The equation of a plane progressive wave is given by
$y = 0.025 sin (100 t + 0.25 x)$
Which of the following represents the velocity of the particle of medium through which the sinusoidal wave is propagating?

2.5 cos (100 t + 0.25 x)

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50 cos (100 t + 0.25 x)

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5 sin (100 t + 0.25 x)

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20 sin (50 t + 0.25 x)

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Solution

The correct option is A $2.5 cos (100 t + 0.25 x)$
The displacement of the particle of medium is given as,
$y = 0.025 sin (100 t + 0.25 x)$
Differentiating the above equation partially w.r.t time, i.e keeping $x$ constant so that it represents the velocity of particle at a fixed position in the medium.
$\Rightarrow v_{P} = \frac{\partial y}{\partial t}$
$\Rightarrow v_{P} = \frac{\partial (0.025 sin (100 t + 0.25 x))}{\partial t}$
$\Rightarrow v_{P} = 100 \times 0.025 cos (100 t + 0.25 x)$
$∴ v_{P} = 2.5 cos (100 t + 0.25 x)$

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