What are coherent source ? why are coherent sources required

What are coherent source? why are coherent sources required to produce sustained interference of light? In Young’s double slit experiment, to produce interference patterns due to a monochromatic sources of light, deduce the expression for the fringe width? Draw the intensity curve for interference in this experiment.

Coherent Sorces of light are the sources which have a phase difference between their wavefronts which do not vary with time and, have same wavelength.

It is required to obtain sustainable interferene pattern because, if the phase difference is not constant and it varies with time, then the pattern will be an average over the time and wont give fringes.

let a system be:
image
now for a bright fringe at P: