Question

Derivation and condition for coherent and incoherent waves addition.

Answer 1

There are two types of sources of waves. One is coherent source of waves and the other is incoherent source of waves.

The sources and the waves are said to be coherent if they have the same frequencies, same wavelength, same speed, almost same amplitude and have no phase difference or a constant phase difference. If any of the properties is lacking, the sources are said to be incoherent.

The examples of the coherent sources and the waves are sound waves from two loud speakers driven by the same audio oscillator, electromagnetic waves from two microwaves horns driven by the same oscillator, light waves generated by Young double-slit experiment, light waves coming from a laser gun.
Let for any other arbitrary point the phase difference between the two displacements produced by the waves $y_1$ and $y_2$ be φ.

Thus, if the displacement produced by $y_1$ is given by

$y_1=acos\left(\omega t\right)$

then, the displacement produced by $y-2$ would be

$y_2=acos(\omega t+\varphi )$

and the resultant displacement will be given by

$y=y_1+y_2$

$y=a[cos\omega t+cos(\omega t+\phi \left)\right]$

$y=2acos\left(\phi /2\right)cos(\omega t+\phi /2)$

$[\because cosA+cosB=2cos(\frac{A+B}{2}\left)cos\right(\frac{A-B}{2}\left)\right]$

The amplitude of the resultant displacement is 2a cos (φ/2) and therefore the intensity at that point will be if the waves have the same intensity $I_o$

$I=4I_{o}co{s}^2\left(\frac{\varphi }{2}\right)$

If the waves are of different intensities, then the resultant intensity is given by

I = I₁ + I₂ + 2 √(I₁ I₂ Cos Θ), where Θ is the phase difference between two waves. For the constructive interference the value of Θ = 0°, so that the Cos Θ = 1 and for the destructive interference the value of Θ = 90°, so that the Cos Θ = 0.

In case of the incoherent sources or waves we can have the intensity simply by algebraic method that means I = I₁ + I₂.

When the phase difference between the two vibrating sources changes rapidly with time, the two sources are incoherent and the intensities just add up. This is indeed what happens when two separate light sources illuminate a wall.