Question

Two coherent waves each of amplitude "a" traveling with a phase difference $\delta$ when superpose with each other the resultant intensity at a given point on the screen is:

• A. $a^2(1+cos\delta )$
• B. $4a^2(1+cos\delta )$
• C. $2a^2(1+cos\delta )$
• D. $(1+cos\delta )$

Answer 1

Answer: (C)

$2a^2(1+cos\delta )$

Let the waves be:

$x_1\left(t\right)=a\cos \omega t$

and $x_2\left(t\right)=a\cos (\omega t+\delta )$
Resultant wave will be:

$x\left(t\right)=x_1\left(t\right)+x_2\left(t\right)=a\cos \omega t+a\cos (wt+\delta )$
$=a\cos \omega t+a\cos \omega t\cos \delta -a\sin \omega t\sin \delta$

$=a(1+cos\delta )\cos \omega t$$-a\sin \delta \sin \omega t$
The amplitude of this wave is
$=\sqrt{a^2(1+cos\delta {)}^2+a^{2}si{n}^2\delta }$
$=\sqrt{a^2+a^{2}co{s}^2\delta +2a^{2}cos\delta +a^{2}si{n}^2\delta }$
$=\sqrt{2a^2+2a^{2}cos\delta }$
Intensity is square of amplitude
$I=2a^2(1+cos\delta )$