(a) (i) The condition for the sustained interference is that both the sources must be coherent (i.e., they must have the same wavelength and the same frequency, and they must have the same phase or constant phase difference).
Two sources are monochromatic if they have the same frequency and wavelength. Since, they are independent, i.e., they have different phases with irregular difference, they are not coherent sources.
Let the displacement of the waves from the sources ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ at point $\mathrm{P}$ on the screen at any time $\mathit{t}$ be given by:
${\mathit{y}}_1=\mathit{a}\cos \mathit{\omega }\mathit{t}$
and ${\mathit{y}}_2=\mathit{a}\cos \left(\mathit{\omega }\mathit{t}+\mathit{\varphi }\right)$
Where, $\mathit{\varphi }$ is the constant phase difference between the two waves.
By the superposition principle, the resultant displacement at point $\mathrm{P}$ is given by:
$\mathit{y}={\mathit{y}}_1+{\mathit{y}}_2$
$\mathit{y}=\mathit{a}\cos \mathit{\omega }\mathit{t}+\mathit{a}\cos \left(\mathit{\omega }\mathit{t}+\mathit{\varphi }\right)$
$\mathit{y}=2\mathit{a}\left[\cos \left(\frac{\mathit{\omega }\mathit{t}+\mathit{\omega }\mathit{t}+\mathit{\varphi }}{2}\right)\cos \left(\frac{\mathit{\omega }\mathit{t}-\mathit{\omega }\mathit{t}-\mathit{\varphi }}{2}\right)\right]$
$\mathit{y}=2\mathit{a}\cos \left(\mathit{\omega }\mathit{t}+\frac{\mathit{\varphi }}{2}\right)\cos \left(\frac{\mathit{\varphi }}{2}\right)$ .......(i)
Let $2\mathit{a}\cos \left(\frac{\mathit{\varphi }}{2}\right)=\mathit{A}$ .......(ii)
Then, equation (i) becomes
$\mathit{y}=\mathit{A}\cos \left(\mathit{\omega }\mathit{t}+\frac{\mathit{\varphi }}{2}\right)$
Now, we have:
${\mathit{A}}^2=4{\mathit{a}}^2{\cos }^2\left(\frac{\mathit{\varphi }}{2}\right)$ .......(iii)
Then intensity of light is directly proportional to the square of the amplitude of the wave. The intensity of light at point on the screen is given by
$\mathit{I}=4{\mathit{a}}^2{\cos }^2\frac{\mathit{\varphi }}{2}$
(b)
When path difference is $\mathit{\lambda }$, phase difference is $2\mathit{\pi }$
$\mathit{I}=4{\mathit{I}}_0{\cos }^2\mathit{\pi }$
$=4{\mathit{I}}_0=\mathit{k}$ (given)
When path difference, $∆=\frac{\mathit{\lambda }}{3}$, the phase difference
${\mathit{\varphi }}_1=\frac{2\mathit{\pi }}{\mathit{\lambda }}∆$
$=\frac{2\mathit{\pi }}{\mathit{\lambda }}\times \frac{\mathit{\lambda }}{3}=\frac{2\ne }{3}$
${\mathit{I}}_1=4{\mathit{I}}_0\cdot {\cos }^2\frac{2\mathit{\pi }}{3}$
$=\mathit{k}{\cos }^2\frac{2\mathit{\pi }}{3}$
$=\mathit{k}\times {\left(-\frac{1}{2}\right)}^2$
$=\frac{1}{4}\mathit{k}$