When a wave travels in a medium, the particle displacements are given by $y = a\, sin\, 2\pi\, (bt -cx)$ where $a, b,$ and $c$ are constants. The maximum particle velocity will be twice the wave velocity if
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$\mathrm{V}_{\max (\mathrm{P})}=\mathrm{A} \omega$ ........$(i)$
Given that $\mathrm{V}_{\max (\mathrm{P})}=2 \mathrm{v} \quad[\because \omega=2 \pi \mathrm{b},$
$\mathrm{A} \omega=2 \mathrm{v}$ and $\mathrm{k}=2 \pi \mathrm{c}$
${\rm{A}}\omega = 2\left( {\frac{{\rm{b}}}{{\rm{c}}}} \right),\quad {\rm{v}} = \frac{\omega }{{\rm{k}}} = \frac{{\rm{b}}}{{\rm{c}}}]$
$\Rightarrow a(2 \pi b)=\frac{2 b}{c}$
$\Rightarrow \mathrm{a} \pi=\frac{1}{\mathrm{c}}$
$\Rightarrow c=\frac{1}{\pi a}$
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