When a wave travels in a medium, particle displacement is given by : y = a,sin ,2π left( {bt

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14.Waves and Sound

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When a wave travels in a medium, the particle displacement is given by : $y = a\,\sin \,2\pi \left( {bt - cx} \right)$ where $a, b$ and $c$ are constants. The maximum particle velocity will be twice the wave velocity if

$c = \frac{1}{{\pi a}}$

$c = \pi a$

$b = ac$

$b = \frac{1}{{ac}}$

Solution

Equation of the harmonic progressive wave given by :

$\mathrm{y}=\mathrm{a} \sin 2 \pi(\mathrm{bt}-\mathrm{cx})$

Here, $\quad 2 \pi \mathrm{v}=\omega=2 \pi \mathrm{b}$

or $\quad \mathrm{v}=\mathrm{b}$

$\mathrm{k}=\frac{2 \pi}{\lambda}=2 \pi \mathrm{c}$ $\left(\frac{1}{\lambda}=c\right)$

(Here $c$ is the symbol given for $\frac{1}{\lambda}$ and not the velocity )

$\therefore $ Velocity of the wave $=\lambda=b \frac{1}{c}=\frac{b}{c}$

$\frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{a} 2 \pi \mathrm{b} \cos 2 \pi(\mathrm{bt}-\mathrm{cx})=\mathrm{a} \omega \cos (\omega \mathrm{t}-\mathrm{kx})$

Maximum particle velocity $=a \omega=a 2 \pi b=2 \pi a b.$

Given this equal to $2 \times \frac{b}{c}$

i.e., $2 \pi \mathrm{a}=\frac{2}{\mathrm{c}} \quad$ or $\quad \mathrm{c}=\frac{1}{\pi \mathrm{a}}$

Standard 11

Physics

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When a wave travels in a medium, the particle displacement is given by : $y = a\,\sin \,2\pi \left( {bt - cx} \right)$ where $a, b$ and $c$ are constants. The maximum particle velocity will be twice the wave velocity if

Solution

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