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Principle of Superposition

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 = al +a12+ 2a1a2cosδ

 

Condition for maximum and minimum intensities

Intensity will be maximum of points where

 

cos δ = + 1 or δ = 2nπ

[n = 0, 1, 2, 3, ... ... ]

 

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

Imax = (al + a2)2

 

The intensity will be minimum, when

 

δ = (2n + 1) π    n = 0, 1, 2, ….. .

 

∆  = (2n + 1) A/2

Imin = (al – a2)2