Question

If two waves of the same frequency and amplitude respectively on superposition produce a resultant disturbance of the same amplitude the waves differ in phase by

• A. $\pi$
• B. zero
• C. $\pi /3$
• D. $2\pi /3$

Answer 1

Answer: (D)

Let $A$ and $v$ be the amplitude and frequency of each wave differing in phase by $\varphi .$ Then

y1​=Asin(kx−2πvt)
and $y_2=A\sin (kx-2\pi vt+\varphi )$
According to the principle of superposition, the resultant wave is
$y=y_1+y_2=A\sin (kx-2\pi vt)+A\sin (kx-2\pi vt+\varphi )$
or $y=\left(2A\cos \frac{\varphi }{2}\right)\sin \left(kx-2\pi vt+\frac{\varphi }{2}\right)$
But as per question $2A\cos \frac{\varphi }{2}=A$
or $\cos \frac{\varphi }{2}=\frac{1}{2}$ or $\frac{\varphi }{2}=\frac{\pi }{3}$
or $\varphi =\frac{2\pi }{3}$