Consider ‘s’ be the sources of monochromatic light wave ‘S1‘ and ‘S2‘ are the two coherent sources emitted from single point ‘s’. Let λ be the wavelength of a monochromatic light, ‘a’ is the amplitude and ϕ is the phase difference between the two waves originating from the source ‘s’. Let the displacement of the wave originating from S1 be y1 = asinωT and the displacement of wave originating from S2 be y2 = asin(ωT + ϕ). After superposition, the displacement of the resultant wave is given by:
y = y1 + y2 = asinωT + asin(ωT + ϕ)
or, y = a(1 + cos ϕ)sinωt + asinϕ.cosωt……(i)
Let a(1 + cos ϕ) = Rcosθ
and asin ϕ = Rsinθ
Here, R is the amplitude of the resultant wave after superposition.
Now, equation (i) becomes,
y = Rsin θ.sinωT + Rsinθ .cosωT
or, y = Rsin(ωT + θ)……(ii)
Equation (ii) is the equation of simple harmonic motion.
Now, squaring and adding equation (i) and (ii),
R2 = 2a2(1 + cos ϕ) = 4a2cos2ϕ/2)
The intensity at a point is given by:
I = R2 = 4a2cos2ϕ/2 = Io cos2ϕ/2
Here, Io = 4a2 .This is the expression for maximum intensity.
When ϕ = 0, 2 π, 4 π…… I = 4a2.
This is the case for bright fringe or constructive interference.
When ϕ = π + 3π, 5 π…… then I = 0.
This is the condition for dark fringe or destructive interference.