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Ch 15: Mechanical Waves
Chapter 15, Problem 35

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 Diagram showing two speakers S1 and S2, distance d, and point P for wave interference analysis.
(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

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1
Determine the path difference between the sound waves from speakers A and B at point P. The path difference is given by |d - 2.20 m|.
For destructive interference (sound cancellation), the path difference should be an odd multiple of half the wavelength, i.e., (2n+1) * (λ/2), where n is an integer.
Given the wavelength λ = 34 cm = 0.34 m, set up the equation for destructive interference: |d - 2.20 m| = (2n+1) * (0.34 m / 2).
Solve for d in terms of n: d = 2.20 m ± (2n+1) * (0.17 m).
Since the person is always 1.50 m from speaker B, use the Pythagorean theorem to relate d, x, and the distance between the speakers: d^2 = x^2 + (1.50 m)^2. Substitute the values of d from the previous step into this equation and solve for 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 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 (e.g., λ/2, 3λ/2). In this scenario, calculating the path difference between the sound waves from speakers A and B as the listener moves is crucial for determining 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 how wavelength relates to frequency and wave speed is essential for analyzing interference patterns and predicting where sound will be canceled or amplified.
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