wave_mechanics mock_test_waves_and_acoustics Question 13
Question: The equations of a travelling and stationary waves are $ y _{1}=a\sin (\omega t-kx) $ and$ y _{2}=a\sin kxcos\omega t $ . The phase differences between two points $ x _{1}=\frac{\pi }{4k}andx _{2}=\frac{4\pi }{3k} $ are $ {\phi _{1}} $ and $ {\phi _{2}} $ respectively for two waves, where k is the wave number. The ratio of $ {\phi _{1}}/{\phi _{2}} $ is
Options:
A) 6/7
B) 16/3
C) 12/13
D) 13/12
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ \Delta x=x _{2}-x _{1}=( \frac{4}{3}-\frac{1}{4} )\frac{\pi }{k}=\frac{13}{12}\frac{\pi }{k} $
$ \sin kx _{1}=\sin k( \frac{\pi }{4k} )=\sin \frac{\pi }{4}\ne 0 $
$ \sin kx _{2}=\sin k( \frac{4\pi }{3k} )=\sin ( \pi +\frac{\pi }{3} )\ne 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( \frac{13\pi }{12k} )=\frac{13\pi }{12} $
$ \frac{{\phi _{1}}}{{\phi _{2}}}=\frac{\pi }{(13\pi /12)}=\frac{12}{13} $
BETA
