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

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 \(\mathrm{v}_{\mathrm{m}}=(2 \pi \mathrm{na})\)
\(\frac{\mathrm{v}_{\mathrm{m}}}{\mathrm{v}}=\frac{2 \pi \mathrm{na}}{5 \mathrm{n}}\)
\(\therefore \quad \frac{\mathrm{v}_{\mathrm{m}}}{\mathrm{v}}=\frac{2 \pi \mathrm{a}}{5}\)