Two speakers are separated by the distance, d1=2.40m, and are in phase. You are standing 3.00m...

Question:

Two speakers are separated by the distance, d1=2.40m, and are in phase. You are standing 3.00m directly in front of one speaker. Each speaker has an output power of 1.20W. Because the speakers are at different distances, there will be frequencies, at which the sound from the speakers interferes destructively. But because the speakers are at different distances, the sound intensities will not be the same, so the destructive interference will not be complete. We want to find the sound level when there is destructive interference. Use 343m/s for the speed of sound in air.

Interference of sound waves taking into account attenuation with distance to sources.

For destructive interference at some point of two coherent in-phase sources, two conditions should be met:

1. Difference between distance from this point to sound sources should be equal to odd number of half-length,

2. The intensity of the waves at this point should be equal to each other.

Answer and Explanation:

1) We figure out the distance between the receiver and the second source and the difference between receiver and sound sources {eq}\Delta D {/eq}, using the Pythagorean theorem.

{eq}d_2=\sqrt{d_1^2+d^2}=3.84m {/eq}

{eq}\Delta D=d_2-d {/eq}=(3.84-3.00)m=0.84m

To fulfill the first condition

{eq}\Delta D=1/2*\lambda*(2n-1)=1/2*v/f*(2n-1) {/eq}

{eq}\lambda,v=343m/s,f -\text{wavelength, speed, and frequency of the sound waves} {/eq}

n=1,2,3,4...

2) The intensity {eq}I_1, I_2 {/eq} of the sound's wave is directly proportional to the source's output power {eq}P_1, P_2 {/eq} and opposite proportional square distance to the source {eq}d, d_2 {/eq}

{eq}P_1/P_2=I_1/I_2*(d/d_2)^2 {/eq}

The second condition means

{eq}I_1=I_2 {/eq}

Considering that maximum output power {eq}P_2= {/eq}1.20 W

So,

{eq}P_1=P_2*(d/d_2)^2=(1.20*(3.00/3.84)^2)W=(1.2*0.61)W=0.73W {/eq}


Learn more about this topic:

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Constructive and Destructive Interference

from CLEP Natural Sciences: Study Guide & Test Prep

Chapter 8 / Lesson 16
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