Two small children are standing in front of you emitting identical piercing 3500 Hz sound waves...

Question:

Two small children are standing in front of you emitting identical piercing 3500 Hz sound waves in the same direction, along the positive x axis directly at you. If the children are screaming in phase, how far back should one child stand from the other in order to produce:

a. perfect destructive interference?

b. an amplitude 1.5 times greater than one child alone?

c. a sound that is half as loud as one child screaming alone would be?

Interference of sound waves from two spatially separated synchronous monochromatic sources.

Superposition of sound waves from two single-frequency sources creates in the surrounding space an interference field of variable intensity from minima to maxima.

The pattern of this interference field will vary depending on the relative position of these sound sources.

Answer and Explanation:

Consider the superposition of the following two sound waves from sources located at different points on the x-axis.

{eq}y_1(x,t)=Acos(2\pi*f((x-x_1)/v-t)) {/eq}

{eq}y_2(x,t)=Acos(2\pi*f((x-x_2)/v-t)) {/eq}

A-amplitude, f-frequency, v- the speed of these waves.

{eq}x_1, x_2 -\text{places of sources on the x-axis} {/eq}

The amplitude of the sum of these waves

{eq}A_1=|y_1(x,t)+y_2(x,t)|=2A*|cos(\pi*f/v*(x_1-x_2))| {/eq}

Assume v=343m/s.

a. Perfect destructive interference means

{eq}A_1=0, |x_1-x_2|=v/f(1/2+n)=0.098(1/2 +n)meters=(0.049+0.098*n)meters, n=0,1,2,3... {/eq}

b.An amplitude of 1.5 times greater than one child alone means

{eq}A_1=1.5A, |x_1-x_2|=v/f*(arccos(0.75)/\pi+n)=0.098*(0.23+n)meters=(0.023+0.098*n)meters {/eq}

c.A sound that is half as loud as one child screaming alone would be means

{eq}A_1=0.5*A, |x_1-x_2|=v/f*(1/\pi*arccos(0.25)+n)=0.098(0.42+n)meters=(0.04+0.098n)meters {/eq}


Learn more about this topic:

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Interference Patterns of Sound Waves

from MTEL Physics (11): Practice & Study Guide

Chapter 16 / Lesson 5
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