Constructive Interference: Conditions & Formula

Constructive interference are the waves that are set to be constructive, when two or more waves have interference at the same wavelength, frequency and in phase.

• In Physics, a wave is defined as the continuous locus of all particles of a medium which are vibrating in the same phase. 
• In case of constructive interference there are two or more waves that are travelling in the same direction and are in the same wavelength, frequency and in phase with each other.
• The displacement of an upward wave happens due to the waves that undergo constructive interference
• The upward displacement is medium and is greater than the displacement of the two interfering pulses.

Key terms: Constructive Interference, Destructive Interference, Young’s Module, Interference of light

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What is Constructive Interference? 

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When two or more waves have interference at same wavelength, frequency and in phase, it results in a mutual reinforcement and produces a single amplitude equal to the sum of the amplitudes of the individual waves. This is known as Constructive Interference. 

• Also, when there are two crests or troughs of a wave that are aligned to each other it gives a resultant amplitude that is higher.
• The two waves that have the same amplitude and frequency (ω) can be described by the following equation:

y₁ = cos (kx – ωt) and y₂ = cos (kx – ωt + δ)

where δ is the phase difference between the waves,

k is the wavenumber,

x is the wave position, and t is time.

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What is Destructive Interference?

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When two or more waves have interference & displacement in the opposite direction i.e.,they are 180 degrees out of phase, where a positive displacement of one wave is cancelled exactly by a negative displacement of the other wave, it is called destructive interference.

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What is Wave Interference?

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Wave Interference is a phenomenon when two coherent waves meet while travelling along the same medium. The resultant of the wave is given by the sum of the individual waves.

The interferences are of two types:

• Constructive Interference
• Destructive Interference

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Models of Constructive Interference

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These are the following phase difference between waves that must be zero for constructive interference:

Models of Constructive Interference

Read More: Electromagnetic Waves

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Condition for Constructive and Destructive Interference

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Let there be two coherent sources of light that is represented as:

y₁ = a sin ꙍ t -------------- 1) 

y₂ = b sin (ꙍ t + Ø) ------------ 2)

 where a and b are the respective amplitudes of the two waves and Ø is the constant phase angle by which the second wave leads the first wave.

According to the superposition principle, the displacement 9y) of the resultant wave at time (t) would be given by 

y = y₁ + y₂ = a sin ꙍ t + b sin (ꙍ t + Ø) = a sin ꙍ t + b sin ꙍ t cos Ø + b cost ꙍ t sin Ø

y = sin ꙍ t (a + b cos Ø) + cos ꙍ t. b sin Ø ------------ 3)

Put (a + b cos Ø) = R cos Ɵ --------------- 4)

b sin Ø = R sin Ɵ ---------------- 5)

y = sin ꙍ t . R cos Ɵ + cos ꙍ t . R sin Ɵ

y = R sin (ꙍ t + Ɵ) ---------------- 6)

Thus the resultant wave is Harmonic wave of amplitude R

Squaring 40 and 5) and adding, we get:

R² (cos² Ɵ + sin² Ɵ) = (a+ b cos Ø)² + (b sin Ø)²

R² x 1 = a² + b² cos² Ø + 2ab cos Ø + b² sin2 Ø

R = √ a² + b² + 2ab cos Ø ---------------- 7)

As resultant intensity I is directly proportional to the square of the amplitude of the resultant wave:

Therefore, 

 I ∝ R²

I ∝ ( a² + b² + 2 ab cos Ø) ---------------- 8)

For constructive interference

I should be maximum, for which 

cos Ø = max = + 1

Therefore Ø = , 2π, 4π….

Ø = 2nπ

Where n = 0,1,2

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Constructive Interference Between Two Plane Waves

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The two waves that have the same frequency and amplitude propagate through vacuum with an angle of 2∝ formed between their wave vectors k₁, k₂

• One wave is travelling horizontally and the other wave is travelling at an angle of Ɵ.
• The first two waves are in phase with point A. 
• The relative phase changes along the x-axis. 

