Question 363: Waves

Question

The equation of a progressive wave is $$Y=a \sin 2 \pi\left(n t-\frac{x}{5}\right)$$. The ratio of maximum particle velocity to wave velocity is

Accepted Answer

General equation of a wave is given by $$y=A \sin (\omega t-k x)$$
Equation for progressive wave is
$$y=a \sin 2 \pi\left(n t-\frac{x}{5}ight)$$
On comparing both equations,
$$A=a, \omega=2 \pi n, k=\frac{2 \pi}{5}$$
Maximum particle velocity is given by
$$\begin{aligned}
& v_m=A \omega \
& \Rightarrow \quad v_m=a(2 \pi n) \
\end{aligned}$$
Wave velocity is given by
$$\begin{aligned}
& v_\omega=\frac{\omega}{k} \Rightarrow v_\omega=\frac{2 \pi n}{\frac{2 \pi}{5}}=5 n \
	herefore \quad & \frac{v_m}{v_\omega}=\frac{a(2 \pi n)}{2 n} \Rightarrow \frac{v_m}{v_\omega}=\frac{2 \pi a}{5}
\end{aligned}$$

Answer

$$\frac{\pi a}{5}$$

Answer

$$\frac{2 \pi a}{5}$$

Answer

$$\frac{3 \pi a}{5}$$

Answer

$$\frac{4 \pi a}{5}$$

Waves

The equation of a progressive wave is $Y=a \sin 2 \pi\left(n t-\frac{x}{5}\right)$. The ratio of maximum particle velocity to wave velocity is

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