Wave Optics - Interference

You might have heard or used the word "interfere" or "interference" in your day to day conversation. In physics, the word interference has a specific meaning.Interference is the interaction or the addition of two waves which gives rise to a new wave pattern.When we talk about interference we generally mean interference of coherent waves.Coherent waves are the waves which come either from the same source or from sources having very close frequencies.

If two sources emit waves of more than one wavelength or frequencies then they are said to be coherent if the they have the same range of wavelengths and the phase difference between two constituent waves of same wavelength is same.That is, the phase difference between the two sources of same wavelength is same for any pair of waves.

If a wave is described as:


Here A is the amplitude (maximum displacement of the wave from the mean position). w is the angular frequency of the wave and  is the phase of the wave and x is the displacement of the wave at time t.

Two coherent waves can interfere in multiple ways depending upon the phase difference between them.The two ways worth noting are:

• Constructive interference: Two waves are said to interfere constructively if the phase difference between them is an integral multiple of  . In other words we can say that the two waves constructively if the crests of one wave coincide (hence troughs too)  with the crest (and troughs)  of the other.

          Mathematically,
           

                                                     
                                                     Constructive Interference.Source:http://theory.uwinnipeg.ca

• Destructive interference:Two waves are said to interfere destructively if the phase difference between them is an odd multiple of .In other words we can say that the two waves interfere destructively if the crests of first wave coincide with the troughs of the other.

          Mathematically,
         
                                                    
                                                   
                                                    Destructive Interference.Source:http://theory.uwinnipeg.ca

The idea involved in the expressions is that if say two waves each of same wavelength start out in phase travel at the same speed through different paths and the distance travelled by them is y₁ and y₂ then, the path difference between the two waves will be y1 - y2.That is, the two crests between which there was no separation initially (as per the convention assumed by us) are now separated by y₁ - y₂.
And for constructive interference we must have,
          . Where, n = Integer, then another crest will coincide with the crest of the first wave.
On similar logic for destructive interference we have,