Interference of Waves: Definition, Derivation and Phase Difference

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What is Interference of Waves?

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Interference of waves is considered to be the phenomenon where two or more waves overlap in order to form a resultant wave which can either be of the same, greater or lesser amplitude.

Disruption can be defined as a condition in which two or more waves intersect and counteract the resulting wavelength which may be of a larger, lower or similar size depending on the frequency or alignment of the peaks and ripples of the colliding waves.

Types of Interference of Waves

When two or more waves come at the same time while moving in the same direction, then they lift each other up or we can also say that the disturbance of the waves passes at the junction. 

Now when all the disturbances are in the same line, the resulting wave is a simple addition of individual wave disturbances similar to the combination of two forces acting in the same way. Therefore, in this case, the height of the two waves also adds to the effect.

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Difference Between Constructive Interference and Destructive Interference

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Constructive interference appears when the lines that denote peaks cross over one another. Destructive interference appears when two waves are totally out of phase. The key differences between Constructive Interference and Destructive Interference include:

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Principle of Linear Superposition

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The principle of linear superposition applies to any number of waves and states that when two or more waves of the same type occur at the same time, the magnitude of the effect at that time will be equal to the total vector strength of each wave to make this easier.

Enhanced and Linear Superposition

For example, a sound that reaches you simultaneously from two different sources, or two pulses that travel to each other through a cable. When these waves meet, the result is large waves and they come together, the largest at any time being an increase in the amplitude of individual waves at that time. Although the waves disturbed the meeting place, they continued to travel as if they had never met before.

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Constructive and Destructive Interference of Waves

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When an eruption of one of the waves falls on the edge of the other waves it leads to a higher elevation. This is called a Constructive Distraction. While if one wave falls into the hole of another wave, its magnitude here will be minimal. This is called a Destructive Distraction.

Constructive and Destructive Interference of Waves Derivation 

Consider a sinusoidal wave that is travelling along the x-axis and given by the equation:

Here,

• A is referred to the amplitude of the wave
• k = 2π/λ is wave number
• ω = 2πf is angular frequency of the wave

Now, we will also be considering another wave of the same frequency and amplitude but that with a different phase which is travelling in the right direction.

⇒ W2 (x, t) = A cos (kx – ωt + Φ)

Here, Φ is referred to the initial phase difference between the waves in the radians

The two waves thus superimpose and add, the resultant wave is given by the equation, W1 + W2 = A

⇒ cos (kx − ωt) + cos (kx − ωt + Φ)

With the help of the cosine rule, cos a + cos b = 2 cos (a − b/2) cos (a + b/2)

Now we will be solving the equation using the formula obtained

⇒ W1 + W2 = 2 A cos Φ/2 {cos (kx – ωt + Φ2)}

Comparing it with the W1 (x, t) = A cos (kx − ωt)

Amplitude of resultant wave will be 2A cos Φ/2

If the phase difference is going to be an even multiple of π (φ = ….., –4π , –2π , 0 , 2π , 4π ,……), then cos φ/2 = 1, then the sum of the two waves is a wave with twice the amplitude and this solution is called the Constructive Interference.

While when the phase difference is going to be an odd multiple of π (φ = …..,–3π , –π , 0 , π , 3π , 5π ,……), then the cos φ/2 = 0, thus the sum of the two waves will be zero. W1 + W2 = 0 and this is the case of Destructive Interference.

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Previous Year Questions

1. In the adjacent diagram, CP represents a wavefront … [JEE Advanced 2003]
2. Which of the following is false for the interference of light? … [JKCET 2012]
3. An isosceles prism of angle of 120 degrees has a … [JEE Advanced 1995]
4. Monochromatic light is incident on a glass prism of angle A … [JEE Mains 2015]
5. An object approaches a convergent lens from the left of the lens … [KCET 2020]
6. The refracting angle of the prism is A and the refractive index … [KCET 2020]
7. A ray of light passes through an equilateral prism … [KEAM]
8. The period of SHM of a particle is 12 s … [JCECE 2004]
9. The ratio of resolving powers of an optical microscope … [NEET 2017]
10. A beam of light of wavelength 600nm from a distant source falls on a single slit … [KCET 2004, UPSEE 2006]
11. The aperture diameter of a telescope is 5 m … [JEE Mains 2020]
12. The angle of minimum deviation for an incident light ray on an equilateral prism … [KCET 2009]
13. In refraction, light waves are bent on passing from one medium … [KCET 2021]
14. Two plane wavefronts of light, one incident on a thin convex lens … [KEAM]
15. The phase difference between two waves … [UPSEE 2019]
16. Which of the following is not the property of laser beams? … [UPSEE 2019]
17. In an interference experiment, when a glass plate of refractive index … [COMEDK UGET 2007]
18. To increase the angular width of interference fringes … [COMEDK UGET 2007]

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Things to Remember

• Interference happens when two or more waves are meeting each other.
• The interference of the waves in the medium taking shape is the one which will be resulting from the net effect of two different waves.
• The constructive interference occurs when the crest of one wave coincides with the crest of another wave.
• Interference can also occur when the two wave trains are moving in the same direction but have different wavelengths or frequencies.

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