This document discusses coherent and incoherent waves, and the addition of waves through interference. It provides the following key points: - Coherent waves have the same frequency and constant phase difference, while incoherent waves have random frequencies and phase differences. - When two coherent waves travel through the same medium, their displacements add through superposition to form a resultant wave with displacement equal to the sum of the individual waves. - Constructive interference occurs when two waves are coherent and in phase, doubling the amplitude and quadrupling the intensity compared to the individual waves. Destructive interference can also occur when waves are out of phase.

1. wave optics _
COHERENT & INCOHERENT
ADDITION OF WAVES .
~ Humera Fatima
II A -57

2. o COHERENT & INCOHERENT
WAVES
▹ Coherent waves
have the same
frequency and
constant phase
difference .
▹ Ex: sound waves ,
light waves ,
electromagnetic
waves etc.
▹ These waves have
random frequencies
and phase
differences .
▹ Ex: Tungsten
filament ,
fluorescent bulbs etc
.

3. _ The addition of wave interference is
based on the superposition principle
according to which ‘two waves
travelling in same medium overlap
each other , the displacement of the
resultant wave is the algebraic sum of
the displacement of each wave .’
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4.  Consider two needles S1 and S2
moving periodically up and down in
an identical fashion 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 , when this happens the two
sources are said to be coherent . Shows
the position of crests & troughs at a
given instant of time .

5.  Consider a point P for which S1P = S2P . Since the distances S1P &
S2P are equal , waves from S1 & S2 will take same time to travel to the point
P & waves that emanate from S1 & S2 in the 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 = 2a
cos wt .
Since 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 of the individual sources , Io is proportional to a2.
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6.  The two sources are said to be
interfere constructively and we have
what we refer to as constructive
interference .
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7. “
▹ We next consider a point Q for which S2Q – S1Q =2λ
▹ The waves emanating from S1 will arrive exactly two
cycles earlier than the waves from S2 & will again be in
phase. Thus if the displacement produced by S1 is given
by y1= a cost wt then the displacement produced by
S2 will be given by y2= a cos wt-4π = a cos wt .
Where we have used the fact that a path difference of
2lambda will corresponds to a phase difference of 4π .
▹ The two displacements are again in phase and the
intensity will be 4Io giving rise to constructive
interference.
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8. .
In the above analysis we have assumed that the distances S1Q &
S2Q are much greater than D ( which represents the distance
between S1 & S2) .
So that although S1Q & S2Q are not equal , the amplitudes of the
displacement produced by each wave are very nearly the same .
.
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