Coherent and Incoherent Addition of Waves

When two or more waves travel through the same medium at the same time, the resultant displacement is the sum of the displacements due to the individual waves.

The redistribution of intensity that occurs when two light waves superpose is called interference.

Sources that emit waves of the same frequency and maintain a constant phase difference are called coherent sources.

Sources for which the phase difference changes randomly with time are called incoherent sources.

When two waves meet in the same phase, the resultant amplitude increases and the intensity becomes maximum.

When two waves meet in opposite phase, the resultant amplitude decreases, and the intensity becomes minimum.

For two waves of equal intensity I₀​, the resultant intensity is:

I = \[4I_0\cos^2\left(\frac{\phi}{2}\right)\]

Constructive Interference

Constructive interference occurs when the path difference between the two waves is: Δx = nλ

where n = 0, 1, 2, 3, …

The corresponding phase difference is: ϕ = 2nπ

At such points:

• The amplitudes reinforce each other.
• The intensity is maximum.
• For equal amplitudes, the maximum intensity is 4I₀.

Destructive Interference

Destructive interference occurs when the path difference is:

where n = 0, 1, 2, 3, …

The corresponding phase difference is: ϕ = (2n + 1)π

At such points:

• The waves oppose each other.
• The intensity is minimal.
• For equal amplitudes, the minimum intensity becomes zero.

When two sources are incoherent, their phase difference changes rapidly and randomly with time. As a result, no stable interference pattern is observed.

The average intensity at any point becomes simply the sum of the individual intensities. For two equal intensities I₀, the average intensity is 2I₀.

Ordinary Bulbs Do Not Show Clear Interference:

Independent light bulbs do not maintain a constant phase difference, so they behave as incoherent sources. Their intensities only add up on average, and no stable visible fringes are formed.