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  4. When a wave travels in a medium, the particle displacement is given by the equation y=a sin 2π (bt-cx) where a, b and c are constants. The maximum particle velocity will be twice the wave velocity, if

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Q. When a wave travels in a medium, the particle displacement is given by the equation $ y=a\sin 2\pi (bt-cx) $ where $a$, $b$ and $c$ are constants. The maximum particle velocity will be twice the wave velocity, if

KEAMKEAM 2009Waves

Solution:

The maximum particle velocity is twice the wave velocity
$ a\omega =2\left( \frac{\omega }{k} \right) $
or $ ak=2 $ .... (i)
Given $ y=a\sin 2\pi (bt-cx) $
or $ y=a\sin (2\pi bt-2\pi cx) $
The general wave equation
$ y=a\sin (\omega t-kx) $ then $ k=2\pi c $
$ \therefore $ $ a2\pi c=2 $ [From Eq. (i)]
or $ c=\frac{1}{\pi a} $