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Ch 35: Interference
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 35: InterferenceProblem 2c
Chapter 35, Problem 2c

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. How far from the center must she walk before she first hears the sound maximally enhanced?

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Step 1: Understand the concept of constructive interference. Constructive interference occurs when two sound waves meet in phase, meaning their crests and troughs align perfectly, resulting in a maximally enhanced sound. This happens when the path difference between the two waves is an integer multiple of the wavelength.
Step 2: Calculate the wavelength of the sound waves using the formula \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound (340.0 m/s) and \( f \) is the frequency of the sound waves (250.0 Hz).
Step 3: Determine the condition for constructive interference. The path difference between the sound waves from the two speakers must equal \( n \lambda \), where \( n \) is an integer (starting from \( n = 1 \) for the first maximum).
Step 4: Set up the geometry of the problem. The woman starts at the midpoint between the two speakers, so the initial path difference is zero. As she moves away from the center, the path difference changes. Use the Pythagorean theorem to express the distances from the woman to each speaker as functions of her position.
Step 5: Solve for the position where the path difference equals \( \lambda \) (the first maximum). This involves equating the difference in distances from the woman to each speaker to \( \lambda \). Simplify the equation and solve for the distance she must walk from the center.

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Key Concepts

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

Interference of Sound Waves

Interference occurs when two or more sound waves overlap, resulting in a new wave pattern. In this scenario, the two speakers produce in-phase sound waves, meaning their peaks and troughs align. This can lead to constructive interference, where the sound waves reinforce each other, creating areas of increased sound intensity, known as antinodes.
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Path Difference

Path difference refers to the difference in distance traveled by sound waves from two sources to a given point. For constructive interference to occur, the path difference must be an integer multiple of the wavelength. In this case, the woman must walk a certain distance from the midpoint to achieve the necessary path difference for the sound waves from both speakers to arrive in phase.
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Wavelength and Frequency Relationship

The wavelength of a sound wave is inversely related to its frequency, as described by the equation v = fλ, where v is the speed of sound, f is the frequency, and λ is the wavelength. For the given frequency of 250 Hz and speed of sound at 340 m/s, calculating the wavelength is essential to determine how far the woman must walk to experience the first maximum in sound intensity.
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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. What does she hear: constructive or destructive interference? Why?

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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 cancelled? Limit your solution to the cases where x ≤ 1.50 m.

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Textbook Question

Two speakers, emitting identical sound waves of wavelength 2.0 m in phase with each other, and an observer are located as shown in Fig. E35.5. At the observer's location, what is the path difference for waves from the two speakers?

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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 maximally reinforced? Limit your solution to the cases where x ≤ 1.50 m.

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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. 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. What is the longest wavelength for which there will be constructive interference at point Q?

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