Question

Equations of two progressive waves at a certain point in a medium are given by $y_1=a\sin (\omega t+{\varphi }_1)$ and $y_2=a\sin (\omega t+{\varphi }_2)$. If amplitude and time period of resultant wave formed by the superposition of these two waves is same as that of both the waves, then ${\varphi }_1-{\varphi }_2$ is?

• A. $\frac{\pi }{3}$
• B. $\frac{2\pi }{3}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is B $\frac{2\pi }{3}$

$y_1=a\sin (\omega t+{\varphi }_1)$

$y_2=a\sin (\omega t+{\varphi }_2)$

Resultant, $y=y_1+y_2$

$\therefore y=a\left[\sin \right(\omega t+{\varphi }_1)+sin(\omega t+{\varphi }_2\left)\right]$

$\Rightarrow y=a\left[2\sin \right(\frac{2\omega t+{\varphi }_1+{\varphi }_2}{2}\left)\cos \right(\frac{{\varphi }_1-{\varphi }_2}{2}\left)\right]$

Given,

$2a\cos \left(\frac{{\varphi }_1-{\varphi }_2}{2}\right)=a$

$\Rightarrow 2\cos \left(\frac{{\varphi }_1-{\varphi }_2}{2}\right)=\frac{1}{2}=\cos \frac{\pi }{3}$

$\therefore ,\frac{{\varphi }_1-{\varphi }_2}{2}=\frac{\pi }{3}$

$\Rightarrow {\varphi }_1-{\varphi }_2=\frac{2\pi }{3}$

Thus, option $B$ is correct.