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#### CHAPTER 10 WAVE OPTICS

Interference

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 $\left(2\mathrm{n}+1\right)$π or path difference is $\left(\frac{2\mathrm{n}+1}{2}\right)$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 $\frac{\phi }{2}$.

Or the intensity I = 4Io cos2 $\frac{\phi }{2}$

If the two sources are not coherent the average intensity will be given by

<I> = 4Io <cos2 $\frac{\phi }{2}$>

where angular brackets represent time averaging.

The function cos2 $\frac{\phi }{2}$ will randomly vary between 0 and 1 and the average value will be $\frac{1}{2}$.

The resultant intensity will be given by,

I = 2Io