Question

Derive an expression for conditions of constructive interference and destructive interference

Answer 1

Conditions of constructive interference and destructive interference.

consider two coherent waves travelling in the same direction along a straight line.

Frequency of each wave is given by \(\frac{\omega}{2\pi},\)

Amplitude of electric field vectors are a₁ and a₂ respectively.

Wave equation is represented by,

y₁ = a₁ sin ωt ....(i)

y₂ = a₂ sin (ωt + ϕ) ....(ii)

Using the theory of superposition,

y = y₁ + y₂ .....(iii)

Here, y₁ and y₂ are the points of electric field.

Putting values from (ii) and (iii) in (i), we have

y = a₁ sin ωt + a₂ sin (ωt + ϕ)

Now using trignometric identities, we have

sin (ωt + ϕ) = sin ωt cos ϕ + cos ωt sin ϕ

we get,

y = a₁ sin ωt + a₂ (sin ωt cos ϕ + cos ωt sin ϕ)

= (a₁ + a₂ cos ϕ) sin ωt + (a₂ sin ϕ) cos ωt ....(iv)

Assume,

a₁ + a₂ cos ϕ = A cos θ

and

a₂ sin ϕ = A sin θ

so, eqn. (iv) gives,

y = A cos θ sin ωt + A sin θ cos ωt

= A sin (ωt + ϕ)

Amplitude of the resultant wave is given by,

Amplitude, \(A = \sqrt{a_1{^2} + a_2{^2} + 2a_1a_2 \cos \phi}\)

Intensity of the wave is proportional to the amplitude of the wave.

Thus, intensity of the resultant wave is given by,

I = A² = a₁² + a₂² + 2a₁a₂ cos ϕ

Constructive interference : For maximum intensity at any point, cos = +1

phase difference, ϕ = 0, 2π, 4π, 6π, .......

= 2nπ(n = 0, 1, 2, ....)

So, maximum intensity is,

I_max = a₁² + a₂² + 2a₁a₂ = (a₁ + a₂)² 

Path difference is,

Δ \(= \frac{\lambda}{2\pi}\) × Phase difference \(= \frac{\lambda}{2\pi}\) × 2nπ = nλ

Constructive interference is obtained when the path difference between the waves is an integral multiple of λ

Destructive interference : For minimum intensity at any point, cos = -1

phase difference is given by,

ϕ = π, 3π, 5π, 7π, .......

= (2n -1) π, n = 1, 2, 3, .....

Minimum intensity is,

I_min = a₁² + a₂² - 2a₁a₂ = (a₁ - a₂)² 

Path difference is,

Δ \(= \frac{\lambda}{2\pi}\) × Phase difference

\(= \frac{\lambda}{2\pi}\) × (2n - 1) π = (2n - 1)\(\frac{\lambda}{2}\)

In destructive interference, path difference is odd multiple of \(\frac{\lambda}{2}.\)