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Question

$$S_{1}$$ and $$S_{2}$$ are two coherent sources of sound of frequency 110 Hz each. They have no initial phase difference. The intensity at a point P due to $$S_{1}$$ is $$I_{0}$$ and due to $$S_{2}$$ is $$ 4I_{0}$$. If the velocity of sound is $$330\ m/s$$ then the resultant intensity at P is

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A
I0
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B
9I0
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C
3I0
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D
8I0
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Solution

The correct option is D $$3I_{0}$$
Wavelength $$\lambda= \dfrac{v}{f}=\dfrac{330}{110}=3 \: m$$ as velocity and frequency are given.
The expression for intensity of two waves after interference is:
$$I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos \phi$$     .....(1) where $$\phi$$ is the phase difference due to path difference ( $$S_2P-S_1P$$)
As seen from the figure,

$$S_2P=\sqrt{3^2 + 4^2}=5$$

$$S_1P=4$$

$$\phi = \dfrac{2\pi}{\lambda}(S_2P-S_1P)= \dfrac{2\pi}{\lambda}(5-4) =  \dfrac{2\pi}{3} 1= \dfrac{2\pi}{3}$$

Substituting $$\phi$$, $$I_1=I_0$$, $$I_2=4I_0$$ in (1)

$$I = I_0 + 4I_0 + 2\sqrt{I_0 \times  4I_0}\cos \dfrac{2\pi}{3}$$

$$\,\,\,= 5I_0 + 4I_0  (-\dfrac{1}{2})=5I_0 -2I_0=3I_0$$

Physics

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