Consider two simple harmonic waves of amplitudes a1 and a2 each of angular frequency ω, both propagating along x-axis, with the same velocity v = ω/k but differing in phase by a constant phase angle φ.
These waves are represented by the equations:
y1 = a1 sin (ωt – kx)
y2 = a2 sin [(ωt – kx) + φ]
where ω= 2π/T is angular frequency and k = 2π/λ is wave number.
Since, the two waves are travelling in the same direction with the same velocity along the same line, they overlap. According to the principle of superposition, the resultant displacement at the given location at the given time is
y = y1 + y2
y = a1 sin (ωt – kx) + a2 sin [(ωt – kx) + φ]
Put (ωt – kx) = θ, then
y = a1 sin θ + a2 sin (θ + φ)
y = a1 sinθ + a2 sinθ cosφ + a2 sinφ cosθ
a2 sinφ = A sinα
a1 + a2 cosφ = A cos α
y = A cosα sinθ + A sinα cosθ
y = A sin (θ + α)
Substituting for θ
y = A sin [(ωt – kx) + α]
Thus, the resultant wave is of angular frequency ω and has an amplitude A given by
A2 = (a1 + a2 cosφ)2 + (a2 sinφ)2
A2 = a12 + a22 + 2a1a2 cosφ
φ is the phase difference between the two superposed waves. If path difference, between the two waves corresponds to phase difference φ , then
φ = 2πp/λ, where 2π/λ is the phase change per unit distance.
When the path difference is an even multiple of λ/2,
p = 2mλ/2
Then phase difference is given by
φ = (2π/λ) × (2m λ/2) = 2mπ
Since cos 2π = +1,
A2 = (a1 + a2)2
When the collinear waves travelling in the same directions are in phase, the amplitude of the resultant wave on superposition is equal to sum of individual amplitudes.
As intensity of wave at a given position is directly proportional to the square of its amplitude,
Imax ∝ (a1 + a2)2
When p = (2m + 1)λ/2, then
φ = (2m + 1)π and cosφ = –1.
A2 = (a1 – a2)2
When phases of two collinear waves travelling in the same direction differ by an odd integral multiple of π, the amplitude of resultant wave generated by their superposition is equal to the difference of their individual amplitudes.
Imin ∝ (a1 – a2)2
If a1 = a2, the intensity of resultant wave is zero. These results show that interference is essentially redistribution of energy in space due to superposition of waves.