Lecture 7 Doppler Effect, Superposition and Interference of Waves

LECTURE 7 DOPPLER EFFECT, SUPERPOSITION AND INTERFERENCE OF WAVES

Instructor: Kazumi Tolich Lecture 7

¨ Reading chapter 14.6 to 14.7 ¤ Doppler effect ¤ Superposition ¤ Interference Quiz: 1

¨ Observers A, B, and C listen to a sound form a moving source. The locations of the sound wave crests at a given moment are shown in the figure. Rank the observer according to the frequency of the sound, smallest first. Quiz: 7-1 answer

¨ A < B < C

¨ The wavelength of the sound wave is indicated by the distance between two neighboring crests.

¨ At C, the crests are closer together, indicating the smaller wavelength. ¨ Thus the observed frequency is highest at C since � = , where � is the speed of the sound which is common for all observers.

¨ This shift in frequency (or wavelength) due to the motion of the wave source and/or observer is called the Doppler effect. The Doppler effect/Demo: 1

5 ¨ Shifted frequency, � , in sound can be observed when the source

is moving, the observer is moving, � or both are moving with respect to the medium.

1 ± � ⁄� � = � 1 ∓ �⁄� �

¨ Demo: siren on a string

� Sonic boom

¨ The Doppler shifts for a moving source and a moving observer cases are similar for low speeds but then diverge.

¨ If the source moves faster then the speed of sound, a sonic boom is created. Applications of the Doppler effect

¨ The Doppler shift is observed in electromagnetic waves as well.

¨ The Doppler effect has many practical applications: weather radar, speed radar, medical diagnostics, astronomical measurements. Quiz: 2

¨ Amy and Zack are both listening to a sound wave from a source that is moving to the right. Rank the frequencies that they hear and the source frequency, smallest first. Quiz: 7-2 answer

¨ � = � < � ±⁄ ¨ � = � ∓⁄ ¨ Since the source is moving toward receiver Zack, and Zack (receiver) is moving ⁄ ⁄ toward the source, he hears � = � > � ⁄ ⁄

¨ Since the source is moving away from the receiver Amy, and Amy (receiver) is ⁄ ⁄ moving toward the source, she hears � = � = � ⁄ ⁄ Example 1

¨ You are standing by a road, and a police car passes by. The frequency of a police siren is 550 Hz as the police car approaches you, and 450 Hz after it passes you and is receding. If the speed of sound is 343 m/s, how fast is the police car traveling? Principle of Superposition

¨ Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition.

¨ If two pulses combine to give a larger pulse, this is constructive interference (a). If they combine to give a smaller pulse, this is destructive interference (b). Quiz: 3

¨ Two identical waves with the same phase are added. Which of the following statements is/are correct about the resultant wave? Choose all that apply. A. Its frequency is the same as that of the original waves. B. Its frequency is doubled. C. Its frequency is zero. D. Its amplitude is the same as that of the original waves. E. Its amplitude is doubled. F. Its amplitude is zero. G. This problem cannot be solved without knowing the wavelengths of the two waves. Quiz: 7-3 answer

¨ Its frequency is the same as the added waves.

¨ Its amplitude is doubled.

Constructive Destructive Interference patterns

14 4λ ¨ Two-dimensional waves exhibit interference pattern.

¨ If the sources are in phase, constructive interference occurs if the path length difference, ∆�, is 0, �, 2�, 3�, ⋯, 5λ and destructive interference occurs if ∆� is �, �, � , ⋯

4λ

4.5λ Quiz: 4

¨ Speakers A and B are right next to each other and emit sound waves with a wavelength of � = 1 m, which interfere constructively at a donkey located far away (say, 200 m). What happens to the sound intensity if speaker A steps back 2.5 m? A. Intensity increases B. Intensity stays the same A C. Intensity goes to zero � D. Impossible to tell B Quiz: 7-4 answer/Demo: 2

¨ Intensity goes to zero

¨ If � = 1 m, then a shift of 2.5 m corresponds to 2.5�, which puts the two waves out of phase, leading to destructive interference.

¨ The sound intensity will therefore go to zero.

¨ Follow-up: What if you move back by 4 m?

¨ Two Speaker Interference ¤ Demonstration of constructive and destructive interference due to path length differences. Chapter 14: Waves and Sound James S. Walker, Physics, 5th Edition

56. Picture the Problem: Pulses A, B, C, and D all travel at 10 m/s on the same string but in opposite directions. The pulses pass through each other, and as they do, their amplitudes sum. Strategy: Each pulse will travel 1.0 m in a time interval of 0.10 s. Draw the pulses at the indicated later times and add the amplitudes in order to determine the displacement of point P at those times. Example: 2 (Walker Ch. 14-57) Solution: 1. (a) At time t = 0.10 s pulses B and C overlap to make the displacement of point P equal to 17 y 2.0 2.0 cm 0 cm : ¨ A pair of in-phase stereo speakers is

placed side by side, separated by a 2. (b) At time t = 0.20 s pulses A and D overlap to distance of � = 0.914 m. You stand make the displacement of point P equal to directly in front of one of the speakers, y 2.0 4.0 cm 2.0 cm : � = 2.44 m from the speaker. What is the lowest frequency that will produce Insight: When particles collide, they bounce off each other. When wave pulses collide, they pass right through each other, and when they do, their amplitudes sum. constructive interference at your location?

57. Picture the Problem: Two in-phase stereo speakers are 0.914 m apart. You are standing 2.44 meters in front of one of the speakers. The figure at the right shows the basic idea but is not drawn to scale. 0.914 m

Strategy: This problem is similar to Example 14-13. Use the Pythagorean theorem to calculate your distance d2 from the second speaker. Subtract from this distance from your distance d to the d2 1 d = 2.44 m nearby speaker in order to find the difference in path length for the 1 two sound waves. Set this path difference equal to one wavelength and solve Equation 14-1 to calculate the frequency.

Solution: 1. Use the Pythagorean 22 d2 2.44 m 0.914 m 2.606 m theorem to find the distance d2:

2. Subtract the distance to the nearby speaker: dd212.606 2.44 m 0.166 m

v 343 m/s 3. Solve Equation 14-1 for the frequency: f 2070 Hz 2.1 kHz 0.166 m Insight: Constructive interference occurs whenever the difference in path length from the two wave sources is an integer multiple of the wavelength.

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