According to this principle the resultant displacement of a particle of medium acted upon by two or more waves simultaneously is equal to the algebraic sum of the displacement of the particle due to individual waves.

Thus if two waves are

y₁ = a₁ sin wt and y₂ = a₂ sin (wt+ δ)

which have same frequency but differing in phase and amplitude, then from the principle of superposition the resultant displacement y will be

Here we assume

and

Y = y₁ + y₂ = a₁ sin wt+ a₂ sin(rot+ δ)

= (a₁ + a₂ cos δ) sin wt + a₂ cos wt sin δ

θa₁ + a₂ cos δ = A cos θ

a2 sin 8 = A sin θ,

y = A cos θ sin wt + A sin θ cos wt

A² =a₁² +a₂² + 2a₁a₂cosδ

Tan θ = a₂sinδ/a₁+a₂cosδ

which gives the amplitude and phase of resultant wave.

The intensity at any point is given by

I= A₂ = a_l +a₁²+ 2a₁a₂cosδ

Condition for maximum and minimum intensities

Intensity will be maximum of points where

cos δ = + 1 or δ = 2nπ

i.e., intensity will be maximum when phase difference is even multiple of nor path difference (∆) is even multiple of λ/2.

δ= 2nπ, n = 0, 1, 2, ….. .

∆ = 2nπ/2

I_max = (al + a₂)²

The intensity will be minimum, when

δ = (2n + 1) π    n = 0, 1, 2, ….. .

∆  = (2n + 1) A/2

I_min = (a_l – a₂)²