The equation of a progressive wave is $\mathrm{Y}=$ $\operatorname{asin} 2 \pi\left(n t-\frac{x}{5}\right)$.…
The equation of a progressive wave is $\mathrm{Y}=$ $\operatorname{asin} 2 \pi\left(n t-\frac{x}{5}\right)$. The ratio of maximum particle velocity of wave velocity is
$\frac{\pi a}{5}$
$\frac{2 \pi \mathrm{a}}{5}$
$\frac{3 \pi \mathrm{a}}{5}$
$\frac{4 \pi \mathrm{a}}{5}$
Solution
The general equation for wave is
$\mathrm{Y}=\mathrm{a} \sin (\omega \mathrm{t}-\mathrm{kx})$
The given equation can be written as
$Y=\operatorname{asin}\left(2 \pi n t-2 \pi \frac{x}{5}\right)$
Comparing both equations,
$\omega=2 \pi n$
$k=\frac{2 \pi}{5}$
Wave velocity is given as $\mathrm{v}=\frac{\omega}{\mathrm{k}}$
$\therefore \quad \mathrm{v}=\frac{2 \pi \mathrm{n}}{\frac{2 \pi}{5}}=5 \mathrm{n}$
Particle velocity can be calculated as follows:
$\frac{d Y}{d x}=a(2 \pi n) \cos \left(2 \pi n t-2 \pi \frac{x}{5}\right)$
Maximum velocity is $\mathrm{v}_{\mathrm{m}}=(2 \pi \mathrm{na})$
$\frac{\mathrm{v}_{\mathrm{m}}}{\mathrm{v}}=\frac{2 \pi \mathrm{na}}{5 \mathrm{n}}$
$\therefore \quad \frac{\mathrm{v}_{\mathrm{m}}}{\mathrm{v}}=\frac{2 \pi \mathrm{a}}{5}$