Question

When a wave travels in a medium, displacement of a particle is given by $y = a\, sin \,2 \,π \,(bt - cx)$ where ‘$a$’, ‘$b$’, ‘$c$’ are constants. The maximum particle velocity will be twice the wave velocity if

• A. $b = ac$
• B. $b=\frac{1}{ac}$
• C. $c = π\,a$
• D. $c=\frac{1}{\pi a}$

Answer 1

Answer: (D)

Given, $y=a \sin 2 \pi(b t-c x)$
Comparing it will general equation
$Y=r \sin \left[\frac{2 \pi t}{T}-\frac{2 \pi}{\lambda} x\right]$
We get, $\,\,\, \frac{2 \pi}{T}=\omega=2 \pi b$.
and $\,\,\,r=a $
$\Rightarrow \lambda=\frac{1}{c} \text { and } T=\frac{1}{b}$
Maximum particle velocity
$\omega r=2 \pi b a$
Wave velocity $v=\frac{\lambda}{T}=\frac{b}{c}$
Given, maximum particle velocity $=2 \times$ wave velocity
$2 \pi b a=2 \times \frac{b}{c} \Rightarrow c=\frac{1}{\pi a}$