Question

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

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

Answer 1

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

$y_1=asin(\omega t+{\theta }_1)$
$y_2=asin(\omega t+{\theta }_2)$
$y=y_1+y_2$
$=asin\omega tcos{\theta }_1+acos\omega tsin{\theta }_1+asin\omega tcos{\theta }_2+acos\omega tsin{\theta }_2$
$=asin\omega t(cos{\theta }_1+cos{\theta }_2)+acos\omega t(sin{\theta }_1+sin{\theta }_2)$
So, for same amplitude and time period.
$cos{\theta }_1+cos{\theta }_2=0$ and $(sin{\theta }_1+sin{\theta }_2{)}_{max}=1$
or, $sin{\theta }_1+sin{\theta }_2=0$ and $(cos{\theta }_1+cos{\theta }_2{)}_{max}=1$
So, ${\theta }_1-{\theta }_2=\frac{2\pi }{3}.$