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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 1b
Chapter 35, Problem 1b

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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Step 1: Understand the problem setup. The two stereo 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 destructive interference (cancellation) occurs.
Step 2: Recall the condition for destructive interference. Destructive interference happens when the path difference between the two waves is an odd multiple of half the wavelength: \( \Delta L = (m + \frac{1}{2}) \lambda \), where \( m \) is an integer (0, 1, 2, ...), \( \Delta L \) is the path difference, and \( \lambda \) is the wavelength.
Step 3: Calculate the path difference \( \Delta L \). The path difference is given by \( \Delta L = |L_A - L_B| \), 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, \( L_A \) will vary as the person moves. Substitute \( L_A = x \) and \( L_B = 1.50 \) into the equation.
Step 4: Set the path difference equal to the destructive interference condition. Substitute \( \Delta L = (m + \frac{1}{2}) \lambda \) into \( \Delta L = |x - 1.50| \). This gives \( |x - 1.50| = (m + \frac{1}{2}) \times 0.34 \). Solve for \( x \) for different values of \( m \) (e.g., \( m = 0, 1, 2 \)) while ensuring \( x \leq 1.50 \).
Step 5: Interpret the results. For each value of \( m \), calculate the corresponding \( x \) values that satisfy the destructive interference condition. Ensure that the values of \( x \) are within the specified range (\( x \leq 1.50 \)). These values represent the positions where the sound is cancelled.

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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 add together to increase amplitude, or destructive interference, where waves cancel each other out. In the context of sound waves from speakers, the listener experiences variations in sound intensity depending on their position relative to the speakers.
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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. For destructive interference to occur, the path difference must be an odd multiple of half the wavelength (λ/2, 3λ/2, etc.). In this scenario, calculating the path difference between the sound waves from speakers A and B to point P is essential to determine where cancellation occurs.
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Wavelength

Wavelength is the distance between successive crests (or troughs) of a wave, and it is a key factor in wave behavior. In this problem, the wavelength of the sound waves is given as 34 cm. Understanding the wavelength helps in calculating the conditions for constructive and destructive interference, as it directly influences the path difference required for cancellation of sound at point P.
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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 maximally reinforced? Limit your solution to the cases where x ≤ 1.50 m.

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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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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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