Resultant Intensity in Interference of two Waves

The superposition principle of the waves states that the resulting displacement of several waves within a medium at a given point can be regarded as the vector sum of the displacement of each wave produced by each particular wave at that point.  Let two waves be considered to be traveling simultaneously concerning the same string. Then the displacement of the elements within the two waves can be represented as y2 (x, t) and y1 (x, t). On overlapping of these two waves, the resultant displacement is obtained. These can be denoted by y(x, t). 

Interference of two waves

The interference of two waves is said to be the phenomenon in which two waves overlap to form a resultant wave. The amplitude and intensity of these resultant waves can be the same, lesser, or greater than the original interfering waves. In this context, the aspect of wave disturbance is important which is defined as a specific condition in which two specific waves hit or meet each other while moving in the same direction. The effect of disruption of the two waves will lead to the medium taking a new shape that will ultimately result in the combined effect of the two waves. 

Resultant intensity in interference of two waves

Suppose two waves having a vertical displacement y2 and y1 superimpose at a particular point of p. Then the displacement of the resultant wave is given as y = y2 + y1. 

S1 P

S2

The two waves are at the specific point P at the given time. The difference only occurs in the phases. Displacement of each separate wave is given by y2 = b sin (θ + ωt ) and y1 = a sin ωt. Here b and a are the amplitude of the waves and θ  is the difference in phase between the two waves which is constant. Applying the principle of superposition stated earlier we get y = b sin (θ + ωt) + a sin ωt. 

If the resultant amplitude is considered as A then y = A sin (θ + ωt) = b sin (θ + ωt) + a sin ωt.

Or, A [ Sin ωt cos θ + Cos ωt Sin θ) =  b[ Cos ωt Sin θ + Sin ωt Cos θ] + a Sin ωt. 

Therefore comparing the coefficients of Sin ωt and Cos ωt on both sides 

b Sin θ = A Sin θ ————-ii)

b Sin θ + a = A Cos θ ———-iii)

Squaring and adding the above two equations we get 

A2 = (b Sin θ + a)2 + (b Sin θ)2

A2 = b2 + 2ab Cos θ + a2 

A = √( b2 + 2ab Cos θ + a2) ——————–iv)

Now dividing equation ii) by iii) we get 

Tan θ = bsin θ/ (bcos θ + a) ——————v)

Now we know that intensity varies directly with the square of the amplitude of the waves 

Let us suppose I1 =  ka2 , I = kA2 and I2= kb2 

Therefore, I = kA2

I = k (b2 + 2ab Cos θ + a2)

I = kb2 + 2kab Cos θ + ka2 

I = I2 + 2√ka√kbCos θ + I1 

I = I2 + I1 + 2√I1I2 Cos θ, which is the resultant intensity when two waves of intensity I1 and I2 interfere. 

Conclusion 

The overall article has been written on the core topic of key notes on resultant intensity in interference of two waves. We are often required to find the intensity of the resultant wave when the intensity of two interfering waves is given. This topic has high importance in the subject of physics and cannot be ignored in the UPSC exam. The core topic has been further discussed through proper discussion on the interference of two waves, the resultant intensity of two interfering waves, the intensity of two waves, and the incoherent addition of waves.