Why are coherent sources necessary to produce a sustained interference pattern?

To produce a sustained interference, coherent sources are compulsory. Reasons are below:

(1) If the sources are not coherent, they cannot emit waves continuously.
(2) Independent sources emit waves which don't have the same phase or a constant phase difference.

In an interference pattern, the intensity I at a point is given by:

I = I₀cos²(π/λ)x

where x = path difference, λ = wavelength, I₀ = intensity of the central bright fringe.

When x = λ, I = K, K = I₀cos²(π/λ)λ = I₀cos²π = I₀

For x = λ, I = I′, I′ = Kcos²(π/λ)λ/3 = Kcos²(π/3) = K × (1/2)² , or I′ = K/4