Two waves have equations x_1=a ( t+ _1 ) and x_2=a ( t+ _2 ). If … - Vidyadip

MCQ

Two waves have equations $x_1=a \sin \left(\omega t+\phi_1\right)$ and $x_2=a \sin \left(\omega t+\phi_2\right)$. If in the resultant wave the frequency and amplitude remain equal to amplitude of superimposing waves, the phase difference between them is ........

A
$\frac{\pi}{6}$
✓
$\frac{2 \pi}{3}$
C
$\frac{\pi}{4}$
D
$\frac{\pi}{3}$

✓

Answer

Correct option: B.

$\frac{2 \pi}{3}$

b
(b)

$x_1=a \sin \left(\omega t+\phi_1\right)$

$x_2=a \sin \left(\omega t+\phi_2\right)$

$x^{\prime}=x_1+x_2$

$=a\left[\sin \left(\omega t+\phi_1\right)+\sin \left(\omega t+\phi_2\right)\right]$

$=2 a \sin \left(\omega t+\frac{\phi_1+\phi_2}{2}\right) \cos \left(\frac{\phi_1-\phi_2}{2}\right)$

Now as given in question

$2 a \cos \frac{\phi_1-\phi_2}{2}=a$

$\cos \left(\frac{\phi_1-\phi_2}{2}\right)=\frac{1}{2}$

$\frac{\phi_1-\phi_2}{2}=\frac{\pi}{3}$

$\phi_1-\phi_2=\frac{2 \pi}{3}$

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