Question

The equation of the progressive wave is $y=asin2\pi \left(nt-\frac{x}{5}\right)$. The ratio of maximum particle velocity to wave velocity is

• A. $\frac{\pi a}{5}$
• B. $\frac{2\pi a}{5}$
• C. $\frac{3\pi a}{5}$
• D. $\frac{4\pi a}{5}$

Answer 1

Answer: (B)

We have
$y=a\sin 2\pi \left(nt-\frac{x}{5}\right)$
or $y=a\sin 2\pi \left(\frac{vt}{\lambda }-\frac{x}{5}\right)\left[\because n=\frac{v}{\lambda }\right]$
Here, $v$ is wave velocity
$u=\frac{dy}{dt}=\frac{2\pi av}{\lambda }\cos 2\pi \left(\frac{vt}{\lambda }-\frac{x}{5}\right)$
$u_{\max }=\frac{2\pi av}{\lambda }$
$\therefore \frac{U_{max}}{V}=\frac{2\pi a}{\lambda }=\frac{2\pi a}{5}\left[\begin{array}{c}\because k=\frac{2\pi }{\lambda }\\ \lambda =\frac{2\pi }{k}=\frac{2\pi }{2\pi /5}=5\end{array}\right]$