CBSE NOTES CLASS 12 PHYSICS
CHAPTER 10 WAVE OPTICS
Superposition of two waves, travelling through the same medium, is called interference.
Constructive interference occurs when the two interfering waves have a displacement in the same direction and the amplitude of the resulting wave is the addition of amplitudes of interfering waves.
Destructive interference occurs when the two interfering waves have a displacement in different direction (out of phase) and the amplitude of the resulting wave is the difference of amplitudes of interfering waves.
Consider two coherent sources S1 and S2 a point P for which S1P = S2P. Since the distances S1P and S2P are equal, waves from S1 and S2 will take the same time to travel to the point P and waves that originate from S1 and S2 in phase will also arrive, at the point P, in phase.
If the displacement produced by the source S1 at the point P is given by
y1 = a cos ωt
then, the displacement produced by the source S2 (at the point P) will also be given by
y2 = a cos ωt
Thus, the resultant of displacement at P would be given by
y = y1 + y2 = 2 a cos ωt
Since the intensity is the proportional to the square of the amplitude, the resultant intensity will be given by,
I = 4Io
where Io represents the intensity produced by each one of the individual sources; Io is proportional to a2.
In fact for all points where the phase difference is 2nπ or path difference is nλ, the interference will be constructive.
If the phase difference is π or path difference is λ, the interference will be destructive.
For any other arbitrary point let the phase difference between the two displacements be φ.
y1 = a cos ωt ⇒ y2 = a cos (ωt + φ)
The resultant displacement will be given by
y = y1 + y2 = a [cos ωt + cos (ωt + φ]
Since φ is constant, the amplitude of the resultant displacement is 2a cos .
Or the intensity I = 4Io cos2
If the two sources are not coherent the average intensity will be given by
<I> = 4Io <cos2 >
where angular brackets represent time averaging.
The function cos2 will randomly vary between 0 and 1 and the average value will be .
The resultant intensity will be given by,
I = 2Io