Questions: Two speakers are driven exactly out of phase with each other at 256 Hz. If a microphone placed on the line between the two speakers, exactly in the middle, detects no sound at all, then what is possibly the distance between the microphone and each speaker? The speed of sound is 340 m / s. Multiple Choice 0.94 m 0.38 m 0.75 m 1.50 m Can be any distance at all

Two speakers are driven exactly out of phase with each other at 256 Hz. If a microphone placed on the line between the two speakers, exactly in the middle, detects no sound at all, then what is possibly the distance between the microphone and each speaker? The speed of sound is 340 m / s.

Multiple Choice
0.94 m
0.38 m
0.75 m
1.50 m
Can be any distance at all

Transcript text: Two speakers are driven exactly out of phase with each other at 256 Hz . If a microphone placed on the line between the two speakers, exactly in the middle, detects no sound at all, then what is possibly the distance between the microphone and each speaker? The speed of sound is $340 \mathrm{~m} / \mathrm{s}$. Multiple Choice 0.94 m 0.38 m 0.75 m 1.50 m Can be any distance at all

Solution

Solution Steps

Step 1: Determine the Wavelength of the Sound

The frequency of the sound is given as $ f = 256 \, \text{Hz} $, and the speed of sound is $ v = 340 \, \text{m/s} $. We can calculate the wavelength ($\lambda$) using the formula:

\[ \lambda = \frac{v}{f} \]

Substituting the given values:

\[ \lambda = \frac{340 \, \text{m/s}}{256 \, \text{Hz}} \approx 1.3281 \, \text{m} \]

Step 2: Determine the Condition for Destructive Interference

For destructive interference to occur at the microphone, which is placed exactly in the middle between the two speakers, the path difference between the sound waves from the two speakers must be an odd multiple of half-wavelengths:

\[ \Delta x = \left(n + \frac{1}{2}\right) \lambda \]

where $ n $ is an integer. Since the microphone is exactly in the middle, the simplest case is when $ n = 0 $, which gives:

\[ \Delta x = \frac{\lambda}{2} = \frac{1.3281 \, \text{m}}{2} \approx 0.6641 \, \text{m} \]

Step 3: Calculate the Possible Distance

Since the microphone is exactly in the middle, the distance from the microphone to each speaker should be half of the path difference for the simplest case of destructive interference:

\[ d = \frac{\Delta x}{2} = \frac{0.6641 \, \text{m}}{2} \approx 0.3320 \, \text{m} \]

However, this calculation does not directly match any of the given options. Let's consider the possibility of a different integer $ n $ that might match one of the options. If we consider $ n = 1 $:

\[ \Delta x = \left(1 + \frac{1}{2}\right) \lambda = \frac{3}{2} \times 1.3281 \, \text{m} \approx 1.9922 \, \text{m} \]

The distance from the microphone to each speaker would then be:

\[ d = \frac{1.9922 \, \text{m}}{2} \approx 0.9961 \, \text{m} \]

This is close to the option 0.94 m, which is the closest match considering possible rounding or measurement approximations.