Superposition of wave

Superposition of wave

Introduction : We are well aware of the parameter of a wave such as amplitude, frequency, wavelength, etc. Superposition of the wave is well understood by amplitude of a wave. Interesting phenomenon of superposition of a wave is due to two or more wave reaching simultaneously at a point. When sound wave travel from one medium to another their wave length as well as velocity changes, but frequency remains constant. 

Superposition of wave : when two or more waves, travelling through a medium, pass through a common point, independent of the presence of other wave. The resultant displacement at the point is equal to the vector sum of the displacements due to the individual wave at that point.

Superposition of wave

Analytical treatment of resultant of two wave.

From diagram above, consider two wave with different amplitude and same frequency  having phase difference of ф at x =0.

Y₁ = A₁ sinωt 

Y₂ = A₂ sin (ωt + ф)

 The resultant displacement due to this two waves is given by,

Y = Y₁ + Y₂ 

Y = A₁ sinωt + A₂ sin (ωt + ф)

Y = A₁ sinωt + A₂ sinωt cosф + A₂ sinф cosωt 

Y = (A₁ + A₂ cosф)sinωt  + A₂ sinф cosωt ----(1)

Let,

 (A₁ + A₂ cosф) = A cosθ ----(2)

and

A₂ sinф = Asinθ----(3)

from (1) , (2) , (3)

Y =  A cosθ sinωt  + Asinθ cosωt

Y =  Asin (ωt + θ)

Adding and squaring  (2) and (3)

A² cos²θ + A² sin²θ = (A₁ + A₂ cosф)² + (A₂² sin²ф )

A²  = A₁ ² + A₂²  cos² ф+ 2A₁ A₂+ A₂² sin²ф 

_{│<<<Expression of Progressive wave│   │velocity of  wave>>>│} 

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