 # Interference of Waves

Interference of waves refers to the phenomenon whose occurrence takes place when two waves meet while travelling along the same medium. Furthermore, the interference of waves makes the medium take on a particular shape. Moreover, the reason for this shape is due to the net effect of the two individual waves upon the medium particles.

## Introduction to Interference of Waves

Interference takes place when two or more waves meet each other. Depending on the alignment of peaks and troughs of the overlapping waves, the waves can cancel each other either partially or entirely. Furthermore, the occurrence of constructive interference and destructive interference takes place from the interaction of waves that have a correlation with each other, either because they have the same or nearly the same frequency or they come from the same source.

One can observe the interference effects with all types of waves, for example, gravity waves, radio, light, surface water waves, or matter waves. Furthermore, interferograms refer to the resulting images or graphs. The principle of superposition of waves explains that when two or more similar type propagating waves are incident on the same point, the resultant amplitude at that point happens to be equal to the vector sum of the amplitudes of the individual waves. Interference Of Waves

### Derivation of Interference of Waves

In accordance with the principle of superposition of waves, if the meeting of a crest of a wave takes place with the crest of another wave of the same frequency at the same point, then the resultant amplitude turns out to be the sum of individual amplitudes. Furthermore, this concept is called constructive interference.

Similarly, if the meeting of a crest of a wave takes place with the trough of another wave, then the resultant amplitude turns out to be equal to the difference in the individual amplitudes. Furthermore, this concept is called destructive interference. Moreover, the derivation of the formula for the sum of two waves is as follows:

The expression of the amplitude of a sinusoidal wave travelling to the right along the x-axis is as,

W1(x,t) = Acos(kx − ωt)

Where A represents the peak amplitude. Furthermore, k = 2π/λ is the wavenumber and ω = 2πf refers to the angular frequency of the wave.

Take the case of another wave with a different phase travelling to the right but with the same frequency and amplitude.

W2(x,t) = Acos(kx – ωt + ϕ)

where φ shows the phase difference that is present between the waves in radians

The superimposition of the two waves takes place and they add; the expression of the resultant wave is shown by the equation,

W1 + W2 = A[cos(kx − ωt) + cos(kx – ωt + ϕ)] (1)

The expression of the sum of two cosines is by the equation,

Cosa + cosb = 2cos(a – b/2)cos(a + b/2)

Solving equation (1) using the formula, one would get

W1 + W2 = 2Acosϕ/2cos(kx – ωt + ϕ/2)

Constructive Interference: When the phase difference turns out to be 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 that happens to be twice the amplitude.

W1 + W2 = 2Acos(kx − ωt)

Destructive Interference: When the phase difference turns out to be an odd multiple of π (φ =….., –3π, –π, 0, π, 3π, 5π,……), then cos φ/2 = 0. So, zero will be the sum of the two waves.

W1 + W2 = 0

## FAQs For Interference of Waves

Question 1: What are the two types of interference?

Answer 1: The two types of interference are constructive interference and destructive interference. Furthermore, constructive interference takes place when the amplitude of the waves increases due to the wave amplitudes reinforcing each other. Furthermore, destructive interference takes place when the amplitude of the waves reduces due to the wave amplitudes opposing each other.

Question 2: What is meant by the interference of light?

Answer 2: Interference of light is a special phenomenon in which there is a reflection of the wave nature of light. Due to the interference of light, there is the interference of two waves with each other. Furthermore, constructive interference happens when the meeting of crest and crest or trough and trough takes place with the amplitude of light and it becomes very large.

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