Derivation of Coherent and Incoherent Addition of Two 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

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.