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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 1a
Chapter 35, Problem 1a

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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Step 1: Understand the problem setup. The speakers A and B are separated by a distance of 1.40 m and emit sound waves in phase with a wavelength of 34 cm (0.34 m). The person at point P is always 1.50 m away from speaker B and moves along a path such that the distance from speaker A changes. We need to find the values of x (distance from speaker A) where the sound is maximally reinforced, meaning constructive interference occurs.
Step 2: Recall the condition for constructive interference. Constructive interference happens when the path difference between the sound waves from the two speakers is an integer multiple of the wavelength. Mathematically, this is expressed as: ΔL = nλ, where ΔL is the path difference, n is an integer (0, 1, 2, ...), and λ is the wavelength.
Step 3: Express the path difference. The path difference ΔL is given by the absolute difference between the distances from the person to each speaker: ΔL = |L_A - L_B|, where L_A is the distance from speaker A to point P and L_B is the distance from speaker B to point P. Since L_B is fixed at 1.50 m, we can write ΔL = |x - 1.50|.
Step 4: Substitute the condition for constructive interference into the path difference equation. For constructive interference, |x - 1.50| = nλ. Substitute λ = 0.34 m into the equation: |x - 1.50| = n(0.34). Solve for x: x = 1.50 ± n(0.34).
Step 5: Limit the solution to the given range. The problem specifies that x ≤ 1.50 m. Therefore, calculate the possible values of x for integer values of n (starting from n = 0) that satisfy this condition. For each value of n, check if x ≤ 1.50 m and list the corresponding values of x.

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Key Concepts

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

Wave Interference

Wave interference occurs when two or more waves overlap and combine to form a new wave pattern. This can result in constructive interference, where waves align to amplify the sound, or destructive interference, where they cancel each other out. In this scenario, the sound waves from speakers A and B will interfere based on their path differences to point P.
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Path Difference

The path difference is the difference in distance traveled by two waves from their sources to a common point. For constructive interference to occur, this path difference must be an integer multiple of the wavelength. In this case, the distances from speakers A and B to point P will determine the path difference and thus the reinforcement of sound at point P.
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Wavelength and Frequency

Wavelength is the distance between successive crests of a wave, and it is inversely related to frequency, which is the number of wave cycles that pass a point per second. In this problem, the wavelength of 34 cm is crucial for calculating the conditions for constructive interference, as it helps determine the specific distances at which the sound will be maximally reinforced.
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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.

3
views
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. How far from the center must she walk before she first hears the sound maximally enhanced?

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