Q. Two waves $y_{1} = A_{1}\sin\left( \omega t - \beta_{1} \right)$ and $y_{2} = A_{2}\sin\left( \omega t - \beta_{2} \right)$ superimpose to form a resultant wave whose amplitude is
A) $\sqrt{A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2}\cos\left( \beta_{1} - \beta_{2} \right)}$
B) $\sqrt{A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2}\sin\left( \beta_{1} - \beta_{2} \right)}$
C) $A_{1} + A_{2}$
D) $\left| A_{1} + \left. \ A_{2} \right| \right.\ $
Explanation
(a) Amplitude $A_{1}\ and\ A_{2}$ are added as vector. Angle between these vectors is the phase difference $\left( \beta_{1} - \beta_{2} \right)$ between them
$\therefore\ \ \ R = \sqrt{A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2}\cos\left( \beta_{1} - \beta_{2} \right)}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.