Question

Consider two wave equation $Y_1=acos\omega t$ & $Y_2=acos(\omega +\varphi )$ the displacement equation producing interference pattern. Show that intensity is given by $l=4a^{2}co{s}^2\left(\varphi /2\right)$

Answer 1

Let the displacement of the wave from the sources $S_1$ and $S_2$ at point $P$ on the screen at any time $t$ be given by:

$y_1=a\cos \omega t$

$y_2=a\cos \omega t+\varphi )$

where, $\varphi$ is the constant phase difference between the two waves

By the superposition principle, the resultant displacement at point $P$ is given by:

$y=y_1+y_2$

$y=a\cos \omega t+a\cos (\omega t+\varphi )$

$y=2a[\cos \left(\frac{\omega t+\omega t+\varphi }{2}\right)\cos \left(\frac{\omega t-\omega t-\varphi }{2}\right)$

$y=2a\cos \left(\frac{\omega t+\varphi }{2}\right)\cos \left(\frac{\varphi }{2}\right)....\left(i\right)$

Let $2a\cos \left(\frac{\varphi }{2}\right)=A....\left(ii\right)$

Then, equation (i) becomes:

$y=A\cos (\omega t+\frac{\varphi }{2})$

Now, we have

$A^2=4a^2{\cos }^2\left(\frac{\varphi }{2}\right)$

The intensity of light is directly proportional to the square of the amplitude of the wave. The intensity of light at point P on the screen is given by:

$I=4a^2{\cos }^2\left(\frac{\varphi }{2}\right)$