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Ch 15: Mechanical Waves
All textbooksYoung & Freedman Calc 14th EditionCh 15: Mechanical WavesProblem 35
Chapter 15, Problem 35

Two speakers, emitting identical sound waves of wavelength 2.0 m in phase with each other, and an observer are located as shown in Diagram showing two speakers 10m apart and an observer 13m away, illustrating wave interference.
Fig. E35.5. (a) At the observer's location, what is the path difference for waves from the two speakers?Illustration of two sound sources 6m apart with an observer 8m away, depicting wave interference patterns.

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Step 1: Identify the positions of the speakers and the observer. The speakers are 6.0 m apart, and the observer is 8.0 m away from the midpoint between the speakers.
Step 2: Calculate the distance from the observer to each speaker using the Pythagorean theorem. For the left speaker, the distance is sqrt((6.0/2)^2 + 8.0^2). For the right speaker, the distance is the same because the setup is symmetrical.
Step 3: Determine the path difference by subtracting the distances calculated in Step 2. Since the distances are equal, the path difference will be zero.
Step 4: Verify the wavelength of the sound waves, which is given as 2.0 m.
Step 5: Conclude that the path difference is zero, meaning the waves from both speakers reach the observer in phase, resulting in constructive interference.

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Related Practice
Textbook Question
Two speakers that are 15.0 m apart produce in-phase sound waves of frequency 250.0 Hz in a room where the speed of sound is 340.0 m>s. A woman starts out at the midpoint between the two speakers. The room's walls and ceiling are covered with absorbers to eliminate reflections, and she listens with only one ear for best precision. (c) How far from the center must she walk before she first hears the sound maximally enhanced?
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Textbook Question
Two radio antennas A and B radiate in phase. Antenna B is 120 m to the right of antenna A. Consider point Q along the extension of the line connecting the antennas, a horizontal distance of 40 m to the right of antenna B. The frequency, and hence the wavelength, of the emitted waves can be varied. (a) What is the longest wavelength for which there will be destructive interference at point Q?
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Textbook Question
Two radio antennas A and B radiate in phase. Antenna B is 120 m to the right of antenna A. Consider point Q along the extension of the line connecting the antennas, a horizontal distance of 40 m to the right of antenna B. The frequency, and hence the wavelength, of the emitted waves can be varied. (b) What is the longest wavelength for which there will be constructive interference at point Q?
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Textbook Question
Two small stereo speakers A and B that are 1.40 m apart are sending out sound of wavelength 34 cm in all directions and all in phase. A person at point P starts out equidistant from both speakers and walks so that he is always 1.50 m from speaker B

(Fig. E35.1). For what values of x will the sound this person hears be (a) maximally reinforced. Limit your solution to the cases where x … 1.50 m
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Textbook Question
Two small stereo speakers A and B that are 1.40 m apart are sending out sound of wavelength 34 cm in all directions and all in phase. A person at point P starts out equidistant from both speakers and walks so that he is always 1.50 m from speaker B

(Fig. E35.1). For what values of x will the sound this person hears be (b) cancelled? Limit your solution to the cases where x … 1.50 m
442
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