Question

Two light waves propagating in the same direction are superposed. The frequency of the two light waves is the same and there is a constant phase difference between them. Obtain an expression for the intensity of the resultant wave.

Answer 1

Let's take two waves $y_1 \ and \ y_2$ and are superposed

$y_1 = a cos ωt$

$y_2 = a cos (ωt + φ )$

and the resultant displacement will be given by

$y = y_1 + y_2$

$y= a [cos ωt + cos (ωt +φ)]$

$y= 2 a cos (φ/2) cos (ωt + φ/2)$

The amplitude of the resultant displacement is 2acos(φ/2) and therefore the intensity at that point will be

$I = 4 I_0 cos^2 (φ/2)$

If φ = 0, ± 2 π, ± 4 π,… we will have constructive interference leading to maximum intensity

If φ = ± π, ± 3π, ± 5π …we will have destructive interference leading to minimum intensity