### Analytical expression of intensity for constructive and destructive interference due to Young's double slit

Analytical expression of intensity for interference due to Young's double slit: Let us consider two waves from slit $S_{1}$ and $S_{2}$ having amplitude $a_{1}$ and $a_{2}$ respectively superimpose on each other at point $P$ . If the displacement of waves is $y_{1}$ and $y_{2}$ and the phase difference is $\phi$ then $y_{1}=a_{1} \: sin \omega t \qquad(1)$ $y_{2}=a_{2} \: sin \left( \omega t + \phi \right) \qquad(2)$ According to the principle of superposition: $y=y_{1}+y_{2} \qquad(3)$ Now substitute the value of $y_{1}$ and $y_{2}$ in the above equation $(3)$ $y=a_{1} \: sin \omega t + a_{2} \: sin \left( \omega t + \phi \right)$ $y=a_{1} \: sin \omega t + a_{2} \left( sin \omega t \: cos \phi + cos \omega t \: sin \phi \right) $ $y=a_{1} \: sin \omega t + a_{2} \: sin \omega t \: cos \phi + a_{2}\: cos \omega t \: sin \phi $ $y= \left( a_{1} + a_{2} \: cos \phi \right) \: sin \omega t + a_{2} \: sin \phi \: cos \omega t \qquad(4)$