents a wave traveling in the +x direction y 2 represents one traveling in the -x direction. Using the identity Then a b a b sin a sin b 2cos sin 2 2 y 1+2 =(2Asin(kx)) cos (wt) This equation represents the wave function of a standing wave. A standing wave is an oscillation pattern with a stationary outline that results from the superposition of two identical waves traveling in opposite directions.

18 Standing Waves

19 Standing Waves

20 Standing Waves

21 Exercise 2: Two waves traveling in opposite directions produce a standing wave. The individual wave functions are y 1 = (4.0 cm) sin(3.0x - 2.0t) y 2 =(4.0 cm) sin(3.0x + 2.0t) where x and y are measured in centimeters. a) Find the amplitude of the simple harmonic motion of the element of the medium located at x = 2.3 cm. b) Find the positions of the nodes and antinodes if one end of the string is at x = 0. c) What is the maximum value of the position in the simple harmonic motion of an element located at an antinode?

22 Standing Waves in a String Fixed at Both Ends When a guitar string of length L is plucked, only certain frequencies can be produced, because only certain wavelengths can sustain themselves. Only standing waves persist. Many harmonics can exist at the same time, but the fundamental (n = 1) usually dominates. As we saw in the wave presentation, a standing wave occurs when a wave reflects off a boundary and interferes with itself in such a way as to produce nodes and antinodes. Destructive interference always occurs at a node. Both types occur at an antinode; they alternate. n = 1 (fundamental) n = 2

23 Notice the pattern is of the form: = 2 L n = 1 = 2 L n = L n = 2 where n = 1, 2, 3,. Thus, only certain wavelengths can exists. To obtain tones corresponding to other wavelengths, one must press on the string to change its length. = = L L n = 3 n = 4

24 Standing Waves in a String Fixed at Both Ends Fundamental (First harmonic) = 2L First Overtone (Second harmonic) = L Second Overtone (Third harmonic) = 2L/3 nth harmonic = 2L/n

25 Harmonics String fixed at both ends f n n v 2L n 1,2,3,4,

26 Harmonics

27 Example: A rope of length L is clamped at both ends. Which one of the following is not a possible wavelength for standing waves on this rope? a) L/2 b) 2L/3 c) L d) 2L e) 4L

28 Exercise 3: Middle C on a piano has a fundamental frequency of 262 Hz, and the first A above middle C has a fundamental frequency of 440 Hz. a) Calculate the frequencies of the next two harmonics of the C string. b) If the A and C strings have the same linear mass density and length L, determine the ratio of tensions in the two strings. What If? What if we look inside a real piano? In this case, the assumption we made in part (b) is only partially true. The string densities are equal, but the length of the A string is only 64 percent of the length of the C string. What is the ratio of their tensions?

29 Standing Waves in Air Columns Standing waves can be set up in a tube of air, such as that inside an organ pipe, as the result of interference between longitudinal sound waves traveling in opposite directions. In a pipe open at both ends, the natural frequencies of oscillation form a harmonic series that includes all integral multiples of the fundamental frequency.

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32 Beats: Interference in Time Beating is the periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies. The number of amplitude maxima one hears per second, or the beat frequency, equals the difference in frequency between the two sources, as we shall show below. When two waves of slightly different frequencies interfere, the interference pattern varies in such a way that a listener hears an alternation between loudness and softness. The variation from soft to loud and back to soft is called a beat. In other words, a beat is the periodic variation in the amplitude of a wave that is the superposition of two waves of slightly different frequencies.

33 Beats: Interference in Time

34 Beats: Interference in Time Figure. Beats are formed by the combination of two waves of slightly different frequencies. (a) The individual waves. (b) The combined wave has an amplitude (broken line) that oscillates in time.

35 Beats: Interference in Time

36 Beats: Interference in Time

37 Exercise 6: Beat Mickey Mouse and Goofy are playing an E note. Mickey s guitar is right on at 330 Hz, but Goofy is slightly out of tune at 332 Hz. a) What frequency will the audience hear? 331 Hz, the average of the frequencies of the two guitars. b) How often will the audience hear the sound getting louder and softer? They will hear it go from loud to soft twice each second. (The beat frequency is 2 Hz, since the two guitars differ in frequency by that amount.)

38 Exercise 6: Beat Two identical piano strings of length m are each tuned exactly to 440 Hz. The tension in one of the strings is then increased by 1.0 %. If they are now struck, what is the beat frequency between the fundamentals of the two strings?

39 Nonsinusoidal Wave Patterns The sound wave patterns produced by the majority of musical instruments are nonsinusoidal. Each instrument has its own characteristic pattern. Note, however, that despite the differences in the patterns, each pattern is periodic. This point is important for our analysis of these waves. Figure. Sound wave patterns produced by (a) a tuning fork, (b) a flute, and (c) a clarinet, each at approximately the same frequency.

40 Nonsinusoidal Wave Patterns If the wave pattern is periodic, it can be represented as closely as desired by the combination of a sufficiently large number of sinusoidal waves that form a harmonic series. The corresponding sum of terms that represents the periodic wave pattern is called a Fourier series.

41 Nonsinusoidal Wave Patterns Figure 7. Harmonics of the wave patterns shown. Note the variations in intensity of the various harmonics. The analysis involves determining the coefficients of the harmonics in the corresponding equation from a knowledge of the wave pattern. The reverse process, called Fourier synthesis.

42 Nonsinusoidal Wave Patterns