Types of superposition of waves

When one wave meets another wave, it goes through the phenomenon of interference. The phenomenon of interference of waves is described as when two or more two waves are superposed to each other for forming a resultant wave of similar, higher, or lower amplitude. The principle of superposition reveals the total number of waves moving in the same medium when they overlap or intersect with each other, that point is the summation of the vector of each displacement which is formed by each wave at the point. 

The superposition of the interface of waves is divided into two different categories, Constructive interference, and Destructive interference –

Constructive interference

constructive interference is defined as when two or more two waves meet with one another, which have their displacements in a similar direction. The two wave’s amplitude is summed up and as a result, the resultant wave comes as the consequence that has more amplitude than other waves which are required for producing the resultant wave. When the similar frequencies of one wave’s crest meet or intersect the other wave’s crest, then the summation of each wave’s amplitudes together forms the resultant amplitude. 

Destructive interference

Destructive interference of waves is defined as, the waves which combine having displacements of each in opposite direction. Suppose two waves are combined to form the resultant wave that has displacements that are opposite to each other. The resultant wave that is formed as a result of a combination of two waves has an amplitude lower than the amplitude of the other two waves that are combined. If the two waves that are combined have similar amplitude, then the resultant wave becomes zero amplitude, i.e. the two waves get completely canceled when they are having same amplitude. 

Conditions required for constructive and destructive interference

It can be considered the light wave from the two sources as 

y1 = a Sin ωt

y2 = b Sin (ωt + φ)

Where, 

a and b are the amplitude of the waves 

φ = constant phase difference

When these two waves superpose on each other the resultant wave that is formed can be written as 

y = y1 + y2

By solving,

• y = a Sin ωt + b Sin (ωt + φ)

• y = a Sin ωt + b [Sin ωt Cos φ + Cos ωt Sin φ]

• y = a Sin ωt + b Sin ωt Cos φ + b Cos ωt Sin φ

• y = Sin ωt [a + b Cos φ] + b Cos ωt Sin φ

From here, it can be considered that 

A + b Cos φ = R Cos ϴ …….. Equation (i)

B Sin φ = R Sin ϴ ……… Equation (ii)

Now, by putting equation (i) and equation (ii)

y = Sin ωt (R Cos ϴ) + Cos ωt (R Sin ϴ)

• y = R [Sin ωt Cos ϴ + Cos ωt Sin ϴ]

• y = R [Sin (ωt + ϴ)]

• y = R Sin (ωt + ϴ) [here, R is the resultant amplitude]

Calculation of resultant wave

Squaring and adding equations (i) and (ii)

(a + b Cos φ)2 + (b Sin φ)2  = (R Cos ϴ)2 + (R Sin ϴ)2

•  a2 + b2 Cos2 φ + 2ab Cos φ + b2 Sin2 φ = R2 Cos2 ϴ + R2 Sin2 ϴ

• a2 + b2 (Cos2 φ + Sin2 φ) + 2ab Cos φ = R2 (Cos2ϴ + Sin2 ϴ)

• a2 + b2 + 2ab Cos φ = R2[Cos2 ϴ + Sin2 ϴ = 1]

Therefore, 

R2 = a2 + b2 + 2ab Cos φ

• R = a2 + b2 + 2ab Cos φ

For R to be maximum, cos φ should be maximum

Therefore, Cos φ = 1

Φ = 0, 2, 4……..

Φ = 2n [Where, n = 0, 1, 2, 3…..]

So, Rmax = a2 + b2 + 2ab 

           = (a+b)2

• R = a +b

For, R to be minimum, cos φ should be minimum

Therefore, Cos φ = -1

Φ = , 3, 5………

Φ = (2n – 1)[Where, n = 1, 2, 3……]

Rmin = a2 + b2 + 2ab (-1) 

        = a2 + b2-2ab 

Rmin = (a-b)2

• Rmin = a – b

Calculation of path difference

Path difference (x) = 2 φ

For maximum, φ = 2n

x = 2 X 2n

x = nλ    

[Where, n = 0, 1, 2, 3……]

For minimum, φ = (2n – 1)

x = 2 X (2n – 1)

x = 2 X (2n – 1) [Where, n = 1, 2, 3……]

Conclusion

It is to be concluded that there are two types of superposition of inference of waves, these are constructive interference and destructive interference. In constructive interference, the resultant amplitude is higher than the other wave’s amplitude but in the case of destructive interference, the resultant interference is lower than that of other wave’s amplitude that are overlapping.