Obtain the conditions for constructive interference and destructive interference.

(N/A) Constructive Interference:
When two waves superimpose at a point,constructive interference occurs if they are in phase. This happens when the path difference between the waves is an integral multiple of the wavelength $\lambda$.
Path difference $\Delta x = n\lambda$,where $n = 0, 1, 2, 3, \dots$
Since a path difference of $\lambda$ corresponds to a phase difference of $2\pi$,the condition for constructive interference in terms of phase difference $\phi$ is:
$\phi = 2n\pi$,where $n = 0, 1, 2, 3, \dots$
At these points,the resultant amplitude is maximum $(2a)$,and the intensity is $I_{max} = (a + a)^2 = 4I_0$.
Destructive Interference:
Destructive interference occurs when the two waves are out of phase by $\pi$ (or an odd multiple of $\pi$). This happens when the path difference is an odd multiple of half the wavelength $\lambda/2$.
Path difference $\Delta x = (2n + 1)\frac{\lambda}{2}$,where $n = 0, 1, 2, 3, \dots$
The condition for destructive interference in terms of phase difference $\phi$ is:
$\phi = (2n + 1)\pi$,where $n = 0, 1, 2, 3, \dots$
At these points,the resultant amplitude is minimum $(a - a = 0)$,and the intensity is $I_{min} = 0$.