Coherent and Incoherent Addition of Waves: We know that light is a wave, but can we get a dark fringe by adding two light waves. How does the dark fringe happen to occur in young’ double-slit experiment? How is standing wave formed? What are Beats? Are these results of interference by coherent waves or by incoherent waves. When can the interference differ with time, and when can it differ with space? In this article, we will learn about the coherent and incoherent addition of waves.

Waves

Waves is a phenomenon that transfers energy and momentum from one place to another. For example, the wave on the surface of the water is generated from a point due to vibrations, and these vibrations are transferred from the source point to any other point. A light wave is generated from the source, and it travels in the form of electromagnetic radiation. Thus it doesn’t require a medium to travel. On the basis of the requirement for medium, waves can be classified into two categories.

1. Mechanical waves:
These types of waves require a medium to Propagate. The medium must have inertia (mass) and should be elastic, for example, waves formed by vibrating stretched strings.

2. Electromagnetic waves:
These type of waves does not require a medium to propagate—for example; radio waves, light waves, etc.

Important Terms Related to Waves

1. Wavelength
The distance travelled by the disturbance in one complete vibration by a medium particle is known as the wavelength \(\left( \lambda \right).\) In a transverse wave, the wavelength can also be defined as the distance between two successive crests or troughs. In a longitudinal wave, the wavelength \(\left( \lambda \right)\) is equal to the distance from the centre of one compression (or rarefaction) to the centre of the next compression (or rarefaction).

2. Amplitude
The amplitude of a wave is the maximum displacement of the particles of the medium from its mean position while the wave passes through it.

3. Frequency
The number of vibrations made by a particle of the medium in one second is called the frequency of the wave. It is represented by \(v.\) Its unit is hertz (Hz), \(v = \frac{1}{T}.\)

4. Time-Period
The time taken by a particle to have one complete vibration is called the time period of the wave. It is the inverse of the frequency.
\(T = \frac{1}{v},\) it is expressed in seconds.

5. Wave Velocity
Wave velocity is the velocity of propagation of the wave in the given medium. It is different from particle velocity. Wave velocity depends upon the nature of the given medium. (Particle velocity keeps on changing as it oscillates with a fixed mean position)
Wave velocity \(\left( v \right) = \) frequency \(\left( v \right) \times \) wavelength \(\left( \lambda \right)\)

Equation of Wave

Any disturbance that keeps on moving forward with time is a wave. But in our world majority of the waves are harmonic waves; that is; the motion of the disturbance of the medium particle is simple harmonic,
\(y = A\,\sin \left( {\omega t} \right)\)
This disturbance gradually moves forward, that is, in the direction of the propagation of the wave. Therefore the equation of the wave is,
\(y = A\,\sin \left( {\omega t – kx} \right)\)
Where,
\(A\) is the maximum amplitude of the wave.
\(\omega \) is the angular frequency of the wave
\(k\) is the wavenumber
Displacement of any particle in space will be a function of time and position.

Superposition of Waves

The superposition principle states that if the if cause adds up linearly, then the effects will also add up linearly. Thus if one wave superimposes another wave, the net displacement of the medium particle will be the sum of displacement due to individual waves.

If \({y_1}\) is the displacement of the particle due to first wave and \({y_2}\) is the displacement of the particle due to the second wave, then the net displacement of the particle will be,
\({y_{{\rm{net}}}} = {y_1} + {y_2}\)

Coherent and Incoherent Waves

Coherent waves are waves that have the same frequency and waveform. In simpler words, they are identical waves. Two waves from the same source are coherent. There may exist some phase differences due to path differences when they superimpose.

Incoherent waves are waves that differ in any of the parameters like frequency or waveform.
Superposition can happen in two types of wave, that is; coherent addition of waves or incoherent addition of waves. An example of coherent addition of waves is young’s double-slit experiment, standing waves and harmonics produced by organic pipes.

Examples of incoherent addition of waves are the production of beats

Youngs Double-slit Experiment

The young double-slit experiment proves the wave nature of the light and contradicts the particle nature of the light. This experiment was done by Thomas Young in \(1801.\)

In this experiment, monochromatic coherent light waves are made to pass through the two slits, and a screen is put in front of the slits. The pattern observed on the screen is alternately bright and dark fringes. This is due to the destructive and constructive interference of the light wave. A dark fringe is formed due to destructive interference, and a bright fringe is formed due to constructive interference. This phenomenon was used to prove the wave nature of the light.

