Consider two coherent sources \( S_{1} \) and \( S_{2} \) producing monochromatic waves to produ...

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Consider two coherent sources \( S_{1} \) and \( S_{2} \) producing monochromatic waves to produce interference pattern. Let the displacement of the wave produced by \( S_{1} \) be given by \( Y_{1}=a \cos \omega t \) and the displacement by \( S_{2} \) be \( Y_{2}=a \cos (\omega t+\phi) \). Find out the expression for the amplitude of the resultant displacement at a point and show that the intensity at that point will be \( I=4 a^{2} \cos ^{2} \phi / 2 \). Hence establish the conditions for constructive and destructive interference.
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