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Question

Two identical sources of sound of same frequency and identical intensities $$\displaystyle I_0$$ are producing sound. If their phases are irregular, then the average intensity of sound at a point where waves from the two sources are superposing is 


A
I0
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B
2I0
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C
4I0
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D
Zero
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Solution

The correct option is B $$\displaystyle 2 I_0$$
Given :   $$I_1 = I_2  = I_o$$

Resultant intensity        $$I = I_1 + I_2 + 2  \sqrt{I_1  I_2}   cos \delta   $$         
where $$\delta $$ is the phase difference.

 $$I = I_o + I_o + 2  \sqrt{I_o \times  I_o}   cos \delta   = 2 (1 + cos  \delta)   $$  

$$\implies  I = 4  I_o  cos^2 \dfrac{\delta}{2}$$

Now average intensity       $$< I > = 4I_o  <cos^2  \dfrac{\delta}{2}>$$

 $$< I > = 4I_o  \times \dfrac{1}{2}  =  2  I_o$$

Physics

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