Question

A wave of wavelength $2\text{cm}$ and velocity $6{\text{cm s}}^{-1}$ is travelling on a string and after collision with a rigid boundary, it forms a standing wave. Assuming that wave is in third mode of vibration, the equation of standing wave formed will be

• A. $2A\sin \frac{\pi x}{3}\cos \left(6\pi t\right)$
• B. $2A\sin \frac{2\pi x}{3}\cos \left(6\pi t\right)$
• C. $2A\sin \pi x\cos \left(6\pi t\right)$
• D. $2A\sin \frac{\pi x}{3}\cos \left(3\pi t\right)$

Answer 1

The correct option is C $2A\sin \pi x\cos \left(6\pi t\right)$

General equation of standing wave can be written as

$y=2A\sin \left(\frac{n\pi x}{L}\right)\cos \omega t$

Given third mode of vibration, number of loop formed $=3$

$\Rightarrow n=3$

$L:$ Length of the string

Given that, $v=6{\text{cm s}}^{-1},\lambda =2\text{cm}$

We will have one and half wave for formation of $3$ loops. So

$L=\lambda +\frac{\lambda }{2}=\frac{3\lambda }{2}=\frac{3\times 2}{2}=3\text{cm}$

We know that, $k=\frac{2\pi }{\lambda }=\frac{\omega }{v}$

$\Rightarrow \frac{2\pi }{2}=\frac{\omega }{6}$

$\therefore \omega =6\pi$

Therefore, the equation of the standing wave is given by,

$y=2A\sin \left(\frac{3\pi x}{3}\right)\cos \left(6\pi t\right)$

$y=2A\sin \left(\pi x\right)\cos \left(6\pi t\right)$

Hence, option $\left(C\right)$ is correct.