You must have seen light coming out from the laser. Let us carry out a small activity. Take two needles and touch the needles on the surface of the water. Here if both the needles move with the same speed then they are said to be coherent. Let us learn more about coherent waves.

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## Coherent and Incoherent Addition of waves

Suppose there is a surface of the water and you take a needle and touch the surface of the water. What will happen? Yes, ripples are formed. Now if you take two needles and you touch the surface of the water with the needles. What do you think will happen?

You will seeÂ a pattern. That pattern is the interference pattern. When you touch both the needles at the surface of the water at the same time, both the needles are in the same phase. Needle 1 will produce a wave. Also, needle 2 will produce its own ripples and they will intersect with waves of the first needle.

Now, if both the needles are moving with the same velocity, the wave formed here are coherent. If the velocity of a 1st needle and 2nd needle are not steady they won’t intersect. This is because one is at a steady speed and other is at variable speed.

**Browse more Topics under Wave Optics**

- Diffraction
- Huygens Principle
- Interference of Light Waves and Youngâ€™sÂ ExperimentÂ
- Polarisation
- Refraction and Reflection of Plane Waves using Huygens Principle

### Coherent Waves

If the potential difference between two waves is zero or is constant w.r.t time, then the two ways are said to be coherent.

### Non-coherent Waves

The waves are non-coherent if the potential difference betweenÂ the two ways keeps on changing. Lightbulb, study lamp are the examples of the coherent waves. They emit waves at random potential difference.

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## Explanation

Now let us consider there are two needles say S_{1Â }and_{Â }S_{2Â }moving up and down on the surface of the water and are pointing at point P. So the path difference here is given asÂ S_{1}P_{Â }–Â S_{2}P. Now the displacement by two needles and S_{1Â }S_{2Â }are:

y_{1Â }= A cos wt ……………… (1)

y_{2}Â = A cos wt …………….. (2)

So the resultant displacement at point P is, y =Â Â y_{1Â +Â }y_{2}. When we substitute the value ofÂ y_{1Â }and_{Â }y_{2}Â we write,

y = A cos wt + A cos wt

y = 2A cos wt……………….. (3)

#### Now, we know the intensity is proportionalÂ to the square of the amplitude waves.

I_{0Â Â }\( \propto\) AÂ²

WhereÂ I_{0Â }is the initial intensity andÂ AÂ² is the amplitude of the wave. From equationÂ 3, we say that A = 2A. So,

I_{0Â Â }\( \propto\) (2A)Â² or I_{0Â }_{Â }\( \propto\) 4 AÂ²

I = 4Â I_{0Â Â Â }

Now, if two needles that areÂ S1Â are S_{2}Â are in the same phase, the potential difference is,

S_{1}PÂ _{Â }– S_{2}P = nÎ»

Where n = 0, 1, 2,3 ……… and Î» = the wavelength of the wave. If the two needlesÂ S1 and S2 are vibrating atÂ its destructive interference then, the potential difference is

S_{1}P_{Â }–Â S_{2}P = (n + 1/2) Î»

Now if the potential difference of the waves isÂ Î¦ then,

y_{1Â } =Â Î± cos wt

_{Â }y_{2 }=Â Î± cos wt

The individual intensity of each wave isÂ I_{0Â }, we get,

y =Â Â y_{1Â +Â }y_{2}

=Â Î± cos wt +Â Î± cos (wt +Î¦)

y = 2Â Î± cos(Î¦/2) cos (wt +Â Î¦/2)

Since, the intensity isÂ I_{0Â }\( \propto\) AÂ²

I_{0Â }\( \propto\) 4Î±Â² cosÂ²Â (Î¦/2)

I = 4Â I_{0Â }Â cosÂ²Â (Î¦/2)

Well, the time-averaged value ofÂ cosÂ²(Î¦_{t}/2) is 1/2. So, the resultant intensity will beÂ I = 2 I_{0} at all the points.

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## Solved Questions For You

Q. Two coherent sources of light can be obtained by

- Two different lamps
- Different lamps having the same power
- Two different lamps of the same power and having the same color
- None of the above

Answer: D. The coherent source cannot be obtained from two different light sources.

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