Question

The equation of the progressive wave is $y=a\,sin\,2\,\pi\left(nt-\frac{x}{5}\right)$. 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(n t-\frac{x}{5}\right) $
or $\,\,\,y=a \sin 2 \pi\left(\frac{v t}{\lambda}-\frac{x}{5}\right) \,\,\,\,\left[\because n=\frac{v}{\lambda}\right]$
Here, $v$ is wave velocity
$u=\frac{d y}{d t}=\frac{2 \pi a v}{\lambda} \cos 2 \pi\left(\frac{v t}{\lambda}-\frac{x}{5}\right)$
$u_{\max }=\frac{2 \pi a v}{\lambda}$
$\therefore \frac{U_{max}}{V}=\frac{2\pi a}{\lambda}=\frac{2\pi a}{5}\begin{bmatrix}\because k =\frac{2\pi}{\lambda}\\ \lambda=\frac{2\pi}{k}=\frac{2\pi}{2\pi/ 5} =5\end{bmatrix}$