#### Question

Light waves each of amplitude "a" and frequency "ω", emanating from two coherent light sources superpose at a point. If the displacements due to these waves are given by y_{1} = a cos ωt and y_{2} = a cos(ωt + ϕ) where ϕ is the phase difference between the two, obtain the expression for the resultant intensity at the point.

#### Solution

Let the displacement of the waves from the sources *S*_{1} and *S*_{2} at point P on the screen at any time *t* be given by:

*y*_{1} = *a* cos *ωt*

*and*

*y _{2} = a cos (ωt + Φ)*

where, Φ is the constant phase difference between the two waves

By the superposition principle, the resultant displacement at point P is given by:

y = y_{1} + y_{2}

y = a cos ωt + a cos (ωt + Φ)

`=2a[cos((omegat+omegat+phi)/2)cos((omegat-omegat-phi)/2)]`

`y=2acos(omegat+phi/2)cos(phi/2)" ...(i)"`

Let 2 `acos(phi/2)=A ...(ii)"`

Then, equation (i) becomes:

`y=Acos(omegat+phi/2)`

Now, we have:

`A^2=4a^2cos^2(phi/2)" ..(iii)"`

The intensity of light is directly proportional to the square of the amplitude of the wave. The intensity of light at point P on the screen is given by:

`I=4a^2cos^2(phi/2)" ...(iv)"`