Let us derive the expression for the fringe pattern formed on the screen.
Suppose the light from a source comes out of the two slits separated by a distance \(d.\)
The distance between the screen and any of the slit is \(D.\)
Approximations:
1. \(D > > d,\) this approximation helps us to assume the angular position of a point on the screen to be the same from both the slits; that is, both the rays are assumed to be parallel.
2. \(d > > \lambda ,\) this approximation helps us to consider the angular position of any of the points to be small. Thus we can use a small-angle approximation.
We know that for constructive interference, the path difference should be an integral multiple of the wavelength.
\(\Delta x = n\lambda \)
The path difference is equal to,
\(\Delta x = d\,\sin \left( \theta \right)\)
As the angle is small,
\(\sin \left( \theta \right) = \tan \left( \theta \right) = \frac{y}{D}\)
\( \Rightarrow \Delta x = \frac{{dy}}{D}\)
\( \Rightarrow n\lambda = \frac{{dy}}{D}\)
Therefore the position of bright fringe is,
\( \Rightarrow y = \frac{{nDy}}{d}\)
The fringe width is given by,
\(\beta = {y_{\left( {n + 1} \right)}} – {y_{\left( n \right)}}\)
\(\beta = \frac{{\lambda D}}{d}\)
For dark fringe, destructive interference should occur.
Destructive interference occurs when the phase difference between the superimposing waves is \(\frac{\pi }{2},\) that is; the path difference will be,
\(\Delta x = \frac{{\left( {2n + 1} \right)\lambda }}{2}\)
As the angle is small,
\(\sin \left( \theta \right) = \tan \left( \theta \right) = \frac{y}{D}\)
\( \Rightarrow \Delta x = \frac{{dy}}{D}\)
\( \Rightarrow \frac{{\left( {2n + 1} \right)\lambda }}{2} = \frac{{dy}}{D}\)
\( \Rightarrow y = \frac{{\left( {2n + 1} \right)D\lambda }}{{2d}}\)

Standing Waves

The standing wave is formed when a wave that is reflected in the same medium superimposes the incident wave. Standing waves contains some points known as nodes and antinodes. Nodes are the point where the destructive interference occurs, and that point has the least amplitude.

Antinode is formed by constructive interference of the two waves. Antinode has the maximum amplitude of the oscillation.

In the case of the standing wave, all the particles of the medium perform Simple Harmonic Motion with different amplitudes ranging from zero at the nodes to a maximum at antinodes.

Let the equation of the light wave be,
\({y_1}\left( {x,\,t} \right) = A\,\sin \left( {\omega t – kx} \right) = A\,\sin \left( {2\pi ft – \frac{{2\pi }}{\lambda }x} \right)\)
Where,
\({y_1}\) is the amplitude of the wave.
\(\omega \) is the angular frequency of the wave.
\(f\) is the frequency of the wave.
\(t\) is the time.
Similarly, the equation of the sound wave from the second source is,
\({y_2}\left( {x,\,t} \right) = A\,\sin \left( {\omega t + kx} \right) = A\,\sin \left( {2\pi ft + \frac{{2\pi }}{\lambda }x} \right)\)
\({y_2}\) is the amplitude of the displacement of the medium particle located at a point.
\(\omega \) is the angular frequency of the wave.
\(f\) is the frequency of the wave
When the waves overlap, the resultant wave is given by,
\({y_{{\rm{net}}}}\left( {x,\,t} \right) = {y_1}\left( {{x_1},\,t} \right) + {y_2}\left( {{x_2},\,t} \right)\)
Putting in the values, we get,
\({y_{{\rm{net}}}} = A\,\sin \left( {2\pi ft – \frac{{2\pi }}{\lambda }x} \right) + A\,\sin \left( {2\pi ft + \frac{{2\pi }}{\lambda }{x_2}} \right)\)
Using the trigonometric property,
\(\sin \left( A \right) + \sin \left( B \right) = 2\cos \left( {\frac{{A – B}}{2}} \right)\sin \left( {\frac{{A + B}}{2}} \right)\)
\( \Rightarrow {y_{{\rm{net}}}} = 2\,A\,\cos \left( {\omega t} \right)\sin \left( {\frac{{2\pi }}{\lambda }x} \right)\)
On comparing with the equation of wave,
\(y = A\,\sin \left( {\omega t – kx} \right) = A\,\sin \left( {2\pi ft – kx} \right)\)
The amplitude of the resultant wave is given by,
\(A\left( x \right) = 2\,A\,\sin \left( {\frac{{2\pi }}{\lambda }x} \right)\)
Thus, the amplitude of the resultant will be a function of the position of the point.
For the maximum amplitude,
\({A_{{\rm{net}}}} = 2\,A\,\sin \left( {\frac{{2\pi }}{\lambda }x} \right)\) It should be maximum.
\(\sin \left( {\frac{{2\pi }}{\lambda }x} \right)\) Should be; \( \pm 1.\)
\(\frac{{2\pi }}{\lambda }x = \left( {2n + 1} \right)\frac{\pi }{2}\)
\( \Rightarrow x = \frac{{\left( {2n + 1} \right)}}{2}\lambda \)
For zero amplitude,
\({A_{{\rm{net}}}} = 2\,A\,\sin \left( {\frac{{2\pi }}{\lambda }x} \right)\) It should be \(0.\)
\(\frac{{2\pi }}{\lambda }\left( {{x_2} – {x_1}} \right) = n\pi \)
\( \Rightarrow x = \frac{\pi }{2}\lambda \)

Beats

Beats are formed by the superposition of incoherent waves. At any particular point, the amplitude of the sound heard keep on change, and periodically the maxima are heard, that is, beats. Beats can be heard at any point in space but at different times instance but the beat frequency will remain the same.
Let us derive the expression for beat frequency.

