Question

Derive an expression of Coherent sources and Theory of interference, (with equal amplitude) arriving at the conditions for constructive and destructive interference.

Answer 1

Interference is based on the superposition principle according to which at a particular point in the medium, the resultant displacement produced by a number of waves is the vector sum of the displacements produced by each of the waves. 

Two sources producing waves are said to be coherent if at a particular point, the phase difference between the displacements produced by each of the waves does not change with time; 

Consider two needles S₁ and S₂ moving periodically up and down in an identical fashion in a trough of water. They produce two water waves, and at a particular point, the phase difference between the displacements produced by each of the waves does not change with time; hence they are coherent. 

The position of crests (solid circles) and troughs (dashed circles) at a given instant of time. Consider a point P for which S₁ P = S₂ P 

Since the distances S₁ P and S₂ P are equal, waves from S₁ and S₂ will take the same time to travel to the point P and waves that emanate from S₁ and S₂ in phase will also arrive, at the point P, in phase. Thus, if the displacement produced by the source S₁ at the point P is given by 

y₁ = a cos wt 

then, the displacement produced by the source S₂ (at the point P) will also be given by 

y₂ = a cos wt 

Thus, the resultant of displacement at P would be given by 

y = y₁ + y₂ = 2 a cos wt 

Since the intensity is the proportional to the square of the amplitude, the resultant intensity will be given by 

I = 4 I₀ 

where I₀ represents the intensity produced by each one of the individual sources; I₀ is proportional to a₂. In fact at any point on the perpendicular bisector of S₁S₂, the intensity will be 4I₀. The two sources are said to interfere constructively and we have what is referred to as constructive interference. 

We next consider a point Q Fig. for which 

S₂Q –S₁Q = 2 λ

The waves emanating from S₁ will arrive exactly two cycles earlier than the waves from S₂ and will again be in phase. Thus, if the displacement produced by S₁ is given by 

y₁ = a cos wt 

then the displacement produced by S₂ will be given by 

y₂ = a cos (wt – 4p) = a cos wt 

where we have used the fact that a path difference of 2λ corresponds to a phase difference of 4p. The two displacements are once again in phase and the intensity will again be 4I₀ giving rise to constructive interference. 

In the above analysis we have assumed that the distances S₁Q and S₂Q are much greater than d (which represents the distance between S₁ and S₂) so that although S₁Q and S₂Q are not equal, the amplitudes of the displacement produced by each wave are very nearly the same. 

We next consider a point R for which S₂R – S₁R = –2.5 λ 

The waves emanating from S₁ will arrive exactly two and a half cycles later than the waves from S₂. Thus if the displacement produced by S₁ is given by 

y₁ = a cos wt 

then the displacement produced by S₂ will be given by 

y₂ = a cos (wt + 5ϕ) = – a cos wt 

where we have used the fact that a path difference of 2.5 λ corresponds to a phase difference of 5p. The two displacements are now out of phase and the two displacements will cancel out to give zero intensity. This is referred to as destructive interference. 

To summarise: 

(1) If we have two coherent sources S₁ and S₂ vibrating in phase, then for an arbitrary point P whenever the path difference, 

S₁P ~ S₂P = nλ (n = 0, 1, 2, 3,...) -------- (1) 

we will have constructive interference and the resultant intensity will be 4I₀; the sign ~ between S₁P and S₂ P represents the difference between S₁P and S₂P. 

On the other hand, if the point P is such that the path difference, 

S₁P ~ S₂P = (n + ½) λ (n = 0, 1, 2, 3, ...) --------- (2) 

we will have destructive interference and the resultant intensity will be zero. Now, for any other arbitrary point G let the phase difference between the two displacements be ϕ. Thus, if the displacement produced by S₁ is given by 

y₁ = a cos wt 

then, the displacement produced by S₂ would be 

y₂ = a cos (wt + ϕ) 

and the resultant displacement will be given by 

y = y₁ + y₂ 

= a [cos wt + cos (wt + ϕ)] 

= 2 a cos ( ϕ/2) cos (wt + ϕ/2) 

The amplitude of the resultant displacement is 2a cos (ϕ/2) and therefore the intensity at that point will be 

I = 4 I₀ cos² ( ϕ/2) ---------- (4) 

If ϕ = 0, ± 2 π, ± 4 π,… 

which corresponds to the condition given by Eq. (1) we will have constructive interference leading to maximum intensity. On the other hand, if ϕ = ± π, ± 3π, ± 5π … [which corresponds to the condition given by Eq. (2)] we will have destructive interference leading to zero intensity. 

(2) Now if the two sources are coherent (i.e., if the two needles are going up and down regularly) then the phase difference f at any point will not change with time and we will have a stable interference pattern; i.e., the positions of maxima and minima will not change with time. However, if the two needles do not maintain a constant phase difference, then the interference pattern will also change with time and, if the phase difference changes very rapidly with time, the positions of maxima and minima will also vary rapidly with time and we will see a “time averaged” intensity distribution. When this happens, we will observe an average intensity that will be given by 

I = 4 I₀ (cos² ϕ /2) 

(3) When the phase difference between the two vibrating sources changes rapidly with time, we say that the two sources are incoherent and when this happens the intensities just add up. The resultant intensity will be given by 

I = 2 I₀ ----------- (4) 

at all points.