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Ch 15: Mechanical Waves
Chapter 15, Problem 35

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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Identify the path difference between the waves reaching point Q from antennas A and B. Since antenna B is 120 m to the right of A and point Q is 40 m to the right of B, the total distance from A to Q is 120 m + 40 m = 160 m. The distance from B to Q is 40 m. Thus, the path difference is 160 m - 40 m = 120 m.
Recall the condition for constructive interference, which occurs when the path difference is an integer multiple of the wavelength, i.e., \(\Delta x = n \lambda\), where \(n\) is an integer (0, 1, 2, ...).
To find the longest wavelength for constructive interference, set \(n = 1\) since the first positive integer will give the maximum wavelength. Thus, the equation becomes \(120 m = 1 \times \lambda\).
Solve for \(\lambda\) to find the longest wavelength for constructive interference. This is done by dividing the path difference by the integer \(n\), i.e., \(\lambda = \frac{120 m}{1}\).
The longest wavelength for constructive interference at point Q is therefore 120 m.

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Key Concepts

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

Constructive Interference

Constructive interference occurs when two or more waves overlap in such a way that their crests and troughs align, resulting in a wave of greater amplitude. This phenomenon is crucial in understanding how waves from antennas combine at a given point. For constructive interference to happen, the path difference between the waves arriving at that point must be an integer multiple of the wavelength.
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Path Difference

Path difference refers to the difference in distance traveled by two waves from their sources to a specific point. In the context of antennas A and B, the path difference at point Q determines whether the waves will interfere constructively or destructively. For constructive interference, this path difference must equal nλ, where n is an integer and λ is the wavelength of the emitted waves.
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Wavelength

Wavelength is the distance between successive crests (or troughs) of a wave and is inversely related to frequency. In this scenario, the wavelength plays a critical role in determining the conditions for constructive interference at point Q. The longest wavelength that allows for constructive interference will correspond to the maximum path difference that still satisfies the condition for 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. (a) What does she hear: constructive or destructive interference? Why?
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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. (c) How far from the center must she walk before she first hears the sound maximally enhanced?
376
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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 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. (a) 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 (a) maximally reinforced. Limit your solution to the cases where x … 1.50 m
584
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1
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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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