Question

Three sinusoidal waves of the same frequency travel along a string in the positive $ x $ -direcdon. Their amplitudes are $ y $ , $ \frac{y}{2} $ and $ \frac{y}{3} $ and their phase constants are $ 0 $ , $ \frac{\pi}{2} $ and $ \pi $ respectively. What is the amplitude of the resultant wave?

• A. 0.63 y
• B. 0.72 y
• C. 0.83 y
• D. 0.52 y

Answer 1

Answer: (C)

$Y_1 = y\,\sin (\omega t + 0)$
$Y_2 = \frac{y}{2} \sin (\omega t + \frac{\pi}{2})$
$Y_3 = \frac{y}{3} \sin (\omega t + \pi)$
According to superposition principle
$Y = Y_1 + Y_2 +Y_3$
$= y\,\sin\,\omega t + \frac{y}{2} \cos\,\omega t + \frac{y}{3} \,\sin (\omega t+ \pi)$
$= y\,\sin\,\omega t - \frac{y}{3}\,\sin\,\omega t + \frac{y}{2} \cos\,\omega t$
$ = \frac{2y}{3} \sin \,\omega t + \frac{y}{2} \cos\,\omega t$
$ = y (\frac{2}{3} \sin \,\omega t + \frac{1}{2} \cos\,\omega t)$
$= \frac{y}{6} (4\,\sin\,\omega t + 3\,\cos\,\omega t)$
$Y = \frac{4y}{6} \sin\,\omega t + \frac{3y}{6} \cos\,\omega t$
$A_1 = \frac{2y}{3}$ and $A_2 = \frac{y}{2}$
$A = \sqrt{A_1^2 + A_2^2}$
$= \sqrt{\frac{4y^2}{9} + \frac{y^2}{4}}$
$ = y \sqrt{\frac{16+9}{36}} = \frac{y}{6} \times 5$
$ = \frac{5y}{6} = 0.83\,y$