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Ch 16: Sound & Hearing
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 16: Sound & HearingProblem 35b
Chapter 16, Problem 35b

Two loudspeakers, A and B (Fig. E16.35), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. Consider point Q along the extension of the line connecting the speakers, 1.00 m to the right of speaker B. Both speakers emit sound waves that travel directly from the speaker to point Q. What is the lowest frequency for which destructive interference occurs at point Q?
Diagram of two speakers A and B connected to an amplifier, with distances labeled for wave interference.

Verified step by step guidance
1
Identify the path difference between the sound waves from speakers A and B to point Q. The distance from A to Q is 3.00 m (2.00 m + 1.00 m), and the distance from B to Q is 1.00 m.
Calculate the path difference: ΔL = L_AQ - L_BQ = 3.00 m - 1.00 m = 2.00 m.
For destructive interference, the path difference should be an odd multiple of half wavelengths: ΔL = (m + 0.5)λ, where m is an integer.
Express the wavelength in terms of frequency: λ = v/f, where v is the speed of sound in air (approximately 343 m/s).
Set up the equation for the lowest frequency (m = 0): 2.00 m = (0.5) * (343 m/s) / f. Solve for f to find the lowest frequency for destructive interference.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Destructive Interference

Destructive interference occurs when two waves meet in such a way that their crests and troughs align oppositely, leading to a reduction in amplitude. For sound waves, this typically happens when the path difference between the waves from two sources is an odd multiple of half the wavelength. In this scenario, understanding how the distance between the speakers and the point of observation affects the phase relationship of the waves is crucial.
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Path Difference

Path difference refers to the difference in distance traveled by two waves from their sources to a common point. In the context of the speakers A and B, the path difference to point Q is essential for determining whether constructive or destructive interference occurs. For destructive interference, the path difference must equal (n + 0.5)λ, where n is an integer and λ is the wavelength of the sound.
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Frequency and Wavelength Relationship

The frequency of a wave is inversely related to its wavelength, described by the equation v = fλ, where v is the speed of sound, f is the frequency, and λ is the wavelength. In this problem, finding the lowest frequency that results in destructive interference at point Q requires calculating the corresponding wavelength based on the path difference. This relationship is fundamental in understanding how sound waves interact in this scenario.
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Related Practice
Textbook Question

Small speakers A and B are driven in phase at 725 Hz by the same audio oscillator. Both speakers start out 4.50 m from the listener, but speaker A is slowly moved away (Fig. E16.34)<IMAGE>. If A is moved even farther away than in part (a), at what distance d will the speakers next produce destructive interference at the listener’s location?

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Textbook Question

Small speakers A and B are driven in phase at 725 Hz by the same audio oscillator. Both speakers start out 4.50 m from the listener, but speaker A is slowly moved away (Fig. E16.34). At what distance d will the sound from the speakers first produce destructive interference at the listener's location?

<Image>

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Textbook Question

The motors that drive airplane propellers are, in some cases, tuned by using beats. The whirring motor produces a sound wave having the same frequency as the propeller. (a) If one single-bladed propeller is turning at 575 rpm and you hear 2.0-Hz beats when you run the second propeller, what are the two possible frequencies (in rpm) of the second propeller? (b) Suppose you increase the speed of the second propeller slightly and find that the beat frequency changes to 2.1 Hz. In part (a), which of the two answers was the correct one for the frequency of the second single-bladed propeller? How do you know?

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Textbook Question

Two small stereo speakers are driven in step by the same variable-frequency oscillator. Their sound is picked up by a microphone arranged as shown in Fig. E16.39. For what frequencies does their sound at the speakers produce constructive interference?

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Textbook Question

Two guitarists attempt to play the same note of wavelength 64.8 cm at the same time, but one of the instruments is slightly out of tune and plays a note of wavelength 65.2 cm instead. What is the frequency of the beats these musicians hear when they play together?

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Textbook Question

Two loudspeakers, A and B (Fig. E16.35), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. Consider point Q along the extension of the line connecting the speakers, 1.00 m to the right of speaker B. Both speakers emit sound waves that travel directly from the speaker to point Q. What is the lowest frequency for which constructive interference occurs at point Q?

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views