Velocity ' $V$ ' of a wave is directly proportional to modulus of elasticity ' $E$ ' and density ' $d$ ' of a medium. The expression of ' $V$ ' using dimensional analysis is:

V = \frac{E}{\sqrt{d}}

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V = \frac{\sqrt{E}}{d}

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V = \sqrt{\frac{E}{d}}

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V = \sqrt{E D}

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Solution

The correct option is C $V = \sqrt{\frac{E}{d}}$
$V \propto E^{α} d^{β}$
$M^{0} L^{1} T^{- 1} = (M^{1} L^{- 1} T^{- 2})^{α} (M L^{- 3})^{β}$
$M^{0} L^{1} T^{- 1} = M^{α + β} L^{- α - 3 β} T^{- 2 d}$
$\Rightarrow α + β = 0$
$\Rightarrow - α - 3 β = 1$

$- 2 α = - 1$ $\Rightarrow α = \frac{1}{2}, β = - \frac{1}{2}$
$\Rightarrow V = \sqrt{\frac{E}{d}}$

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