Question

The equations of a travelling and stationary waves are $y_{1}=a \sin (\omega t-k x)$ and $y_{2}=a \sin k x \cos \omega t .$ The phase difference between two points $x_{1}=\frac{\pi}{4 k}$ and $x_{2}=\frac{4 \pi}{3 k}$ are $\phi_{1}$ and $\phi_{2}$ respectively for two. waves, where $k$ is the wave number. The ratio of $\phi_{1} / \phi_{2}$ is:

• A. 6/7
• B. 16/3
• C. 12/13
• D. 13/12

Answer 1

Answer: (C)

$\Delta x=x_{2}-x_{1}=\left(\frac{4}{3}-\frac{1}{4}\right) \frac{\pi}{k}$
$=\frac{13}{12} \frac{\pi}{k}$
$\sin k x_{1}=\sin k\left(\frac{\pi}{4 k}\right)=\sin \frac{\pi}{4} \neq 0$
$\sin k x_{2}=\sin k\left(\frac{4 \pi}{3 k}\right)$
$=\sin \left(\pi+\frac{\pi}{3}\right) \neq 0$
$x_{1}$ and $x_{2}$ are not the nodes
$\frac{2 \pi}{k}>\Delta x>\frac{\pi}{k}$
$\Rightarrow \lambda>\Delta x>\frac{\lambda}{2}$
For $\phi_{1}=\pi, \phi_{2}=k(\Delta x)$
$=k\left(\frac{13 \pi}{12 k}\right)=\frac{13 \pi}{12}$
$\frac{\phi_{1}}{\phi_{2}}=\frac{\pi}{(13 \pi / 12)}=\frac{12}{13}$