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Q.
If the equation of motion of standing wave is $ y=0.3\sin $ $ (314t-1.57x), $ the velocity of standing wave is :
Bihar CECEBihar CECE 2002Waves
Solution:
Key Idea: Standard equation of standing wave is $ y=a\sin (\omega t=kx). $
When two identical transverse, progressive waves travel in a bounded medium with the same speed but in opposite directions then by their superposition a new type of wave is produced which appears stationary in the medium. This is called standing wave. Let a be the amplitude, $ \omega $ angular velocity, k constant and $ x $ displacement, then equation of standing wave is
$ y=a\sin (\omega t-kx) $
Given, equation is
$ y=0.3\sin (314t-1.57x) $
On comparing the two equations, we have
$ \omega t=314t $ and $ kx=1.57x $
$ \Rightarrow $ $ \omega =314 $ and $ k=1.57 $
$ \therefore $ velocity $ v=\frac{\omega }{k}=\frac{314}{1.57}=200\,\text{unit} $