Question

Three components of sinusoidal progressive waves travelling in the same direction along the same path having the same period, but with amplitudes $A,A/2$ and $A/3$. The phase of the vibration at any position $x$ on their path at time $t=0$ are $0,-\pi /2$ and $\pi$ respectively. Find the amplitude and phase of the resultant wave

• A. $\frac{5A}{2},{\tan }^{-1}\left(4/3\right)$
• B. $\frac{6A}{5},{\tan }^{-1}\left(2/3\right)$
• C. $\frac{5A}{6},{\tan }^{-1}\left(3/4\right)$
• D. $\frac{5A}{6},{\tan }^{-1}\left(4/3\right)$

Answer 1

The correct option is C $\frac{5A}{6},{\tan }^{-1}\left(3/4\right)$

Let $A_r$ be the resultant amplitude and $\varphi$ be the phase of resultant wave with respect to $A$.
From the data given in the question,


From the figure,
$\Rightarrow A_r=\sqrt{{\left(\frac{2A}{3}\right)}^2+{\left(\frac{A}{2}\right)}^2}=\frac{5A}{6}$
and for phase,
$\tan \varphi =\frac{\text{opposite side}}{\text{adjacent side}}=\frac{A/2}{2A/3}=\frac{3}{4}$
$\Rightarrow \varphi ={\tan }^{-1}\left(3/4\right)$ with respect to $A$

Hence, option (c) is the correct answer.