Q. Two coherent waves are represented by $y_{1} = a_{1}\ \cos{\omega t}$ and$y_{2} = a_{1}\ \sin{\omega t}$, superimposed on each other. The resultant intensity is
A) $\left( a_{1} + a_{2} \right)$
B) $\left( a_{1} - a_{2} \right)$
C) $\left( a_{1}^{2} + a_{2}^{2} \right)$
D) $\left( a_{1}^{2} - a_{2}^{2} \right)$
Explanation
(c) $y_{1} = a_{1}\cos{\omega t} = a_{1}\sin(\omega t + 90{^\circ})$
$y_{2} = a_{2}\sin{\omega t}$
$\therefore\ $Phase difference$= \phi = 90{^\circ}$
$R = \sqrt{a_{1}^{2} + a_{2}^{2} + 2a_{1}a_{2}\cos\phi} = \sqrt{a_{1}^{2} + a_{2}^{2}}$
$\therefore$ Resultant intensity$I = R^{2} = \left( a_{1}^{2} + a_{2}^{2} \right)$.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.