## Coherent and Incoherent Addition of Waves

### Table of Contents

## Coherent Source of Waves

The waves are said to be coherent if they have same wavelength, frequency and constant phase difference. **For Example**, sound waves from two loud speakers by same audio oscillator produces coherent waves. Light produced by laser is another example. There are two ways to produce coherent sources. The first method is by the division of wavefront. In this method, the wavefront is divided into two parts or more parts by the usage of prisms, lenses and mirrors. **Wavefront **is a surface (which is either real or imaginary) for which an optical wave comprises a constant phase. It can be the crest or the trough of the same wave. The second method is by dividing the amplitude of the wave by two parts or more parts through partial reflection or refraction. These divided parts may travel through different paths and finally joins together to produce interference.

### Interference of light waves

For the interference phenomenon, the sources should be coherent. **Interference** of light waves is defined as the modification in the distribution of light energy when two or more waves superimpose each other. There are two types of Interference. They are **constructive interference** and **destructive interference**. In constructive interference, the amplitude of the waves adds together when the light waves are at same place and time. In this case, the waves are in phase. In destructive interference the waves are out of phase. So when the waves add together the resultant wave becomes zero.

### Intensity of light at the point of interference

We know that intensity of any light wave is directly proportional to the square of its amplitude. In case of coherent sources, let’s consider the factor of frequency to be a constant. Thus the intensity for the wave is considered as KA^{2}.

I = KA^{2}, where K is the constant which depends on what medium the wave is in.

Consider the resultant amplitude as ‘R’ at the point of interference. Now the resultant intensity at this point can be written as

I_{R }= KR^{2}

= K [A_{1}^{2 }+ A_{2}^{2 }+ 2A_{1 }A_{2} Cos ϕ]

= KA_{1}^{2} + KA_{2}^{2}+ 2KA_{1 }A_{2 }Cos ϕ

Substitute KA_{1}^{2} = I_{1 }and KA_{2}^{2 }= I_{2}

Thus I_{R }= I_{1 }+ I_{2 }+ 2 √(I_{1 }I_{2 }Cos ϕ)

So the resultant intensity of two coherent source of waves after the interference phenomenon is

**I _{R }= I_{1 }+ I_{2 }+ 2 **√

**(I**

_{1 }I_{2 }Cos ϕ)Here ϕ is the phase difference between two waves.

In case of constructive interference, the value of ϕ = 0 and so Cos ϕ =1.

Then I_{R }= I_{1 }+ I_{2 }+ 2 (_{√}I_{1 }I_{2 }= (√ I_{1 }+ _{√}I_{2})^{2} where the waves are superposed in same phase. Here the resultant intensity is maximum. For destructive interference, the waves superpose in opposite direction.

Then Cos ϕ = -1 Thus the resultant intensity is minimum. So, I_{R }= I_{1 }+ I_{2 }- 2 √( I_{1 }I_{2 }= (√I_{1 }- √I_{2})^{2}.

### Interference of equal Intensity waves

As mentioned earlier, the resultant intensity of waves at the point of interference I_{R }= (I_{1 }+ I_{2 }+ 2 √(I_{1 }I_{2 }Cos ϕ). Let us consider that the two waves have same and equal intensity, which means I_{1 }= I_{2}= I_{0}.

Thus I_{R }= 2I_{0 }+ 2I_{0} Cos ϕ

= 2I_{0} (1+ Cos ϕ)

1+ Cos ϕ = 2cos^{2} ϕ/2 (Cos ϕ in half angle form)

Then I_{R }= 4I_{0} cos^{2} ϕ/2.

Hence for constructive interference, intensity will be maximum, I_{R }= ( √I_{1 }+ √I_{2})^{2} = 4 I_{0.}

For destructive interference, the intensity is minimum and I_{R }= (√I_{1 }- √I_{2})^{2} = 0.

### Path difference for constructive and destructive interference

In constructive interference, the phase difference is considered as 2n π, where n being an integer. Now to find the path difference between the waves,

Δ= λ/2π* ϕ

= λ/2π*2nπ

Δ = nλ (n = 0, 1, 2……)

Hence, we can say that two waves interfere constructively, when their path difference Δ = λ, 2λ, 2.... nλ .

