The displacement $y$ of a particle in a medium can be expressed as $y = 10^{-6} \sin \left(100t + 20x +…
The displacement $y$ of a particle in a medium can be expressed as $y = 10^{-6} \sin \left(100t + 20x + \frac{\pi}{4}\right)$, where $t$ is in second and $x$ in metre. The speed of the wave is
$2000\text{ ms}^{-1}$
$5\text{ ms}^{-1}$
$20\text{ ms}^{-1}$
$5\pi\text{ ms}^{-1}$
Solution
The displacement is given by
$y = 10^{-6}\sin \left(100t + 20x + \frac{\pi}{4}\right)$
Comparing with standard equation $y = A\sin (\omega t + kx + \phi)$, we get
$\omega = 100$, $k = 20$
where, $\omega$ is angular frequency and $k$ is angular wave number.
Now, $k = \frac{2\pi}{\lambda} = \frac{2\pi\nu}{\lambda\nu} = \frac{\omega}{v} = 20$
$\Rightarrow v = \frac{\omega}{20} = \frac{100}{20} = 5\text{ ms}^{-1}$
So, the speed of the wave is $5\text{ ms}^{-1}$.