Explain the intensity at the point of superposition of waves emanating from coherent and incoherent sources.

(N/A) If light sources have a constant initial phase difference or their phase difference does not change with time,these sources are called coherent sources.
In an interference pattern,the intensity at any point does not change with time. This type of interference is known as stationary interference.
For stationary interference,there must be two coherent sources,and their amplitudes should be the same.
The positions of maxima and minima in stationary interference do not change with time.
When the phase difference between two vibrating sources changes very rapidly over time,these sources are known as incoherent sources.
The light intensities are added to each other due to the superposition of waves emanating from incoherent sources; hence,two different light sources illuminate the wall independently.
When the path difference of the two sources is not constant,the interference pattern also changes with time. If the path difference changes very rapidly with time,the positions of maxima and minima will also change rapidly,and we will observe the time-averaged distribution of intensity.
This average intensity is given by:
$\langle I \rangle = 4 I_{0} \langle \cos^{2} \left( \frac{\phi}{2} \right) \rangle$
where $\langle \cos^{2} \left( \frac{\phi}{2} \right) \rangle$ represents the time-averaging term.
If $\phi(t)$ varies randomly with time,the time-averaged quantity $\langle \cos^{2} \left( \frac{\phi}{2} \right) \rangle$ will be $\frac{1}{2}$,and the resulting intensity at all points is:
$I = 4 I_{0} \times \frac{1}{2}$
$\therefore I = 2 I_{0} \text{ at all points.}$