Interference of Waves

Interference of waves is the phenomenon when two waves meet or superimpose each while traveling along the same medium. This resultant interference of two or more waves causes the medium to take a new shape that results from the net effect of the two individual waves upon the particles of the medium. Thus we can say that when two waves meet or come together, the result is the sum of the individual waves.

What is Interference?

You may wonder what will happen when two waves travelling in the same medium are meeting each other? Will there be any change in frequency of resultant wave or will there be change in amplitude? Also questions may arise will there be any change in the nature of two waves meeting?

To answer all these questions we need to understand the phenomenon of superposition and interference of two waves.

Interference can be defined as the phenomenon in which two or more waves meet each other and superpose to form a resultant wave than maybe of greater, lower or the same amplitude depending upon the nature of superimposition or alignment of peaks and troughs of the overlapping waves. 

When two or more waves arrive at the same point while travelling through the same medium, they superimpose themselves on one another or more specifically we can say that, the disturbances of the waves superimpose when they come together. Each of these disturbances corresponds to a force, and we know that forces add. Now if the disturbances are along the same line, then the resulting wave is a simple addition of the disturbances of the individual waves similar two how two forces acting in the same direction add up.Thus,in this case the amplitude of two waves also adds up to give resultant amplitude.

The Principle of Linear Superposition

The principle of linear superposition applies to any number of waves states that when two or more waves of the same type are incident on the same point, the resultant amplitude at that point is equal to the vector sum of the amplitudes of the individual waves and to simplify matters just consider what happens when two waves come together.

For example, sound reaching you simultaneously from two different sources, or two pulses traveling towards each other along a string. When these waves come together, the result is superimposed waves and they add together, with the amplitude at any point being the addition of the amplitudes of the individual waves at that point. Although these waves interfere with each other when they meet, they continue traveling as if they had never encountered each other. 

Constructive and Destructive Interference 

If the crest of one of the waves falls on the crest of the other wave resulting in maximum amplitude. This is constructive interference. While if the crest of one wave falls on the trough of another wave, then the amplitude here is minimum. This is destructive interference. 

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Further we will derive conditions for constructive and destructive interference in the below section.

Derivation

Consider a sinusoidal wave travelling along x-axis and given by equation

W1(x,t)=Acos(kx−ωt)

Where A is the amplitude of the wave

k = 2π/λ is the wavenumber 

ω = 2πf is the angular frequency of the wave.

Now consider another wave of the same frequency and amplitude but with a different phase travelling to the right direction.

W2(x,t)=Acos(kx−ωt+ϕ)

Here φ is the initial phase difference between the waves in radians

The two waves superimpose and add; the resultant wave is given by the equation,W1+W2=A[cos(kx−ωt)+cos(kx−ωt+ϕ)] 

Using cosine rule, cosa+cosb=2cos(a−b/2)cos(a+b/2)

Solving equation (1) using the formula obtained

W1+W2=2Acosϕ/2{cos(kx−ωt+ϕ2)}

Comparing it with W1(x,t)=Acos(kx−ωt)  

Amplitude of resultant wave will be 2Acosϕ/2

If the phase difference is an even multiple of π (φ = …..,–4π, –2π, 0, 2π, 4π,……), then cos φ/2 =1, so the sum of the two waves is a wave with twice the amplitude and this situation is called constructive interference.

W1+W2=2Acos(kx−ωt)

While when the phase difference is an odd multiple of π (φ =…..,–3π, –π, 0, π, 3π, 5π,……), then

cos φ/2 = 0, so the sum of the two waves will be zero.

W1+W2=0 and this is the case of destructive interference.

Path difference for Constructive and Destructive Interference

We know that phase difference and path difference of waves are related by ΔX=λ⋅Δϕ/2π

where, ΔX is the path difference and Δϕ is the phase difference.

Thus using the above derivation of constructive and destructive interference we can say that, the constructive path difference will be 0, λ, 2λ……. and the destructive path difference will be λ/2, 3λ/2, 5λ/2……