Question 357: Waves

Question

The equation of simple harmonic progressive wave is given by $$y=a \sin 2 \pi(b t-c x)$$. The maximum particle velocity will be half the wave velocity, if $$\mathrm{c}=$$

Accepted Answer

General equation of a simple harmonic progressive wave,
$$y=A \sin 2 \pi\left[\frac{t}{T}-\frac{x}{\lambda}ight]$$
Given: $$\mathrm{y}=\mathrm{a} \sin 2 \pi(b t-c x)$$
$$\Rightarrow \mathrm{A}=\mathrm{a}, \frac{1}{\mathrm{~T}}=\mathrm{b} 	ext { and } \frac{1}{\lambda}=\mathrm{cx}$$
Also, $$\left(\mathrm{v}_{\mathrm{p}}ight)_{\max }=\mathrm{a} \omega=\mathrm{a}(2 \pi \mathrm{n})=\frac{\mathrm{a} 2 \pi}{\mathrm{T}}$$
$$=\frac{\mathrm{A} 2 \pi}{\mathrm{T}}=\mathrm{a} 2 \pi \mathrm{b}$$
From $$\mathrm{v}=\frac{\lambda}{\mathrm{T}}=\frac{1 / \mathrm{c}}{1 / \mathrm{b}}=\frac{\mathrm{b}}{\mathrm{c}}$$
Given: $$\left(\mathrm{v}_{\mathrm{p}}ight)_{\max }=\frac{1}{2} \mathrm{v}$$
$$\begin{aligned}
& \Rightarrow 2 \pi a b=\frac{1}{2} 	imes \frac{b}{c} \
	herefore \quad & c=\frac{1}{4 \pi a}
\end{aligned}$$

Answer

$$2 \pi \mathrm{a}$$

Answer

$$\frac{1}{4 \pi \mathrm{a}}$$

Answer

$$\frac{1}{2 \pi \mathrm{a}}$$

Answer

$$4 \pi \mathrm{a}$$

Waves

The equation of simple harmonic progressive wave is given by $y=a \sin 2 \pi(b t-c x). The maximum particle velocity will be half the wave velocity, if \mathrm{c}=

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