Interference of waves

Interference of waves

When two waves of same frequency travelling in the same direction in a medium superpose with each other, their resultant intensity is maximum at some points and minimum at some other points. This phenomenon of superposition is called interference.

Let us consider two simple harmonic waves of same frequency travelling in the same direction. If a₁ and a₂ are the amplitudes of the waves and φ is the phase difference between them, then their instantaneous displacements are

y₁ = a₁ sin ωt    ?.(1)

y₂ = a₂ sin (ωt + φ)   ?.(2)

According to the principle of superposition, the resultant displacement is represented by

y = y₁ + y₂

 = a₁ sin ωt + a₂ sin (ωt + φ)

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

= (a₁ + a₂ cos φ) sin ωt + a₂ sin φ cos ωt ...(3)

Put a₁ + a₂ cos φ = A cos θ ...(4)

a₂ sin φ = A sin θ ...(5)

where A and θ are constants, then

y = A sin ωt. cos θ + A cos ωt. sin θ

or

y = A sin (ωt + θ) ...(6)

This equation gives the resultant displacement with amplitude A. From eqn. (4) and (5)

A²cos ² θ + A² sin ² θ

= (a₁ +a₂ cos φ) ² + (a² sin φ) ²

 ∴A² = a₁ ² + a₂ ² + 2a₁a₂ cos φ

∴ A = root[  a₁² + a₂² + 2a₁ a₂ cosφ ] ... (7)

Also tan θ = ( a₂ sin φ )   + ( a₁ +a₂ cos φ ) ...(8)

We know that intensity is directly proportional to the square of the amplitude

(i.e) I α A²

I α (a₁ ² + a₂ ² + 2a₁a₂ cos φ) ... (9)

Special cases

The resultant amplitude A is maximum, when cos φ = 1 or φ = 2mπ where m is an integer (i.e) I_max α (a1+ a2) ²

The resultant amplitude A is minimum when

cos φ = ?1 or φ = (2m + 1)π

I _min α (a1 ? a2) ²

 The points at which interfering waves meet in the same phase φ = 2mπ i.e 0, 2π, 4π, ... are points of maximum intensity, where constructive interference takes place. The points at which two interfering waves meet out of phase φ = (2m + 1)π i.e π, 3π, ... are called points of minimum intensity, where destructive interference takes place.