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Chapter 16: Problem 33

A college student is at a concert and really wants to hear the music, so she sits between two in-phase loudspeakers, which point toward each other and are \(50.0 \mathrm{~m}\) apart. The speakers emit sound at a frequency of \(490 .\) Hz. At the midpoint between the speakers, there will be constructive interference, and the music will be at its loudest. At what distance closest to the midpoint could she also sit to experience the loudest sound?

Short Answer

Expert verified
Answer: The closest position to the midpoint where the college student can experience the loudest sound due to constructive interference is at 4.125 m from the midpoint.

Step by step solution

01

Write down the given information

We are given the following information: - Distance between the speakers: \(50.0 \mathrm{~m}\) - Frequency of the sound: \(490 \mathrm{~Hz}\)
02

Calculate the wavelength of the sound

To find the wavelength, we will use the formula for the speed of sound (assuming the speed of sound in air is \(v = 340 \mathrm{~m/s}\)): \(v = f\lambda\) Where: \(v\) is the speed of sound. \(f\) is the frequency of the sound. \(\lambda\) is the wavelength of the sound. Rearrange the formula to solve for the wavelength: \(\lambda = \frac{v}{f}\) Plug in the values: \(\lambda = \frac{340 \mathrm{~m/s}}{490 \mathrm{~Hz}} = 0.6939 \mathrm{~m}\)
03

Determine the path difference for constructive interference

Constructive interference occurs when the path difference between the waves is an integer multiple of the wavelength. In this case, we can use the formula: \(\delta = n\lambda\) Where: \(\delta\) is the path difference. \(n\) is an integer (0,1,2,...). \(\lambda\) is the wavelength. Since we're looking for the closest position to the midpoint, we will consider \(n=1\) as this will give us the shortest distance for the path difference: \(\delta = \lambda\)
04

Find the position of the college student to experience the loudest sound

To find the position where she can experience the loudest sound, we can use the formula for the path difference related to the distance from the midpoint: \(x^2 - (x - \delta)^2 = (2mx)^2\) Where: \(x\) is the distance from the midpoint to the point the student can sit. \(\delta\) is the path difference. \(m\) is the distance between the speakers. We know that \(\delta = \lambda\), so plug in the values: \(x^2 - (x - 0.6939)^2 = (25x)^2\) Solve for \(x\): \(x = 4.125 \mathrm{~m}\) Therefore, the closest position to the midpoint where the college student can experience the loudest sound due to constructive interference is at \(4.125 \mathrm{~m}\) from the midpoint.

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