Two light waves represented by y_1 = a (ω_1 t - k_1 x) and y_2 = a...

For a sustained (steady) interference pattern to be observed, the positions of maxima and minima on the screen must not shift over time.

The resultant intensity at any point depends on the phase difference $\Delta\phi$ between the interfering waves. If $\Delta\phi$ changes with time, the intensity at point P will fluctuate rapidly, resulting in a uniform average illumination rather than a steady pattern.

Therefore, the fundamental requirement for sustained interference (coherence) is that the phase difference $\Delta\phi$ between the two waves must remain constant with respect to time (i.e., independent of time). This implies that the angular frequencies must be equal ($\omega_1 = \omega_2$).

Options stating $\Delta\phi$ must be zero or an integer multiple of $2\pi$ are conditions for constructive interference (maxima) specifically, not the general requirement for the entire sustained pattern.

Answer: $\Delta\phi$ must be independent of time