SWI-1 Two speakers are in phase and are both playing a tone with a frequency of 250.0 Hz. A listener starts at the location of speaker 2 and moves along the x axis. At what values of x are the first 3 locations where he will hear an intensity minimum? Hint: these are the n = 0, 1, 2 points of destructive interference. Take the speed of sound in air to be 343 m/s. speaker 1 (0, 4.50m) speaker 2 (0, 0)
SWI-1 Two speakers are in phase and are both playing a tone with a frequency of 250.0 Hz. A listener starts at the location of speaker 2 and moves along the x axis. At what values of x are the first 3 locations where he will hear an intensity minimum? Hint: these are the n = 0, 1, 2 points of destructive interference. Take the speed of sound in air to be 343 m/s. speaker 1 (0, 4.50m) speaker 2 (0, 0)
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![**Problem: Sound Interference and Minimum Intensity Points**
**Description:**
Two speakers, in phase, are emitting a tone at a frequency of 250.0 Hz. A listener starts at the position of speaker 2 and moves along the x-axis. We need to find the x-values corresponding to the first three points where the listener hears an intensity minimum due to destructive interference, defined by n = 0, 1, and 2. Assume the speed of sound in air is 343 m/s.
**Diagram Explanation:**
- **Speakers' Positions:**
- Speaker 1 is located at coordinates (0, 4.50 m).
- Speaker 2 is located at the origin (0, 0).
- **Listener's Path:**
- The listener begins at speaker 2 and moves along the positive x-axis.
**Objective:**
To determine the specific x-values where the listener experiences the first three points of destructive interference.
**Concept:**
Destructive interference occurs when the path difference between the sound waves from the two speakers is an odd multiple of half wavelengths. For destructive interference, the condition is:
\[
\Delta L = (n + \frac{1}{2})\lambda
\]
where:
- \( \Delta L \) is the path difference,
- \( n \) is an integer (0, 1, 2,...),
- \( \lambda \) is the wavelength of the sound.
**Calculation Steps:**
1. Calculate the wavelength (\(\lambda\)) using the formula:
\[
\lambda = \frac{v}{f}
\]
where:
- \( v = 343 \, \text{m/s} \) (speed of sound),
- \( f = 250.0 \, \text{Hz} \).
2. Find the x-values for the first three intensity minima (n = 0, 1, 2) using the condition for destructive interference.
**Solution:**
1. Calculate \(\lambda\):
\[
\lambda = \frac{343}{250.0} \, \text{m}
\]
2. Solve for x where the condition \(\Delta L = (n + \frac{1}{2})\lambda\) holds for n = 0, 1, and 2.
This will allow for finding the positions (x-values) where the listener experiences destructive interference.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9496fb66-15f0-4d2e-bc7e-ca3bb9b70577%2F13b1ac84-6bcf-433e-af8a-63189f870059%2Fgkzyvz_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Sound Interference and Minimum Intensity Points**
**Description:**
Two speakers, in phase, are emitting a tone at a frequency of 250.0 Hz. A listener starts at the position of speaker 2 and moves along the x-axis. We need to find the x-values corresponding to the first three points where the listener hears an intensity minimum due to destructive interference, defined by n = 0, 1, and 2. Assume the speed of sound in air is 343 m/s.
**Diagram Explanation:**
- **Speakers' Positions:**
- Speaker 1 is located at coordinates (0, 4.50 m).
- Speaker 2 is located at the origin (0, 0).
- **Listener's Path:**
- The listener begins at speaker 2 and moves along the positive x-axis.
**Objective:**
To determine the specific x-values where the listener experiences the first three points of destructive interference.
**Concept:**
Destructive interference occurs when the path difference between the sound waves from the two speakers is an odd multiple of half wavelengths. For destructive interference, the condition is:
\[
\Delta L = (n + \frac{1}{2})\lambda
\]
where:
- \( \Delta L \) is the path difference,
- \( n \) is an integer (0, 1, 2,...),
- \( \lambda \) is the wavelength of the sound.
**Calculation Steps:**
1. Calculate the wavelength (\(\lambda\)) using the formula:
\[
\lambda = \frac{v}{f}
\]
where:
- \( v = 343 \, \text{m/s} \) (speed of sound),
- \( f = 250.0 \, \text{Hz} \).
2. Find the x-values for the first three intensity minima (n = 0, 1, 2) using the condition for destructive interference.
**Solution:**
1. Calculate \(\lambda\):
\[
\lambda = \frac{343}{250.0} \, \text{m}
\]
2. Solve for x where the condition \(\Delta L = (n + \frac{1}{2})\lambda\) holds for n = 0, 1, and 2.
This will allow for finding the positions (x-values) where the listener experiences destructive interference.
Expert Solution
Step 1
Given:
Frequency of the sound = 250 Hz
Speed of the sound = 343 m/s
We have to find at what three values of x, we get the minimum sound intensity.
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