Let the equation sound wave from the first source be,
\({y_1} = A\,\sin \left( {{\omega _1}t – kx} \right) = A\,\sin \left( {2\pi {f_1}t – kx} \right)\)
Where,
\({y_1}\) is the amplitude of the displacement of the medium particle located at a point.
\({{\omega _1}}\) is the angular frequency of the wave.
\({{f_1}}\) is the frequency of the first sound wave.
\(t\) is the time.
\(A\) is the maximum displacement of the medium particle from the mean position.
Similarly, the equation of the sound wave from the second source is,
\({y_2} = A\,\sin \left( {{\omega _2}t – kx} \right) = A\,\sin \left( {2\pi {f_2}t – kx} \right)\)
\({y_2}\) is the amplitude of the displacement of the medium particle located at a point.
\({{\omega _2}}\) is the angular frequency of the second sound wave.
\({{f_2}}\) is the frequency of the wave
When the sound waves from the two sources overlap, the resultant wave is given by,
\({y_{{\rm{net}}}} = {y_1} + {y_2}\) Putting in the values we get,
\({y_{{\rm{net}}}} = A\,\sin \left( {{\omega _2}t – kx} \right) + A\,\sin \left( {{\omega _2}t – kx} \right)\)
\( \Rightarrow {y_{{\rm{net}}}} = A\,\sin \left( {2\pi {f_1}t – kx} \right) + A\,\sin \left( {2\pi {f_2}t – kx} \right)\)
Using the trigonometric property,
\(\sin \left( A \right) + \sin \left( B \right) = 2\cos \left( {\frac{{A – B}}{2}} \right)\sin \left( {\frac{{A + B}}{2}} \right)\)
\( \Rightarrow {y_{{\rm{net}}}} = 2A\cos \left( {\left( {{f_1} – {f_2}} \right)\pi t} \right)\sin \left( {\left( {{f_1} + {f_2}} \right)\pi t – kx} \right)\)
On comparing with the equation of wave,
\(y = A\sin \left( {\omega t – kx} \right) = A\sin \left( {2\pi ft – kx} \right)\)
The amplitude of the resultant wave is given by,
\(A = 2A\cos \left( {\left( {{f_1} – {f_2}} \right)\pi t} \right)\)
Thus, the amplitude of the resultant will vary with time, therefore for the beat to occur, the amplitude of the wave should be maximum,
\({A_{{\rm{net}}}} = 2A\cos \left( {\left( {{f_1} – {f_2}} \right)\pi t} \right)\) should be maximum.
\(\cos \left( {\left( {{f_1} – {f_2}} \right)\pi t} \right)\) should be maximum.
\(\left( {{f_1} – {f_2}} \right)\pi {t_{{\rm{beats}}}} = k\pi \)
\(\frac{1}{{{f_{{\rm{beats}}}}}} = {t_{{\rm{beats}}}}\)
\( \Rightarrow {f_{{\rm{beats}}}} = \left| {{f_1} – {f_2}} \right|\)

Sample Problems

Q.1. Two sound sources having frequencies \(201\,{\rm{Hz}}\) and \(203\,{\rm{Hz}}\) are present near each other Find the beat frequency of the beats produced by the overlap of the two sound waves.
Sol:
Given,
The frequency of the first sound wave is,
\({f_1} = 201\,{\rm{Hz}}\)
The frequency of the second wave is,
\({f_2} = 203\,{\rm{Hz}}\)
We know that the beat frequency is given by,
\( \Rightarrow {f_{{\rm{beats}}}} = \left| {{f_1} – {f_2}} \right|\)
Where,
\({{f_1}}\) is the frequency of the first sound wave.
\({{f_2}}\) is the frequency of the second sound wave.
\( \Rightarrow {f_{{\rm{beats}}}} = 2\,{\rm{Hz}}\)
Therefore the beat frequency comes out to be \(2\,{\rm{Hz}}.\)

Summary

Waves are disturbances that propagate in space, transferring energy and momentum. If two waves intercept, then the results will the given by the superposition principle.

Coherent waves are identical waves. They can have a phase difference. In the case of coherent waves, constructive and destructive interference occur with space or position and not with time. This means that if the constructive or destructive interference occurs at a particular position, then it will remain so at all time instances. Incoherent waves differ in more than one parameter of the wave, especially frequency. In this case, at any position, constructive or destructive interference occurs with time.

Learn Transverse and Longitudinal Waves

FAQs

Q.1. What are coherent waves?
Ans: Waves that are identical and can only differ in phase are known as coherent waves.

Q.2. What are incoherent waves?
Ans: Incoherent waves differ in more than one parameter of the wave.

Q.3. Is standing wave formed by interference of coherent waves?
Ans: Yes, the standing waves are formed by interference of the coherent waves.

Q.4. What are beats?
Ans: Beats are formed by the superposition of incoherent waves. At any particular point, the amplitude of the sound heard keep on change, and periodically the maxima are heard, that is known as beats.

Q.5. Is the fringe width in young’s double-slit experiment the same for dark and bright fringe?
Ans: Yes, the fringe width is the same for both dark and bright fringe.

We hope this detailed article on the coherent and incoherent addition of waves helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!