Consider two light sources S_{1 }and S_{2} which are coherent. Assume that the light sources are switched on at the same time. Consider a point P where the two light waves emitted from the coherent sources are superposed. Now the point P is located at a distance a_{1} from source S_{1 }and a distance a_{2} from source a_{2}. If the two sources are in same phase, then the path difference, Δ = a_{2}- a_{1.} If the path difference at point P with respect to the two sources is multiple of λ, then the intensity at point p is maximum and will be equal to 4I_{0}, if the two waves are of equal intensity. Otherwise it is shown as (√I_{1 }+ √I_{2})^{2}. In case of destructive interference, we know that Cos ϕ = -1 and it happens when the phase difference, ϕ = (2n+1) π which corresponds to opposite phase. To calculate the path difference between the waves in destructive interference, Δ= λ/2 π * ϕ

= λ/2π*(2n+1) π

= (2n + 1) λ/2

Hence we can say that two waves interfere destructively, when their path difference Δ = λ/2, 3λ/2.....(2n+1) λ/2.

For destructive interference, the waves superpose in opposite direction. Then Cos ϕ = -1 Thus the resultant intensity is minimum. So I_{R }= I_{1 }+ I_{2 }- 2 √( I_{1 }I_{2 }= √( I_{1 }- I_{2})^{2}.

## Incoherent Sources of waves

The waves are said to be incoherent if they do not have constant phase difference. These sources will produce light with random and frequent changes of phase between the photons. For example, ordinary fluorescent tubes and Tungsten filament lamps produces incoherent light waves. All conventional sources will be incoherent sources. We do not get interference pattern with incoherent sources of waves.

Consider two waves which is emitted from two sources of intensities I_{1 }and I_{2. }The intensity of these two waves is

**I = I _{1} + I_{2.} **

Coherent light waves are normally stronger when compared to incoherent source of light waves. Incoherent light waves are usually weak. Coherent light waves are uni-directional. Incoherent light waves are omni-directional.

**Scattering of light waves**

When light wave confronts a matter, it re-emits the light in all direction other than forward direction. This process is called scattering. All molecules of matter scatter light. Scattering can be coherent and incoherent. We consider the phase delays of the waves to mention the scattering. The scattering of wave is found to be coherent and constructive, if the phase delay is the same for all scattered waves. If the phase delay is varying uniformly from 0 to 2 π, then it is said to be coherent and destructive. If it varies in random, then it is incoherent. Coherent scattering occurs in one or a few directions. Coherent destructive scattering occurs in all other direction.

## Summary

- The source which emits light with constant phase difference is called a
**Coherent Source**. For the phenomenon of interference, the source must be a coherent source. - Coherent Source of waves have same wavelength, frequency and constant phase difference. The resultant intensity is
**I = I**_{1}+ I_{2}+ 2 √ (I_{1}I_{2}Cos Θ). - For constructive interference, I
_{R}= √( I_{1 }+ I_{2})^{2}where the waves are superposed in same phase. In this case, the resultant intensity is maximum. Waves that combine together in phase add up each together and gets high intensity. This is Coherent constructive addition. - For destructive interference, the resultant intensity, I
_{R}= (√I_{1 }- √I_{2})^{2}. Here the waves superpose in opposite direction. The light waves that combine out of phase are destructive, added to zero is coherent destructive addition. - In case of interference of waves with equal intensity, intensity will be maximum for constructive interference, I
_{R }= (√I_{1 }+ √I_{2})^{2}= 4 I_{0 }and for destructive interference, the intensity is minimum and - (√I
_{1 }- √I_{2})^{2}= 0. - Two waves interfere constructively, when their path difference Δ = λ, 2λ,...... nλ .
- Two waves interfere destructively, when their path difference Δ = λ/2, 3λ/2...(2n+1)λ/2.
- Incoherent waves do not have constant phase difference. For Incoherent waves the intensity is
**I =**I_{1}+ I_{2.}The waves that add at random phase is incoherent addition of waves.- Scattering of light can be coherent and incoherent. The scattering of wave is coherent and constructive, if the phase delay is the same for all scattered waves. If it varies randomly, it is incoherent.

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