# Coherent and non-coherent sources

As we know, if we superpose two coherent sources of intensities $$I_1$$ and $$I_2$$, the resultant intensity is not merely equal to $$I_1 + I_2$$ but sometimes can be much less and much more than that value. On the other hand, most practical and common sources - the non coherent ones - if we superpose them, the resultant intensity is simply the algebraic sum of the respective intensities ($$I_1 + I_2$$).

Say there are two coherent sources producing some waves with a phase difference of $$\pi/2$$. The resultant amplitude is the square root of sum of squares of respective amplitudes. By this we can conclude that these coherent sources superimpose to produce a new intensity which is algebraic sum of respective intensities (just like the non-coherent sources).

My question is mathematically the reason of such an occurrence is justified, but how to physically justify and analyse the reason why two coherent sources suddenly behaved like a non coherent one?

Physically, you could say that this happens because in the case when there is a phase difference of $$\frac{\pi}{2}$$, the cosine of the phase difference is equal to the average of the cosine of the phase difference for incoherent sources.
Net intensity produced by two coherent sources with a phase difference of $$\phi$$ is given by: $$I = I_1 + I_2+2\sqrt{I_1 I_2} \cos\phi$$ When $$\phi = \frac{\pi}{2}$$, $$\cos\phi=0$$, which leads to net intensity being the sum of the two intensities.
For incoherent sources, the phase difference takes all possible values. This is why the average of all such intensities is taken. And the average of $$\cos\phi$$ is $$0$$ which leads to both cases having the same net intensity.