The phase difference at the first point is given as,

∆ = \(\frac{2 \pi d}{\lambda}= \frac{2 \pi xsin \theta}{\lambda}\)

The waves are in phase with each other, then,

\(\frac{xsin \theta}{\lambda}\) = ± 0,1,2…..

They are a half cycle out of phase, then,

\(\frac{xsin \theta}{\lambda}\) = ± \(\frac{1}{2}\) ± \(\frac{3}{2}\)

The constructive interference occurs when the waveforms are in phase whereas in destructive interference they are out of phase.

\(d_f = \frac{\lambda}{sin \theta}\)

Where, df – fringe spacing, with the increase in fringe spacing, wavelength also increases.

Read More: Faraday's Law

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Difference between Constructive Interference and Destructive Interference

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These are the following differences between constructive and destructive interference:

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Formula of Constructive Interference

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The resultant wave which is a harmonic wave of amplitude R where in R = √ a² + b² + 2ab cos Ø.

As resultant intensity I is directly proportional to the square of the amplitude of the resultant wave.

 Therefore I ∝ R²

i.e. I ∝ ( a² + b²+ 2ab cos Ø)

For constructive interference 

I should be maximum, for which 

cos Ø = max = + 1

Therefore Ø = 0, 2π, 4π, ……..

Ø = 2nπ

Where n = 0,1,2………..

If x is the path difference between the two waves reaching point P, cross ponding to phase difference Ø, then

\(x = \frac{\lambda}{2\pi} \theta = \frac{\lambda}{2\pi} (2n\pi) = n\lambda\)

x = nʎ

Hence the condition for constructive interference at a point is the phase difference between the two waves reaching the point should be zero or an even integral multiple of π.

Equivalently, path difference between two waves reaching the point should be zero or an integral multiple of full wavelength.

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Examples of Constructive Interference

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There are several examples where constructive interference occurs, where two or more waves have interference at same wavelength, frequency and in phase which means that crests and troughs aligns:

• Sound Waves: When two speakers emit sound waves of the same frequency and are placed together, they produce waves that are of higher sound intensity, resulting in a louder sound.
• Musical Instruments: There are instruments that have strings such as guitar or violin, plucked or bowed to produce waves that can interfere constructively.
• Double-Slit Experiment: This is the experiment that demonstrates constructive interference with light waves where a beam of light passes through two closely spaced slits which creates a pattern of alternating light and dark bands on a screen.
• Soap Bubbles: There are bubbles that are obtained due to the result of constructive interference. The surface of the bubbles leads to the colourful pattern.
• Laser Interference: Lasers emit coherent light waves used in holography to create three-dimensional images.
• Radio Broadcasting: These are the radio waves that transmit the same signal from different antennas.

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Application of Constructive Interference

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The application of constructive interference is mentioned below:

• In the case of Acoustics, a beat is produced as an interference pattern which produces sounds of slightly different waves.
• Constructive interference measures the physical properties of a sound wave.
• Constructive interference is a periodic variation in volume whose rate is the difference between two frequencies and when two tones are in close pitch, the difference in frequency generates a beat.
• Contractive interference used to calibrate the wavelength standards and also as an Optical interferometry, in Michelson Morley experiment.
• In the case of a telescope, it is used to create a combined image.
• In the case of Radio Interferometry, two waves in the same phase, add each other up.

Also Read:

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Things to Remember

• Interference is the phenomenon of redistribution of light energy in a medium on account of superimposition of light waves from two coherent sources. 
• The waves are said to be constructive interference when the crest of one wave coincides with the crest of another wave.
• Young's double slit experiment produces bright and dark fringes that are placed alternately and they are equally spaced. These bands are called Interference fridges.
• Constructive interference measures the physical properties of a sound wave.
• The resultant of the wave is given by the sum of the individual waves.
• In the case of Radio Interferometry, two waves in the same phase, add each other up.
• When there are two crests or troughs of a wave that are aligned to each other that gives a resultant amplitude that is higher

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Previous Year Questions

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