Coherent and Incoherent Addition of Waves

Coherent and Incoherent addition of waves are used in various fields of physics like acoustics, optics, and quantum mechanics. The waves adding up with each other can interfere either constructively or destructively, depending upon their phase differences and amplitudes. This phenomenon is known as the superposition of waves, which results from the coherent or incoherent addition of waves.

When two or more waves (i.e, light or sound) overlap with each other, the resultant displacement at any point and at any instant of time may be found by adding the instantaneous displacements that would be produced at the point by the individual waves if each were present alone this phenomenon is known as superposition of waves.

• Resultant wave = Sum of disturbances
• For electromagnetic waves, displacement is the Amplitude of electric field

The equation for the superposition of two waves with the same frequency and amplitude, but with a phase difference φ, is given by:

y = 2A cos(φ/2) cos(ωt + φ/2)

Where,

• y is the total displacement
• A is the amplitude of each individual wave
• ω is the angular frequency, and t is the time.

Coherent sources are those sources that generate waves having a constant phase relationship with one another which in turn implies that the crests and the troughs of the waves stay in proper alignment with each other. The coherence sources can also be regarded as “in step” waves.

Characteristics of Coherent Sources

The following are the characteristics of coherent sources:

• The waves have a constant phase difference over time.
• The waves have a fixed frequency and wavelength.
• These waves are characterized by the fact that they have a constant amplitude and a direction of propagation.
• The waves are monochromatic – that is, they have a single well-defined frequency.
• The waves are spatially and temporally correlated, meaning they have a fixed relationship in space and time.

Examples of Coherent Sources

• LASER
• MASER (Microwave Amplification by Stimulated Emission of Radiation)
• Certain types of radio transmitters.

Incoherent sources are those that produce waves with no fixed phase relationship between them. This implies that the waves are in different phases and their crests and troughs are not always in alignment. Incoherent sources may be tought of as waves that are ‘out of phase’ from each other. Waves coming from incoherent sources have multiple frequencies, amplitudes, and propagation directions. Such waves are not ‘monochromatic’.

Another notable point is that the waves lack spatial as well as temporal correlation which tells that there is no fixed connection between them in space and time. Typical examples of the incoherent sources are all natural light that we get, such as the sun, the incandescent bulbs and the fluorescent lamps.

Scattering of Light

When the light interacts with a rough or irregular surface, it can be scattered in different directions. The ray can be scattered either coherently or incoherently determined by the surface property and light wavelength. The coherent scattering happens on the surfaces that are smooth and regular, with a wavelength of light much larger than the surface irregularities. Here, the light which is scattered have a fixed phase relationship with the incident light so the interference patterns can be observed.

If roughness and irregularities on the surface are roughly the same as the light’s wavelength, incoherent scattering does take place. In this case, the scattered light will have random phase, and no interference pattern would be observed.

Also Read: Scattering of Light

Constructive Interference

The constructive interference happens when the two waves that are propagating in the same direction, with the similar phase and the same frequency, superimpose to result in a wave with a larger amplitude. This takes place when the wave crests and troughs all are in the same line.

Destructive Interference

When two or more waves of the same frequency but with opposite phases come together, their destructive interference occurs and as a result of their addition, a wave of less amplitude or even their complete cancellation is produced. These are often observed when the crests of one wave match the troughs of others.

Locus Point

The locus point is the location in space where waves from two or more coherent sources meet constructively or destructively to form an interference pattern. The location of the locus point is based on the relative phase and amplitude of the waves.

Consider two waves with the same frequency and amplitude but with a phase difference φ between them. The displacement of the first wave can be written as:

y₁ = A cos(ωt)

The displacement of the second wave can be written as:

y₂ = A cos(ωt + φ)

The total displacement of the two waves is the sum of the individual displacements:

y = y₁ + y₂

y = A cos(ωt) + A cos(ωt + φ)

Using the trigonometric identity:

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

we can rewrite the total displacement as:

y = 2A cos(φ/2) cos(ωt + φ/2)

This equation shows that the resulting wave has an amplitude of 2A cos(φ/2) and a phase of ωt + φ/2.

Coherent Addition of Waves

If the two waves are in phase (φ = 0), then cos(φ/2) = 1, and the total displacement becomes:

y = 2A cos(ωt)

This wave has the same frequency and amplitude as the individual waves but with a doubled amplitude. This is an example of constructive interference.

If the two waves are out of phase (φ = π), then cos(φ/2) = 0, and the total displacement becomes:

y = 0

This is an example of destructive interference, where the two waves cancel each other.

Incoherent Addition of Waves

If the two waves have a random phase difference (φ is random), then cos(φ/2) is also random, and the total displacement becomes:

y = A cos(ωt) + A cos(ωt + φ)

In this case, the waves add incoherently, and the resulting wave has an amplitude that is the root-mean-square (RMS) of the individual amplitudes, which is A√2.

One of the fundamental physics principles is the superposition principle which is the key to understanding many physical phenomena and the distinction of coherent and incoherent sources is one of the most important things in understanding this physical processes. The waves emitted by any coherent emitter have a fixed phase relationship with each other which results in the formation of either constructive or destructive interference patterns. On the other hand, the incoherent sources produce random phases that lead to simple addition of their wave amplitudes. Understanding their behaviour have importance in fields like waves, sound, and quantum mechanics where it helps to give solutions to many technological and engineering problems.

Also, Check

• Superposition and Interference
• Wave Optics
• Properties of Waves

What is the difference between coherent and incoherent sources?

Coherent sources produce the waves of light with a fixed phase relationship, whereas incoherent sources produce the light waves with a random phase relation.

What is the coherent addition of waves?

Coherent addition of waves occurs when waves with the same frequency and phase are combined, leading to constructive interference.

What is the addition of waves called?

The concept of interference can be defined when two waves are added to each other. They can either combine constructively or destructively.

Can the scattering of waves be coherent and incoherent?

Scattering of waves can be either coherent, like in Rayleigh scattering, or incoherent, like in Mie scattering, depending on the phase relationships between the scattered waves.

Is sunlight coherent or incoherent?

The sunlight can be named as the incoherent source because it combines waves with different frequencies and phases, resulting in random interference.