Mathematical Analysis Of Superposition

Description:

Two waves with a phase difference of Φ can be mathematically represented as :

Y₁ = a Cos 𝑤t

Y₂ = a Cos (𝑤t + Φ)

Where,

• Y₁ = Equation of 1^st wave.

• Y₂ = Equation of 2^nd wave, which is Φ phase ahead of 1^st wave.

• a = Amplitude of wave.

Resultant displacement:

Y = Y₁ + Y₂

Y = a Cos 𝑤t + a Cos (𝑤t + Φ)

Y = 2a Cos (Φ/2)  Cos (𝑤t + Φ/2)

Constructive Superposition

For constructive superposition, intensity and amplitude should be maximum. Hence, 2a Cos (Φ/2) should be maximum, i.e.,

Cos (Φ/2) = 1

or Φ/2 = 0, Π, 2Π, 3Π, ……

∴ Φ = 0, 2Π, 4Π, . . . . . . ., n2Π {n = 0, 1, 2, ……}

Path difference = 0, λ, 2λ,………, nλ {n = 0, 1, 2, ……}

Destructive Position

For destructive superposition, intensity and amplitude should be zero. Hence, 2a Cos (Φ/2) should be zero, i.e.,

Cos (Φ/2) = 0

or  Φ/2 = Π/2, 3Π/2, 5Π/2, ……

Φ = Π, 3Π, 5Π, . . . . . . ., (2n+1)Π  {n = 0, 1, 2, ……}

Path difference = λ/2, 3λ/2, 5λ/2………, (2n+1)λ/2  {n = 0, 1, 2, ……}